One-factor confirmatory factor analysis model

Description

Returns the MLE estimates and the estimated asymptotic covariance matrix of parameter estimates for one-factor confirmatory factor analysis model

Usage

CFA.1(S, N, equal.loading = FALSE, equal.error = FALSE, package="lavaan", 
	se="standard", ...)

Arguments

Value

Note

The output will differ slightly, both in form and potentially values, based on which package lavaan or sem is used.

Author(s)

Keke Lai (University of California-Merced) and Ken Kelley (University of Notre Dame)

See Also

sem, covmat.from.cfm

Examples

## Not run: 
cov.mat<- matrix(
c(1.384, 1.484, 1.988, 2.429, 3.031,
1.484, 2.756, 2.874, 3.588, 4.390,
1.988, 2.874, 4.845, 4.894, 6.080,
2.429, 3.588, 4.894, 6.951, 7.476,
3.031, 4.390, 6.080, 7.476, 10.313), nrow=5)


CFA.1(N=300, S=cov.mat, package="lavaan")

CFA.1(N=300, S=cov.mat, package="sem")


## End(Not run)

Correlation matrix for Lomax (1983) data set

Description

Correlation matrix for Lomax (1983) data set

Usage

data(Cor.Mat.Lomax)

Details

Variables 1 through 14 in the correlation matrix are, respectively:

DRS refers to Diagnostic Reading Scales, WRAT refers to Wide Range Achievement Test, and CTBS refers to Comprehensive Tests of basic skills.

The model was designed to study the causal relationship between the phonological, word recognition, reading rate, and comprehension components of the reading process. There are four latent variables in the model: (a) phonological; (b) word recognition; (c) reading rate; (d) reading comprehension.

Phonological is indicated by (a) DRS-consonant sounds; (b) DRS-consonant blends and diagraphs; (c) DRS-common syllables or phonograms; (d) DRS-blending.

Word recognition is indicated by (a) WRAT-total raw score; (b) DRS-total correct both lists; (c) DRS-total words read correct oral

Reading rate is indicated by (a) DRS-wpm first oral passage; (b) DRS-wpm first silent passage; (c) DRS-mean wpm oral passages read; (d) DRS-mean wpm silent passages read.

Reading comprehension is indicated by (a) DRS-total correct oral comprehension; (b) DRS-total correct silent comprehension; (c) CTBS-comprehension ESS scores.

Source

Lomax, R. G. (1983). Applying structural modeling to some component processes of reading comprehension development. Journal of Experimental Education, 52 (1), 33–40.

References

Lomax, R. G. (1983). Applying structural modeling to some component processes of reading comprehension development. Journal of Experimental Education, 52 (1), 33–40.

Correlation matrix for Maruyama & McGarvey (1980) data set

Description

Correlation matrix for Maruyama & McGarvey (1980) data set

Usage

data(Cor.Mat.MM)

Details

Variables 1 through 13 in the correlation matrix are, respectively:

The model was designed to examine whether acceptance by significant others (i.e., parents, teachers, and peers) causes improved scholastic achievement. There are five latent variables in the model: (a) SES, socio-economic status; (b) ABL, academic ability; (c) ACH, achievement; (d) ASA, acceptance by significant adults; (e) APR, acceptance by peers.

SES is indicated by (a) SEI, Duncan Socioeconomic Index of Occupations; (b) EDHH, educational attainment of the head of the household; (c) R/P, ratio of rooms in the house to persons living in the house.

ACH is indicated by (a) VACH, standardized verbal test scores; (b) VGR, verbal grades.

ABL is indicated by (a) PEA, Peabody Picture Vocabulary Test; (b) RAV, Raven Progressive Matrices.

ASA is indicated by (a) FEV, father's evaluation; (b) MEV, mother's evaluation; (c) TEV, teacher's evaluation.

APR is indicated by (a) PPOP, playground popularity; (b) SPOP, seating popularity; (c) WPOP, schoolwork popularity.

Source

Maruyama, G., & McGarvey, B. (1980). Evaluating causal models: An application of maximum-likelihood analysis of structural equations. Psychological Bulletin, 87 (3), 502–512.

References

Maruyama, G., & McGarvey, B. (1980). Evaluating causal models: An application of maximum-likelihood analysis of structural equations. Psychological Bulletin, 87 (3), 502–512.

Expected value of the squared multiple correlation coefficient

Description

Returns the expected value of the squared multiple correlation coefficient given the population squared multiple correlation coefficient, sample size, and the number of predictors

Usage

Expected.R2(Population.R2, N, p)

Arguments

Details

Uses the hypergeometric function as discussed in section 28 of Stuart, Ord, and Arnold (1999) in order to obtain the correct value for the squared multiple correlation coefficient. Many times an exact value is given that ignores the hypergeometric function. This function yields the correct value.

Value

Returns the expected value of the squared multiple correlation coefficient.

Note

Uses package gsl and its hyperg_2F1 function.

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

References

Olkin, I. & Pratt, J. W. (1958). Unbiased estimation of certain correlation coefficients. Annals of Mathematical statistics, 29, 201–211.

Stuart, A., Ord, J. K., & Arnold, S. (1999). Kendall's advanced theory of statistics: Classical inference and the linear model (Volume 2A, 2nd Edition). New York, NY: Oxford University Press.

See Also

ss.aipe.R2, ci.R2, Variance.R2

Examples

# library(gsl)
# Expected.R2(.5, 10, 5)
# Expected.R2(.5, 25, 5)
# Expected.R2(.5, 50, 5)
# Expected.R2(.5, 100, 5)
# Expected.R2(.5, 1000, 5)
# Expected.R2(.5, 10000, 5)

Conversion functions from noncentral noncentral values to their corresponding and vice versa, for those related to the F-test and R Square.

Description

Given values of test statistics (and the appropriate additional information) the value of the noncentral values can be obtained. Likewise, given noncentral values (and the appropriate additional information) the value of the test statistic can be obtained.

Usage

Rsquare2F(R2 = NULL, df.1 = NULL, df.2 = NULL, p = NULL, N = NULL)

F2Rsquare(F.value = NULL, df.1 = NULL, df.2 = NULL)

Lambda2Rsquare(Lambda = NULL, N = NULL)

Rsquare2Lambda(R2 = NULL, N = NULL)

Arguments

Details

These functions are especially helpful in the search for confidence intervals for noncentral parameters, as they convert to and from related quantities.

Value

Returns the converted value from the specified function.

Author(s)

Ken Kelley (University of Notre Dame, KKelley@ND.Edu)

See Also

ss.aipe.R2, ci.R2, conf.limits.nct, conf.limits.ncf

Examples

Rsquare2Lambda(R2=.5, N=100)

The Gardner learning data, which was used by L.R. Tucker

Description

Repeated measures data on 24 participants, each with 21 trials (each trial based on 20 replications).

Usage

data(Gardner.LD)

Format

A data frame where the rows represent the timepoints for the individuals.

ID

: a numeric vector

Trial

: a numeric vector

Score

: a numeric vector

Group

: a numeric vector

Details

The 24 participants of this study were presented with 420 presentations of four letters where the task was to identify the next letter that was to be presented. Twelve of the participants (Group 1) were presented the letters S, L, N, and D with probabilities .70, .10, .10, and .10, respectively. The other 12 participants (Group 2) were presented the letter L with probability .70 and three other letters, each with a probability of .10. The 420 presentations were (arbitrarily it seems) grouped into 21 trials of 20 presentations. The score for each trial was the number of times the individual correctly guessed the dominant letter. The participants were naive to the probability that the letters would be presented. Other groups of individuals (although the data is not available) were tested under a different probability structure. The data given here is thus known as the 70-10-10-10 group from Gardner's paper. L. R. Tucker used this data set to illustrate methods for understanding change.

Source

Tucker, L. R. (1960). Determination of Generalized Learning Curves by Factor Analysis, Educational Testing Services, Princeton, NJ.

References

Gardner, R. A., (1958). Multiple-choice decision-behavior, American Journal of Psychology, 71, 710–717.

Complete Data Set of Holzinger and Swineford's (1939) Study

Description

The complete data set of scores of 301 participants in 26 tests in Holzinger and Swineford's (1939) study.

Usage

data(HS)

Format

A data frame with 301 observations on the following 34 variables.

id

case number of participants (note there are skips)

sex

sex of participants

grade

grade in school of the participants with levels Female Male

age

the age (ignoring months into the year) of the participants

month_since_birthday

the number of months since the last birthday

age_months

age in months

age_years

age in years and months combined (more fine grained measure of years)

school

the school the participant is from with levels Grant-White Pasteur

t1_visual_perception

scores on visual perception test, test 1

t2_cubes

scores on cubes test, test 2

t3_paper_form_board

scores on paper form board test, test 3

t4_lozenges

scores on lozenges test, test 4

t5_general_information

scores on general information test, test 5

t6_paragraph_comprehension

scores on paragraph comprehension test, test 6

t7_sentence

scores on sentence completion test, test 7

t8_word_classification

scores on word classification test, test 8

t9_word_meaning

scores on word meaning test, test 9

t10_addition

scores on add test, test 10

t11_code

scores on code test, test 11

t12_counting_groups_of_dots

scores on counting groups of dots test, test 12

t13_straight_and_curved_capitals

scores on straight and curved capitals test, test 13

t14_word_recognition

scores on word recognition test, test 14

t15_number_recognition

scores on number recognition test, test 15

t16_figure_recognition

scores on figure recognition test, test 16

t17_object_number

scores on object-number test, test 17

t18_number_figure

scores on number-figure test, test 18

t19_figure_word

scores on figure-word test, test 19

t20_deduction

scores on deduction test, test 20

t21_numerical_puzzles

scores on numerical puzzles test, test 21

t22_problem_reasoning

scores on problem reasoning test, test 22

t23_series_completion

scores on series completion test, test 23

t24_woody_mccall

scores on Woody-McCall mixed fundamentals, form I test, test 24

t25_paper_form_board_r

scores on additional paper form board test, test 25

t26_flags

scores on flags test, test 26

Details

Holzinger and Swineford (1939) data is widely cited, but generally only the Grant-White School data is used. The present dataset contains the complete data of Holzinger and Swineford (1939).

A total number of 301 pupils, coming from Paster School and Grant-White School, who participated in Holzinger and Swineford's (1939) study. This study consists of 26 tests, which are used to measure the participants' spatial, verbal, mental speed, memory, and mathematical ability.

The spatial tests consist of t1_visual_perception, t2_cubes, t3_paper_form_board, t4_lozenges. Additional spatial tests are t25_paper_form_board_r (revised test 3) and t26_flags. t25_paper_form_board_r can (potentially) be used as a substitute for t3_paper_form_board. t26_flags is thought to be a possible substitute for t4_lozenges.

The verbal tests consist of t5_general_information, t6_paragraph_comprehension, t7_sentence, t8_word_classification, and t9_word_meaning.

The speed tests consist of t10_addition, t11_code, t12_counting_groups_of_dots, and t13_straight_and_curved_capitals.

The memory tests consist of t14_word_recognition, t15_number_recognition, t16_figure_recognition, t17_object_number, t18_number_figure, and t19_figure_word.

The mathematical-ability tests consist of t20_deduction, t21_numerical_puzzles, t22_problem_reasoning, t23_series_completion, and t24_woody_mccall.

Source

Holzinger, K. J. and Swineford, F. A. (1939). A study in factor analysis: The stability of a bi-factor solution. Supplementary Education Monographs, 48. University of Chicago.

References

Holzinger, K. J. and Swineford, F. A. (1939). A study in factor analysis: The stability of a bi-factor solution. Supplementary Education Monographs, 48. University of Chicago.

Examples

data(HS)
summary(HS)

MBESS

Description

MBESS Implements methods that are useful in designing research studies and analyzing data, with particular emphasis on methods that are developed for or used within the behavioral, educational, and social sciences (broadly defined). That being said, many of the methods implemented within MBESS are applicable to a wide variety of disciplines. MBESS has a suite of functions for a variety of related topics, such as effect sizes, confidence intervals for effect sizes (including standardized effect sizes and noncentral effect sizes), sample size planning (from the accuracy in parameter estimation [AIPE], power analytic, equivalence, and minimum-risk point estimation perspectives), mediation analysis, various properties of distributions, and a variety of utility functions. MBESS (pronounced 'em-bes') was originally an acronym for 'Methods for the Behavioral, Educational, and Social Sciences,' but MBESS became more general and now contains methods applicable and used in a wide variety of fields and is an orphan acronym, in the sense that what was an acronym is now literally its name. MBESS has greatly benefited from others, see <https://www3.nd.edu/~kkelley/site/MBESS.html> for a detailed list of those that have contributed and other details.

Details

Please read the manual and visit the corresponding web site https://www3.nd.edu/~kkelley/site/MBESS.html for information on the capabilities of the MBESS package. Feel free to contact me if there is a feature you would like to see added if it would complement the goals of the MBESS package. Beginning with version 4.8.0, the package also has a home on GitHub https://github.com/yelleKneK/MBESS. Over the years, multiple people have contributed functions to the package. See individual functions for details.

Author(s)

Ken Kelley <KKelley@ND.Edu; https://www3.nd.edu/~kkelley/>

Maintainer: Ken Kelley <KKelley@ND.Edu; https://www3.nd.edu/~kkelley/>

Construct a covariance matrix with specified error of approximation

Description

This function implements Cudeck & Browne's (1992) method to construct a covariance matrix in the structural equation modeling (SEM) context. Given an SEM model and its model parameters, a covariance matrix is obtained so that (a) the population discrepancy due to approximation equals a certain specified value; and (b) the population model parameter vector is the minimizer of the discrepancy function.

Usage

Sigma.2.SigmaStar(model, model.par, latent.var, discrep, ML = TRUE)

Arguments

Details

This function constructs a covariance matrix \Sigma^{*} such that \Sigma^{*} = \Sigma( \theta ) + E , where \Sigma(\theta) is the population model-implied covariance matrix, and E is a matrix containing the errors due to approximation. The matrix E is chosen so that the discrepancy function F( \Sigma^{*}, \Sigma (\theta) ) has the specified discrepancy value.

This function uses the same notation to specify SEM models as does sem. Please refer to sem for more detailed documentation about model specification and the RAM notation. For technical discussion on how to obtain the model implied covariance matrix in the RAM notation given model parameters, see McArdle and McDonald (1984).

Value

Author(s)

Keke Lai (University of California-Merced)

References

Cudeck, R., & Browne, M. W. (1992). Constructing a covariance matrix that yields a specified minimizer and a specified minimum discrepancy function value. Psychometrika, 57, 357–369.

Fox, J. (2006). Structural equation modeling with the sem package in R. Structural Equation Modeling, 13, 465–486.

McArdle, J. J., & McDonald, R. P. (1984). Some algebraic properties of the reticular action model. British Journal of Mathematical and Statistical Psychology, 37, 234–251.

See Also

sem; specify.model; theta.2.Sigma.theta

Examples

## Not run: 
library(sem)

###############
## EXAMPLE 1; a CFA model with three latent variables and nine indicators.
###############

# To specify the model
model.cfa<-specify.model()
xi1 -> x1, lambda1, 0.6
xi1 -> x2, lambda2, 0.7
xi1 -> x3, lambda3, 0.8
xi2 -> x4, lambda4, 0.65
xi2 -> x5, lambda5, 0.75
xi2 -> x6, lambda6, 0.85
xi3 -> x7, lambda7, 0.5
xi3 -> x8, lambda8, 0.7
xi3 -> x9, lambda9, 0.9
xi1 <-> xi1, NA, 1
xi2 <-> xi2, NA, 1
xi3 <-> xi3, NA, 1
xi1 <-> xi2, phi21, 0.5
xi1 <-> xi3, phi31, 0.4
xi2 <-> xi3, phi32, 0.6
x1 <-> x1, delta11, 0.36
x2 <-> x2, delta22, 0.5
x3 <-> x3, delta33, 0.9
x4 <-> x4, delta44, 0.4
x5 <-> x5, delta55, 0.5
x6 <-> x6, delta66, 0.6
x7 <-> x7, delta77, 0.6
x8 <-> x8, delta88, 0.7
x9 <-> x9, delta99, 0.7


# To specify model parameters
theta <- c(0.6, 0.7, 0.8,
0.65, 0.75, 0.85,
0.5, 0.7, 0.9,
0.5, 0.4, 0.6,
0.8, 0.6, 0.5,
0.6, 0.5, 0.4,
0.7, 0.7, 0.6)

names(theta) <- c("lambda1", "lambda2", "lambda3", 
"lambda4","lambda5", "lambda6", 
"lambda7", "lambda8","lambda9",
"phi21", "phi31", "phi32", 
"delta11", "delta22","delta33",
"delta44", "delta55","delta66",
"delta77", "delta88","delta99")

res.matrix <- Sigma.2.SigmaStar(model=model.cfa, model.par=theta, 
latent.var=c("xi1", "xi2", "xi3"), discrep=0.06)

# res.matrix

# To verify the returned covariance matrix; the model chi-square
# should be equal to (N-1) times the specified discrepancy value.
# Also the "point estimates" of model parameters should be 
# equal to the specified model parameters

# res.sem<-sem(model.cfa, res.matrix$Sigma.star, 1001)
# summary(res.sem)

# To construct a covariance matrix so that the model has
# a desired population RMSEA value, one can transform the RMSEA
# value to the discrepancy value

res.matrix <- Sigma.2.SigmaStar(model=model.cfa, model.par=theta, 
latent.var=c("xi1", "xi2", "xi3"), discrep=0.075*0.075*24)

# To verify the population RMSEA value
# res.sem<-sem(model.cfa, res.matrix$Sigma.star, 1000000)
# summary(res.sem)

###############
## EXAMPLE 2; an SEM model with five latent variables
###############

model.5f <- specify.model()
eta1 -> y4, NA, 1
eta1 -> y5, lambda5, NA
eta2 -> y1, NA, 1
eta2 -> y2, lambda2, NA 
eta2 -> y3, lambda3, NA
xi1 -> x1, NA, 1
xi1 -> x2, lambda6, NA 
xi1 -> x3, lambda7, NA
xi2 -> x4, NA, 1
xi2 -> x5, lambda8, NA 
xi3 -> x6, NA, 1
xi3 -> x7, lambda9, NA 
xi3 -> x8, lambda10, NA
xi1 -> eta1, gamma11, NA
xi2 -> eta1, gamma12, NA
xi3 -> eta1, gamma13, NA
xi3 -> eta2, gamma23, NA
eta1 -> eta2, beta21, NA
xi1 <-> xi2, phi21, NA
xi1 <-> xi3, phi31, NA
xi3 <-> xi2, phi32, NA
xi1 <-> xi1, phi11, NA
xi2 <-> xi2, phi22, NA
xi3 <-> xi3, phi33, NA
eta1 <-> eta1, psi11, NA
eta2 <-> eta2, psi22, NA
y1 <-> y1, eplison11, NA
y2 <-> y2, eplison22, NA
y3 <-> y3, eplison33, NA
y4 <-> y4, eplison44, NA
y5 <-> y5, eplison55, NA
x1 <-> x1, delta11, NA
x2 <-> x2, delta22, NA
x3 <-> x3, delta33, NA
x4 <-> x4, delta44, NA
x5 <-> x5, delta55, NA
x6 <-> x6, delta66, NA
x7 <-> x7, delta77, NA
x8 <-> x8, delta88, NA


theta <- c(0.84, 0.8, 0.9, 
1.26, 0.75, 1.43, 1.58, 0.83, 
0.4, 0.98, 0.52, 0.6,0.47, 
0.12, 0.14, 0.07,
0.44, 0.22, 0.25, 
0.3, 0.47, 
0.37, 0.5, 0.4, 0.4, 0.58, 
0.56,0.3, 0.6, 0.77, 0.54, 0.75, 0.37, 0.6)

names(theta) <- c(
"lambda5","lambda2","lambda3",
"lambda6","lambda7","lambda8","lambda9","lambda10" , 
"gamma11",  "gamma12","gamma13" ,  "gamma23" ,  "beta21",
"phi21","phi31", "phi32", 
"phi11","phi22",  "phi33",     
"psi11" ,    "psi22"   ,  
"eplison11","eplison22" ,"eplison33", "eplison44" ,"eplison55", 
  "delta11"  , "delta22" ,  "delta33" ,  "delta44" ,  "delta55" ,  "delta66",  
"delta77" , "delta88")

# To construct a covariance matrix so that the model has 
# a population RMSEA of 0.08

res.matrix <- Sigma.2.SigmaStar(model=model.5f, model.par=theta, 
latent.var=c("xi1", "xi2", "xi3", "eta1","eta2"), discrep=0.08*0.08*57)

# To verify
# res.sem<- sem(model.5f, res.matrix$Sigma.star, 1000000)
# summary(res.sem)

## End(Not run)

Variance of squared multiple correlation coefficient

Description

Function to determine the variance of the squared multiple correlation coefficient given the population squared multiple correlation coefficient, sample size, and the number of predictors.

Usage

Variance.R2(Population.R2, N, p)

Arguments

Details

Uses the hypergeometric function as discussed in and section 28 of Stuart, Ord, and Arnold (1999) in order to obtain the correct value for the variance of the squared multiple correlation coefficient.

Value

Returns the variance of the of the squared multiple correlation coefficient.

Note

Uses package gsl and its hyperg_2F1 function.

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

References

Stuart, A., Ord, J. K., & Arnold, S. (1999). Kendall's advanced theory of statistics: Classical inference and the linear model (Volume 2A, 2nd Edition). New York, NY: Oxford University Press.

See Also

Expected.R2, ci.R2, ss.aipe.R2

Examples

# library(gsl)
# Variance.R2(.5, 10, 5)
# Variance.R2(.5, 25, 5)
# Variance.R2(.5, 50, 5)
# Variance.R2(.5, 100, 5)

Sample size planning for the standardized mean different from the accuracy in parameter estimation approach

Description

A set of functions that ss.aipe.smd calls upon to calculate the appropriate sample size for the standardized mean difference such that the expected value of the confidence interval is sufficiently narrow.

Usage

ss.aipe.smd.full(delta, conf.level, width, ...)
ss.aipe.smd.lower(delta, conf.level, width, ...)
ss.aipe.smd.upper(delta, conf.level, width, ...)

Arguments

Value

Warning

The returned value is the sample size per group. Currently only ss.aipe.smd.full returns the exact value. However, ss.aipe.smd.lower and ss.aipe.smd.upper provide approximate sample size values.

Note

The function ss.aipe.smd is the function users should generally use. The function ss.aipe.smd calls upon these functions as needed. They can be thought of loosely as internal MBESS functions.

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

References

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum.

Cumming, G. & Finch, S. (2001). A primer on the understanding, use, and calculation of confidence intervals that are based on central and noncentral distributions, Educational and Psychological Measurement, 61, 532–574.

Hedges, L. V. (1981). Distribution theory for Glass's Estimator of effect size and related estimators. Journal of Educational Statistics, 2, 107–128.

Kelley, K. (2005). The effects of nonnormal distributions on confidence intervals around the standardized mean difference: Bootstrap and parametric confidence intervals, Educational and Psychological Measurement, 65, 51–69.

Kelley, K., Maxwell, S. E., & Rausch, J. R. (2003). Obtaining Power or Obtaining Precision: Delineating Methods of Sample-Size Planning, Evaluation and the Health Professions, 26, 258–287.

Kelley, K., & Rausch, J. R. (2006). Sample size planning for the standardized mean difference: Accuracy in Parameter Estimation via narrow confidence intervals. Psychological Methods, 11(4), 363–385.

Steiger, J. H., & Fouladi, R. T. (1997). Noncentrality interval estimation and the evaluation of statistical methods. In L. L. Harlow, S. A. Mulaik, & J.H. Steiger (Eds.), What if there were no significance tests? (pp. 221–257). Mahwah, NJ: Lawrence Erlbaum.

See Also

ss.aipe.smd

Generate random data for an ANCOVA model

Description

Generate random data for a simple (one-response-one-covariate) ANCOVA model considering the covariate as random. Data can be generated in the contexts of both randomized design (same population covariate mean across groups) and non-randomized design (different population covariate means across groups).

Usage

ancova.random.data(mu.y, mu.x, sigma.y, sigma.x, rho, J, n, randomized = TRUE)

Arguments

Details

This function uses a multivariate normal distribution to generate the random data; the covariate is considered as a random variable in the model. This function uses mvrnorm in the MASS package in an internal function, and thus it requires the MASS package be installed.

This function assumes homogeneous covariance matrix among groups, in both the randomized design and non-randomized design contexts.

Value

This function returns an n by J2 matrix, where n and J are as defined in the argument. The first J columns of the matrix contains the random data for the response, and the second J columns of the matrix contains the random data for the covariate.

Author(s)

Keke Lai (University of California-Merced) and Ken Kelley (University of Notre Dame) <kkelley@nd.edu>

See Also

mvrnorm in the MASS package

Examples

random.data <- ancova.random.data(mu.y=c(3,5), mu.x=10, sigma.y=1, 
sigma.x=2, rho=.8, J=2, n=20)

Confidence interval for the multiple correlation coefficient

Description

A function to obtain the confidence interval for the population multiple correlation coefficient when predictors are random (the default) or fixed.

Usage

ci.R(R = NULL, df.1 = NULL, df.2 = NULL, conf.level = 0.95, 
Random.Predictors = TRUE, Random.Regressors, F.value = NULL, 
N = NULL, K=NULL, alpha.lower = NULL, alpha.upper = NULL, ...)

Arguments

Details

This function is based on the function ci.R2 in MBESS package.

This function can be used with random predictor variables (Random.Predictors=TRUE) or when predictor variables are fixed (Random.Predictors=FALSE). In many applications in the behavioral, educational, and social sciences, predictor variables are random, which is the default for this function.

For random predictors, the function implements the procedure of Lee (1971), which was implemented by Algina and Olejnik (2000; specifically in their ci.smcc.bisec.sas SAS script). When Random.Predictors=TRUE, the function implements code that is in part based on the Alginia and Olejnik (2000) SAS script.

When Random.Predictors=FALSE, and thus the predictors are planned and thus fixed in hypothetical replications of the study, the confidence limits are based on a noncentral F-distribution (see conf.limits.ncf).

Value

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

References

Algina, J. & Olejnik, S. (2000). Determining sample size for accurate estimation of the squared multiple correlation coefficient. Multivariate Behavioral Research, 35, 119–136.

Lee, Y. S. (1971). Some results on the sampling distribution of the multiple correlation coefficient. Journal of the Royal Statistical Society, B, 33, 117–130.

Smithson, M. (2003). Confidence intervals. New York, NY: Sage Publications.

Steiger, J. H. (2004). Beyond the F Test: Effect size confidence intervals and tests of close fit in the Analysis of Variance and Contrast Analysis. Psychological Methods, 9, 164–182.

Steiger, J. H. & Fouladi, R. T. (1992). R2: A computer program for interval estimation, power calculation, and hypothesis testing for the squared multiple correlation. Behavior research methods, instruments and computers, 4, 581–582.

See Also

ci.R2, ss.aipe.R2, conf.limits.nct

Confidence interval for the population squared multiple correlation coefficient

Description

A function to calculate the confidence interval for the population squared multiple correlation coefficient.

Usage

ci.R2(R2 = NULL, df.1 = NULL, df.2 = NULL, conf.level = .95, 
Random.Predictors=TRUE, Random.Regressors, F.value = NULL, N = NULL, 
p = NULL, K, alpha.lower = NULL, alpha.upper = NULL, tol = 1e-09)

Arguments

Details

This function can be used with random predictor variables (Random.Predictors=TRUE) or when predictor variables are fixed (Random.Predictors=FALSE). In many applications of multiple regression, predictor variables are random, which is the default in this function.

For random predictors, the function implements the procedure of Lee (1971), which was implemented by Algina and Olejnik (2000; specifically in their ci.smcc.bisec.sas SAS script). When Random.Predictors=TRUE, the function implements code that is in part based on the Alginia and Olejnik (2000) SAS script.

When Random.Predictors=FALSE, and thus the predictors are planned and thus fixed in hypothetical replications of the study, the confidence limits are based on a noncentral F-distribution (see conf.limits.ncf).

Value

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

References

Algina, J. & Olejnik, S. (2000). Determining Sample Size for Accurate Estimation of the Squared Multiple Correlation Coefficient. Multivariate Behavioral Research, 35, 119–136.

Kelley, K. (2007). Constructing confidence intervals for standardized effect sizes: Theory, application, and implementation. Journal of Statistical Software, 20 (8), 1–24.

Lee, Y. S. (1971). Some results on the sampling distribution of the multiple correlation coefficient. Journal of the Royal Statistical Society, B, 33, 117–130.

Smithson, M. (2003). Confidence intervals. New York, NY: Sage Publications.

Steiger, J. H. & Fouladi, R. T. (1992) R2: A computer program for interval estimation, power calculation, and hypothesis testing for the squared multiple correlation. Behavior research methods, instruments and computers, 4, 581–582.

See Also

ss.aipe.R2, conf.limits.ncf

Examples

# For random predictor variables.
# ci.R2(R2=.25, N=100, K=5, conf.level=.95, Random.Predictors=TRUE)

# ci.R2(F.value=6.266667, N=100, K=5, conf.level=.95, Random.Predictors=TRUE)

# For fixed predictor variables.
# ci.R2(R2=.25, N=100, K=5, conf.level=.95, Random.Predictors=TRUE)

# ci.R2(F.value=6.266667, N=100, K=5, conf.level=.95, Random.Predictors=TRUE)

# One sided confidence intervals when predictors are random.
# ci.R2(R2=.25, N=100, K=5, alpha.lower=.05, alpha.upper=0, conf.level=NULL,
# Random.Predictors=TRUE)

# ci.R2(R2=.25, N=100, K=5, alpha.lower=0, alpha.upper=.05, conf.level=NULL,
# Random.Predictors=TRUE)

# One sided confidence intervals when predictors are fixed.
# ci.R2(R2=.25, N=100, K=5, alpha.lower=.05, alpha.upper=0, conf.level=NULL,
# Random.Predictors=FALSE)

# ci.R2(R2=.25, N=100, K=5, alpha.lower=0, alpha.upper=.05, conf.level=NULL,
# Random.Predictors=FALSE)

Confidence interval for a contrast in a fixed effects ANOVA

Description

Function to calculate the exact confidence interval for a contrast in a fixed effects analysis of variance context. This function assumes homogeneity of variance (as does the ANOVA upon which 's.anova' is based).

Usage

ci.c(means = NULL, s.anova = NULL, c.weights = NULL, n = NULL, 
N = NULL, Psi = NULL, conf.level = 0.95, alpha.lower = NULL, 
alpha.upper = NULL, df.error = NULL, ...)

Arguments

Value

Returns the confidence limits for the contrast:

Note

Be sure to use the standard deviation and not the error variance for s.anova, not the square of this value (the error variance) which would come from the source table (i.e., do not use the variance of the error but rather use its square root, the standard deviation).

Be sure to use fractional c-weights when doing complex contrasts (not integers) to specify c.weights. For example, in an ANCOVA of four groups, if the user wants to compare the mean of group 1 and 2 with the mean of group 3 and 4, c.weights should be specified as c(0.5, 0.5, -0.5, -0.5) rather than c(1, 1, -1, -1). Make sure the sum of the contrast weights is zero.

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

References

Kelley, K. (2007). Constructing confidence intervals for standardized effect sizes: Theory, application, and implementation. Journal of Statistical Software, 20 (8), 1–24.

Steiger, J. H. (2004). Beyond the F Test: Effect size confidence intervals and tests of close fit in the analysis of variance and contrast analysis. Psychological Methods, 9, 164–182.

See Also

conf.limits.nct, ci.sc, ci.src, ci.smd, ci.smd.c, ci.sm

Examples

# Here is a four group example. Suppose that the means of groups 1--4 are 2, 4, 9, 
# and 13, respectively. Further, let the error variance be .64 and thus the standard
# deviation would be .80 (note we use the standard deviation in the function, not the 
# variance). The contrast of interest here is the average of groups 1 and 4 versus the 
# average of groups 2 and 3. 

ci.c(means=c(2, 4, 9, 13), s.anova=.80, c.weights=c(.5, -.5, -.5, .5), 
n=c(3, 3, 3, 3), N=12, conf.level=.95)


# Here is an example with two groups. 
ci.c(means=c(1.6, 0), s.anova=.80, c.weights=c(1, -1), n=c(10, 10), N=20, conf.level=.95)


# An example given by Maxwell and Delaney (2004, pp. 155--171) :
# 24 subjects of mild hypertensives are assigned to one of four treatments: drug 
# therapy, biofeedback, dietary modification, and a treatment combining all the 
# three previous treatments. Subjects' blood pressure is measured two weeks
# after the termination of treatment. Now we want to form a 95% level
# confidence interval for the difference in blood pressure between subjects
# who received drug treatment and those who received biofeedback treatment 

## Drug group's mean = 94; group size=4
## Biofeedback group's mean = 91; group size=6 
## Diet group's mean = 92; group size=5
## Combination group's mean = 83; group size=5
## Mean Square Within (i.e., 'error.variance') = 67.375

ci.c(means=c(94, 91, 92, 83), s.anova=sqrt(67.375), c.weights=c(1, -1, 0, 0), 
n=c(4, 6, 5, 5), N=20, conf.level=.95)

Confidence interval for an (unstandardized) contrast in ANCOVA with one covariate

Description

To calculate the confidence interval for an unstandardized contrast in the one-covariate ANCOVA.

Usage

ci.c.ancova(Psi, adj.means, s.ancova = NULL, c.weights, n,
cov.means, SSwithin.x, conf.level = 0.95, ...)

Arguments

Value

Note

Be sure to use the standard deviation and not the error variance for s.ancova, not the square of this value which would come from the source table (i.e., do not use the variance of the error but rather use the square root).

If n receives a single number, that number is considered as the sample size per group. If n receives a vector, the vector is considered as the sample size of each group.

Be sure to use fractions not the integers to specify c.weights. For example, in an ANCOVA of four groups, if the user wants to compare the mean of group 1 and 2 with the mean of group 3 and 4, c.weights should be specified as c(0.5, 0.5, -0.5, -0.5) rather than c(1, 1, -1, -1). Make sure the sum of the contrast weights are zero.

Author(s)

Keke Lai (University of California–Merced) and Ken Kelley (University of Notre Dame; kkelley@nd.edu)

References

Kelley, K. (2007). Constructing confidence intervals for standardized effect sizes: Theory, application, and implementation. Journal of Statistical Software, 20 (8), 1–24.

Maxwell, S. E., & Delaney, H. D. (2004). Designing experiments and analyzing data: A model comparison perspective. Mahwah, NJ: Erlbaum.

See Also

ci.c, ci.sc.ancova

Examples

# Maxwell & Delaney (2004, pp. 428-468) offer an example that 30 depressive 
# individuals are randomly assigned to three groups, 10 in each, and ANCOVA 
# is performed on the posttest scores using the participants' pretest 
# scores as the covariate. The means of pretest scores of group 1 to 3 are 
# 17, 17.7, and 17.4, respectively, and the adjusted means of groups 1 to 3 
# are 7.5, 12, and 14, respectively. The error variance in ANCOVA is 29, 
# and the sum of squares within groups from ANOVA on the covariate is 
# 313.37. 

# To obtain the confidence interval for adjusted mean of group 1 versus 
# group 2:
ci.c.ancova(adj.means=c(7.5, 12, 14), s.ancova=sqrt(29), c.weights=c(1, -1, 0), 
n=10, cov.means=c(17, 17.7, 17.4), SSwithin.x=313.37)

Confidence interval for the population correlation coefficient

Description

This function is used to form a confidence interval for the population correlation coefficient. Note that this appraoch assumes that the variables the sample correlation coefficient are based are assumed to be bivariate normally distributed (e.g., Hays, 1994, Chapter 14).

Usage

ci.cc(r, n, conf.level = 0.95, alpha.lower = NULL, alpha.upper = NULL)

Arguments

Details

Note that this appraoch to confidence intervals does will not generally lead to a symmetric confidence interval. The function first transforms r into Z^\prime , forms a confidence interval for the population value (i.e., \zeta), and then transforms the confidence limits for \zeta into the scale of the correlation coefficient.

Value

Note

This confidence interval assumes that the two variables the correlation is based are bivariate normal. See Hays (2004, Chapter 14) for details.

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

References

Kelley, K. (2007). Confidence intervals for standardized effect sizes: Theory, application, and implementation. Journal of Statistical Software, 20(8), 1–24.

Hays, W. L. (1994). Statistics (5th ed). Fort Worth, TX: Harcourt Brace College Publishers)

See Also

transform_Z.r, transform_r.Z

Examples

# Example, from Hayes. Suppose n=100 and r=.35. 
ci.cc(r=.35, n=100, conf.level=.95)

# Here is another way to enter the above example. 
ci.cc(r=.35, n=100, conf.level=NULL, alpha.lower=.025, alpha.upper=.025)

# Here are examples of one-sided confidence intervals. 
ci.cc(r=.35, n=100, conf.level=NULL, alpha.lower=0, alpha.upper=.05)
ci.cc(r=.35, n=100, conf.level=NULL, alpha.lower=.05, alpha.upper=0)

Confidence interval for the coefficient of variation

Description

Function to calculate the confidence interval for the population coefficient of variation using the noncentral t-distribution.

Usage

ci.cv(cv=NULL, mean = NULL, sd = NULL, n = NULL, data = NULL, 
conf.level = 0.95, alpha.lower = NULL, alpha.upper = NULL, ...)

Arguments

Details

Uses the noncentral t-distribution to calculate the confidence interval for the population coefficient of variation.

Value

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

References

Johnson, B. L., & Welch, B. L. (1940). Applications of the non-central t-distribution. Biometrika, 31, 362–389.

Kelley, K. (2007). Sample size planning for the coefficient of variation from the accuracy in parameter estimation approach. Behavior Research Methods, 39 (4), 755–766.

Kelley, K. (2007). Constructing confidence intervals for standardized effect sizes: Theory, application, and implementation. Journal of Statistical Software, 20 (8), 1–24.

McKay, A. T. (1932). Distribution of the coefficient of variation and the extended t distribution, Journal of the Royal Statistical Society, 95, 695–698.

See Also

cv

Examples

set.seed(113)
N <- 15
X <- rnorm(N, 5, 1)
mean.X <- mean(X)
sd.X <- var(X)^.5

ci.cv(mean=mean.X, sd=sd.X, n=N, alpha.lower=.025, alpha.upper=.025,
conf.level=NULL)
ci.cv(data=X, conf.level=.95)
ci.cv(cv=sd.X/mean.X, n=N, conf.level=.95)

Confidence Interval for omega-squared (\omega^2) for between-subject fixed-effects ANOVA and ANCOVA designs (and partial omega-squared \omega^2_p for between-subject multifactor ANOVA and ANCOVA designs)

Description

Function to obtain the exact confidence interval using the non-central F-distribution for omega-squared or partial omega-squared in between-subject fixed-effects ANOVA and ANCOVA designs.

Usage

ci.omega2(F.value = NULL, df.1 = NULL, df.2 = NULL, N = NULL, conf.level = 0.95, 
alpha.lower = NULL, alpha.upper = NULL, ...)

Arguments

Details

The confidence level must be specified in one of following two ways: using confidence interval coverage (conf.level), or lower and upper confidence limits (alpha.lower and alpha.upper). The value returned is the confidence interval limits for the population \omega^2 (or partial \omega^2).

This function uses the confidence interval transformation principle (Steiger, 2004) to transform the confidence limits for the noncentality parameter to the confidence limits for the population's (partial) omega-squared (\omega^2). The confidence interval for the noncentral F-parameter can be obtained from the conf.limits.ncf function in MBESS, which is used internally within this function.

Value

Returns the confidence limits for (partial) omega-sqaured.

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

References

Fleishman, A. I. (1980). Confidence intervals for correlation ratios. Educational and Psychological Measurement, 40, 659–670.

Kelley, K. (2007). Constructing confidence intervals for standardized effect sizes: Theory, application, and implementation. Journal of Statistical Software, 20 (8), 1–24.

Steiger, J. H. (2004). Beyond the F Test: Effect size confidence intervals and tests of close fit in the Analysis of Variance and Contrast Analysis. Psychological Methods, 9, 164–182.

See Also

ci.srsnr, ci.snr, conf.limits.ncf

Examples

## To illustrate the calculation of the confidence interval for noncentral 
## F parameter,Bargman (1970) gave an example in which a 5-group ANOVA with 
## 11 subjects in each group is conducted and the observed F value is 11.2213. 
## This exmaple continued to be used in Venables (1975),  Fleishman (1980), 
## and Steiger (2004). If one wants to calculate the exact confidence interval 
## for omega-squared of that example, this function can be used.

ci.omega2(F.value=11.221, df.1=4, df.2=50, N=55)

ci.omega2(F.value=11.221, df.1=4, df.2=50, N=55, conf.level=.90)

  

Confidence Interval for the Proportion of Variance Accounted for (in the dependent variable by knowing the levels of the factor)

Description

Function to obtain the exact confidence limits for the proportion of variance of the dependent variable accounted for by knowing the levels of the factor (or the grouping factor in a single factor design) group status in a fixed factor analysis of variance.

Usage

ci.pvaf(F.value = NULL, df.1 = NULL, df.2 = NULL, N = NULL, 
conf.level = 0.95, alpha.lower = NULL, alpha.upper = NULL, ...)

Arguments

Details

The confidence level must be specified in one of following two ways: using confidence interval coverage (conf.level), or lower and upper confidence limits (alpha.lower and alpha.upper).

This function uses the confidence interval transformation principle (Steiger, 2004) to transform the confidence limits for the noncentrality parameter to the confidence limits for the population proportion of variance accounted for by knowing the group status. The confidence interval for the noncentral F parameter can be obtained from the function conf.limits.ncf in MBESS, which is used within this function.

Value

Returns the confidence interval for the proportion of variance of the dependent variable accounted for by knowing group status in a fixed factor analysis of variance (using a noncentral F-distribution).

Note

This function can be used for single or factorial ANOVA designs.

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

References

Fleishman, A. I. (1980). Confidence intervals for correlation ratios. Educational and Psychological Measurement, 40, 659–670.

Kelley, K. (2007). Constructing confidence intervals for standardized effect sizes: Theory, application, and implementation. Journal of Statistical Software, 20 (8), 1–24.

Steiger, J. H. (2004). Beyond the F Test: Effect size confidence intervals and tests of close fit in the Analysis of Variance and Contrast Analysis. Psychological Methods, 9, 164–182.

See Also

conf.limits.ncf

Examples

## Not run: 
## Bargman (1970) gave an example in which a 5-group ANOVA with 11 subjects in each 
## group is conducted and the observed F value is 11.2213. This example was used 
## in Venables (1975),  Fleishman (1980), and Steiger (2004). If one wants to calculate the 
## exact confidence interval for the proportion of variance accounted for in that example, 
## this function can be used.

ci.pvaf(F.value=11.221, df.1=4, df.2=50, N=55)

ci.pvaf(F.value=11.221, df.1=4, df.2=50, N=55, conf.level=.90)

ci.pvaf(F.value=11.221, df.1=4, df.2=50, N=55,  alpha.lower=0, alpha.upper=.05)

## End(Not run)

Confidence Interval for a Regression Coefficient

Description

A function to calculate a confidence interval for the population regression coefficient of interest using the standard approach and the noncentral approach when the regression coefficients are standardized.

Usage

ci.rc(b.k, SE.b.k = NULL, s.Y = NULL, s.X = NULL, N, K, R2.Y_X = NULL, 
R2.k_X.without.k = NULL, conf.level = 0.95, R2.Y_X.without.k = NULL, 
t.value = NULL, alpha.lower = NULL, alpha.upper = NULL, 
Noncentral = FALSE, Suppress.Statement = FALSE, ...)

Arguments

Details

This function calls upon ci.reg.coef in MBESS, but has a different naming system. See ci.reg.coef for more details.

For standardized variables, do not specify the standard deviation of the variables and input the standardized regression coefficient for b.k.

Value

Returns the confidence limits for the standardized regression coefficients of interest from the standard approach to confidence interval formation or from the noncentral approach to confidence interval formation using the noncentral t-distribution.

Note

Not all of the values need to be specified, only those that contain all of the necessary information in order to compute the confidence interval (options are thus given for the values that need to be specified).

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

References

Kelley, K. (2007). Confidence intervals for standardized effect sizes: Theory, application, and implementation. Journal of Statistical Software, 20(8), 1–24.

Kelley, K. & Maxwell, S. E. (2003). Sample size for Multiple Regression: Obtaining regression coefficients that are accurate, not simply significant. Psychological Methods, 8, 305–321.

Kelley, K. & Maxwell, S. E. (2008). Power and accuracy for omnibus and targeted effects: Issues of sample size planning with applications to Multiple Regression. Handbook of Social Research Methods, J. Brannon, P. Alasuutari, and L. Bickman (Eds.). New York, NY: Sage Publications.

Smithson, M. (2003). Confidence intervals. New York, NY: Sage Publications.

Steiger, J. H. (2004). Beyond the F Test: Effect size confidence intervals and tests of close fit in the Analysis of Variance and Contrast Analysis. Psychological Methods, 9, 164–182.

See Also

ss.aipe.reg.coef, conf.limits.nct, ci.reg.coef, ci.src

Confidence interval for a regression coefficient

Description

A function to calculate a confidence interval around the population regression coefficient of interest using the standard approach and the noncentral approach when the regression coefficients are standardized.

Usage

ci.reg.coef(b.j, SE.b.j=NULL, s.Y=NULL, s.X=NULL, N, p, R2.Y_X=NULL,
R2.j_X.without.j=NULL, conf.level=0.95, R2.Y_X.without.j=NULL, 
t.value=NULL, alpha.lower=NULL, alpha.upper=NULL, Noncentral=FALSE, 
Suppress.Statement=FALSE, ...)

Arguments

Details

For standardized variables, do not specify the standard deviation of the variables and input the standardized regression coefficient for b.j.

Value

Returns the confidence limits specified for the regression coefficient of interest from the standard approach to confidence interval formation or from the noncentral approach to confidence interval formation using the noncentral t-distribution.

Note

Not all of the values need to be specified, only those that contain all of the necessary information in order to compute the confidence interval (options are thus given for the values that need to be specified).

The function ci.rc in MBESS also calculates the confidence interval for the population (unstandardized) regression coefficient. The function ci.src also calculates the confidence interval for the population (standardized) regression coefficient. These two functions perform the same tasks as ci.reg.coef does and are preferred to it because of simpler arguments.

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

References

Kelley, K. & Maxwell, S. E. (2003). Sample size for Multiple Regression: Obtaining regression coefficients that are accurate, not simply significant. Psychological Methods, 8, 305–321.

Kelley, K. & Maxwell, S. E. (2008). Sample Size Planning with applications to multiple regression: Power and accuracy for omnibus and targeted effects. In P. Alasuuta, J. Brannen, & L. Bickman (Eds.), The Sage handbook of social research methods (pp. 166–192). Newbury Park, CA: Sage.

Smithson, M. (2003). Confidence intervals. New York, NY: Sage Publications.

See Also

ss.aipe.reg.coef, conf.limits.nct, ci.rc, ci.src

Confidence Interval for a Reliability Coefficient

Description

A function to calculate the point estimate and confidence interval for a reliability coefficient (alpha, omega, and variations thereof). Please see the many options; the defaults may not be best for your situation. See Kelley and Pornprasertmanit (2016) for recommendation and a discussion of the methods, where they ultimately recommend the bias-corrected and accelerated bootstrap (interval.type="bca" with hierarchical omega (type="hierarchical") for continuous items.

Usage

ci.reliability(data = NULL, S = NULL, N = NULL, aux = NULL, 
type = "omega", interval.type = "default", B = 10000, conf.level = 0.95)

Arguments

Details

When coefficient alpha is used, the measurement model is assumed to be true-score equivalent (or tau equivalent) model such that factor loadings are equal across items. When the coefficient omega, hierarchical omega, and categorical omega are used, the measurement model is assumed to be congeneric model (i.e., one-factor confirmatory factor analysis model). Coefficient omega assumes that a model fits data perfectly so the variance of the composite scores is calculated from model-implied covariance matrix. However, hierarchical omega allows a model to not fit data perfectly (Kelley and Pornprasertmanit, in press). Categorical omega is a method to calculate coefficient omega for categorical items (Green and Yang, 2009). That is, categorical omega is estimated by the parameter estimates from CFA for categorical items. If coefficient omega or hierarchical omega is used, CFA for continuous items is used, which is not appropriate for categorical items.

If researchers wish to make the measurement model with all parallel items (equal factor loadings and equal error variances), users can specify it by setting interval.type = "parallel" and type = "alpha" or type = "alpha-cfa". See McDonald (1999) for the assumptions of each of these models.

The list below shows all methods to find the confidence interval of reliability.

1. "none" or 0 to not find any confidence interval

2. "parallel" or 11 to assume that the items are parallel and analyze confidence interval based on Wald confidence interval (see van Zyl, Neudecker, & Nel, 2000, Equation 22; also referred as the asymptotic method of Koning & Franses, 2003).

3. "feldt" or 12 is based on that \frac{1 - \alpha}{1 - \hat{\alpha}} is distributed as F distribution with the degree of freedoms of N - 1 and (N - 1) \times (p - 1) (Feldt, 1965).

4. "siotani" or 13 is the same as the "feldt" method but using the degree of freedoms of N and N \times (p - 1) (Siotani, Hayakawa, & Fujikoshi, 1985; van Zyl et al., 2000, Equations 7 and 8; also referred as the exact method of Koning & Franses, 2003).

5. "fisher" or 21 for the Fisher's z transformation on the correlation coefficient approach, z = 0.5 \times \log{\frac{1 + \alpha}{1 - \alpha}}, directly on the coefficient alpha and find confidence interval of transformed scale (Fisher, 1950). The variance of the z is \frac{1}{N - 3} where N is the total sample size.

6. "bonett" or 22 for the Fisher's z transformation on the intraclass correlation approach with the variance of \frac{2p}{(N - 2)(p - 1)} (Bonett, 2002, Equation 6).

7. "hakstian" or 23 uses the cube root transformation and assumes normal distribution on the cube root transformation (Hakstian & Whalen, 1976). The variance of the transformed reliability is based on the degrees of freedom in the "feldt" method.

8. "hakstianbarchard" or 24 uses a correction of the violation of compound symmetry of covariance matrix by adjusting the degrees of freedom in the "hakstian". This correction is used for the inference in type 12 sampling (both persons and items are sampled from the population of persons and items) See Hakstian and Barchard (2000) for further details.

9. "icc" or 25 for the Fisher's z transformation on the intraclass correlation approach, z = \log{1 - \alpha}. The variance of the z is \frac{2p}{N(p - 1)} where p is the number of items (Fisher, 1991, p. 221; van Zyl et al., 2000, p. 277).

10. "ml" or 31 or normal-theory to analyze the confidence interval based on normal-theory approach (or multivariate delta method). See van Zyl, Neudecker, & Nel (2000, Equation 21) for the confidence interval of coefficient alpha (also be referred as Iacobucci & Duhachek's, 2003, method). See Raykov (2002) for details for coefficient omega. If users use type="alpha-cfa", the sem package will be used to obtain parameter estimates and standard errors used for the formula proposed by Raykov (2002).

11. "mll" or 32 to analyze the confidence interval based on normal-theory approach as above. However, the point estimate and standard error were used to build confidence interval using logistic transformation as the note below.

12. "mlr" or 33 to analyze the confidence interval based on normal-theory approach (or multivariate delta method). However, the estimation method uses robust standard errors (Satorra and Bentler, 2000). This is the default estimation approach (but see Kelley and Pornprasertmanit (2016) who recommend the BCa bootstrap [which is bca])

13. "mlrl" or 34 to analyze he confidence interval based on normal-theory approach using robust standard error and logistic transformation (see below).

14. "adf" or 35 for asymptotic distribution-free method (see Maydeu-Olivares, Coffman, & Hartman, 2007 for further details for coefficient omega; we use phantom variable approach, Cheung, 2009, and "WLS" estimator for coefficient omega, Browne, 1984, in the lavaan package, Rosseel, 2012).

15. "adfl" or 36 to use asymptotic distribution-free method to derive standard error and parameter estimate. Then, logistic transformation is used to build confidence interval (see below).

16. "ll" or 37 for profile likelihood-based confidence interval of both reliability coefficients (Cheung, 2009) analyzed by the OpenMx package (Boker et al., 2011)

17. "bsi" or 41 for standard bootstrap confidence interval which finds the standard deviation across the bootstrap estimates, multiply the standard deviation by critical value, and add and subtract from the reliability estimate.

18. "bsil" or 42 to use standard bootstrap confidence interval. However, logistic transformation is used to build confidence interval.

19. "perc" or 43 for percentile bootstrap confidence interval.

20. "bca" or 44 for bias-corrected and accelerated bootstrap confidence interval.

The logistic transformation (Browne, 1982) is applicable for "ml", "mlr", "adf", and "bsi" as "mll", "mlrl", "adfl", and "bsil". The logistic transformation does not assume that the sampling distribution of reliability is symmetric. It acknowledges the fact that reliability ranges from 0 and 1. Logistic transformation is applied to the reliability estimates. Confidence interval is established for the transformed value. The lower and upper bounds of the transformed value is translated back to the reliability estimates. See Browne (1982) or Kelley and Pornprasertmanit (in press) for further details.

Note that not all confidence interval methods are available for all types of reliability and all types of input. For example, bootstrap confidence intervals are not available for covariance matrix input. Parallel confidence intervals are not available for hierarchical omega. We provided appropriate error messages for all impossible combinations.

Value

Note

This function is not compatible with code from MBESS Version 3.

Author(s)

Sunthud Pornprasertmanit (Texas Tech University; sunthud.pornprasertmanit@ttu.edu) and Ken Kelley (University of Notre Dame; kkelley@nd.edu. The previous version was written by Keke Lai (University of California-Merced), Leann J. Terry (while at Indiana University), and Ken Kelley

References

Boker, S., M., N., Maes, H., Wilde, M., Spiegel, M., Brick, T., et al. (2011). OpenMx: An open source extended structural equation modeling framework. Psychometrika, 76, 306–317.

Bonett, D. G. (2002). Sample size requirements for testing and estimating coefficient alpha. Journal of Educational and Behavioral Statistics, 27, 335–340.

Browne, M. W. (1982). Covariance structures. In D. M. Hawkins (Ed.), Topics in applied multivariate analysis (pp. 72–141). Cambridge, UK: Cambridge University Press.

Browne, M. W. (1984). Asymptotic distribution free methods in the analysis of covariance structures. British Journal of Mathematical and Statistical Psychology, 24, 445–455.

Cheung, M. W.-L. (2009). Constructing approximate confidence intervals for parameters with structural constructing approximate confidence intervals for parameters with structural equation models. Structural Equation Modeling, 16, 267–294.

Feldt, L.S. (1965). The approximate sampling distribution of Kuder-Richardson reliability coefficient twenty. Psychometrika, 30, 357–370.

Fisher, R. A. (1950). Statistical methods for research workers. Edinburgh, UK: Oliver & Boyd.

Fisher, R. A. (1991). Statistical methods for research workers. In J.H. Bennett (Ed.), Statistical methods, experimental design, and scientific inference. Oxford: Oxford University Press.

Green, S. B., & Yang, Y. (2009). Reliability of summed item scores using structural equation modeling: An alternative to coefficient alpha. Psychometrika, 74, 155–167.

Hakstian, A. R., & Whalen, T. E. (1976). A k-sample significance test for independent alpha coefficients. Psychometrika, 41, 219–231.

Iacobucci, D., & Duhachek, A. (2003). Advancing alpha: measuring reliability with confidence. Journal of Consumer Psychology, 13, 478–487.

Kelley, K. & Pornprasertmanit, P. (2016). Confidence intervals for population reliability coefficients: Evaluation of methods, recommendations, and software for homogeneous composite measures. Psychological Methods, 21, 69–92.

Koning, A. J., & Franses, P. H. (2003). Confidence intervals for Cronbach's coefficient alpha values (ERIM Report Series Ref. No. ERS-2003-041-MKT). Rotterdam, The Netherlands: Erasmus Research Institute of Management.

Maydeu-Olivares, A., Coffman, D. L., & Hartmann, W. M. (2007). Asymptotically distribution-free (ADF) interval estimation of coefficient alpha. Psychological Methods, 12, 157–176.

McDonald, R. P. (1999). Test theory: A unified approach. Mahwah, New Jersey: Lawrence Erlbaum Associates, Publishers.

Raykov, T. (2002). Analytic estimation of standard error and confidence interval for scale reliability. Multivariate Behavioral Research, 37, 89–103.

Rosseel, Y. (2012). lavaan: An R package for structural equation modeling. Journal of Statistical Software, 48, 1–36.

Satorra, A. & Bentler, P. M. (2001). A scaled difference chi-square test statistic for moment structure analysis. Psychometrika, 66, 507–514.

Siotani, M., Hayakawa, T., & Fujikoshi, Y. (1985). Modem multivariate statistical analysis: A graduate course and handbook. Columbus, Ohio: American Sciences Press.

van Zyl, J. M., Neudecker, H., & Nel, D. G. (2000). On the distribution of the maximum likelihood estimator of Cronbach's alpha. Psychometrika, 65 (3), 271–280.

Yuan, K. & Bentler, P. M. (2002). On robustness of the normal-theory based asymptotic distributions of three reliability coefficient estimates. Psychometrika, 67 (2), 251–259.

See Also

CFA.1; sem; lavaan

Examples

# Use this function for the attitude dataset (ignoring the overall rating variable)
# ci.reliability(data=attitude[,-1], type = "omega", interval.type = "mlrl")

# ci.reliability(data=attitude[,-1], type = "alpha", interval.type = "ll")


## Forming a hypothetical population covariance matrix
# Pop.Cov.Mat <- matrix(.3, 9, 9)
# diag(Pop.Cov.Mat) <- 1
# ci.reliability(S=Pop.Cov.Mat, N=50, type="alpha", interval.type = "bonett")

Confidence interval for the population root mean square error of approximation

Description

Confidence interval for the population root mean square error of approximation (RMSEA).

Usage

ci.rmsea(rmsea, df, N, conf.level = 0.95, alpha.lower = NULL, 
alpha.upper = NULL)

Arguments

Details

Provides a confidence interval for the population root mean square error of approximation (RMSEA) using the noncentral chi-square distribution (e.g., Steiger & Lind, 1980).

Value

returns the upper and lower limit as well as the observed value of the RMSEA.

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

References

Steiger, J. H., & Lind, J. C. (1980). Statistically-based tests for the number of common factors. Paper presented at the annual Spring meeting of the Psychometric Society, Iowa City, IA.

Confidence Interval for a Standardized Contrast in a Fixed Effects ANOVA

Description

Function to obtain the confidence interval for a standardized contrast in a fixed effects analysis of variance context.

Usage

ci.sc(means = NULL, s.anova = NULL, c.weights = NULL, n = NULL, 
N = NULL, Psi = NULL, ncp = NULL, conf.level = 0.95, 
alpha.lower = NULL, alpha.upper = NULL, df.error = NULL, ...)

Arguments

Value

Note

Be sure to use the standard deviation and not the error variance for s.anova, not the square of this value (the error variance) which would come from the source table (i.e., do not use the variance of the error but rather use its square root, the standard deviation).

Be sure to use the error variance and not its square root (i.e., use the variance of the standard deviation of the errors). Be sure to use the standard deviations of errors for s.anova and s.ancova, not the square of these values (i.e., do not use the variance of the errors).

Be sure to use fractional c-weights when doing complex contrasts (not integers) to specify c.weights. For exmaple, in an ANCOVA of four groups, if the user wants to compare the mean of group 1 and 2 with the mean of group 3 and 4, c.weights should be specified as c(0.5, 0.5, -0.5, -0.5) rather than c(1, 1, -1, -1). Make sure the sum of the contrast weights are zero.

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

References

Kelley, K. (2007). Constructing confidence intervals for standardized effect sizes: Theory, application, and implementation. Journal of Statistical Software, 20 (8), 1–24.

Lai, K., & Kelley, K. (2007). Sample size planning for standardized ANCOVA and ANOVA contrasts: Obtaining narrow confidence intervals. Manuscript submitted for publication.

Steiger, J. H. (2004). Beyond the F Test: Effect size confidence intervals and tests of close fit in the Analysis of Variance and Contrast Analysis. Psychological Methods, 9, 164–182.

See Also

conf.limits.nct, ci.src, ci.smd, ci.smd.c, ci.sm, ci.c

Examples

# Here is a four group example. Suppose that the means of groups 1--4 are 2, 4, 9, 
# and 13, respectively. Further, let the error variance be .64 and thus the standard
# deviation would be .80 (note we use the standard deviation in the function, not the 
# variance). The standardized contrast of interest here is the average of groups 1 and 4
# versus the average of groups 2 and 3. 

ci.sc(means=c(2, 4, 9, 13), s.anova=.80, c.weights=c(.5, -.5, -.5, .5), 
n=c(3, 3, 3, 3), N=12, conf.level=.95)


# Here is an example with two groups. 
ci.sc(means=c(1.6, 0), s.anova=.80, c.weights=c(1, -1), n=c(10, 10), N=20, conf.level=.95)

Confidence interval for a standardized contrast in ANCOVA with one covariate

Description

Calculate the confidence interval for a standardized contrast in ANCOVA with one covariate. The standardizer (i.e., the divisor) can be either the error standard deviation of the ANOVA model (i.e., the model excluding the covariate) or of the ANCOVA model.

Usage

ci.sc.ancova(Psi=NULL, adj.means=NULL, s.anova = NULL, s.ancova, 
standardizer = "s.ancova", c.weights, n, cov.means, SSwithin.x, 
conf.level = 0.95)

Arguments

Value

Note

Be sure to use the standard deviations and not the error variances for s.anova and s.ancova, not the squares of these values which would come from the source tables (i.e., do not use the variance of the errors but rather use its square root, the standard deviation).

If n receives a single number, that number is considered as the sample size per group. If n is assigned to a vector, the vector is considered as the sample size of each group.

Be sure to use fractional c-weights when doing complex contrasts (not integers) to specify c.weights. For example, in an ANCOVA of four groups, if the user wants to compare the mean of group 1 and 2 with the mean of group 3 and 4, c.weights should be specified as c(0.5, 0.5, -0.5, -0.5) rather than c(1, 1, -1, -1). Make sure the sum of the contrast weights are zero.

The argument to be assigned to standardizer must be either "s.ancova" or "s.anova".

Author(s)

Keke Lai (University of California–Merced) and Ken Kelley kkelley@nd.edu

References

Kelley, K. (2007). Constructing confidence intervals for standardized effect sizes: Theory, application, and implementation. Journal of Statistical Software, 20 (8), 1–24.

Kelley, K., & Rausch, J. R. (2006). Sample size planning for the standardized mean difference: Accuracy in Parameter Estimation via narrow confidence intervals. Psychological Methods, 11, 363–385.

Lai, K., & Kelley, K. (2012). Accuracy in parameter estimation for ANCOVA and ANOVA contrasts: Sample size planning via narrow confidence intervals. British Journal of Mathematical and Statistical Psychology, 65, 350–370.

Steiger, J. H., & Fouladi, R. T. (1997). Noncentrality interval estimation and the evaluation of statistical methods. In L. L. Harlow, S. A. Mulaik, & J.H. Steiger (Eds.), What if there were no significance tests? (pp. 221–257). Mahwah, NJ: Lawrence Erlbaum.

See Also

ci.c.ancova, ci.sc

Examples

# Maxwell & Delaney (2004, pp. 428--468) offer an example that 30 depressive 
# individuals are randomly assigned to three groups, 10 in each, and ANCOVA 
# is performed on the posttest scores using the participants' pretest 
# scores as the covariate. The means of pretest scores of group 1, 2, and 3 are 
# 17, 17.7, and 17.4, respectively, whereas the adjusted means of groups 1, 2, and 3 
# are 7.5, 12, and 14, respectively. The error variance in ANCOVA is 29 and thus 
# 5.385165 is the error standard deviation, with the sum of squares within groups 
# from an ANOVA on the covariate is 752.5. 

# To obtained the confidence interval for the standardized adjusted
# mean difference between group 1 and 2, using the ANCOVA error standard
# deviation:
ci.sc.ancova(adj.means=c(7.5, 12, 14), s.ancova=5.385165, c.weights=c(1,-1,0), 
n=10, cov.means=c(17, 17.7, 17.4), SSwithin.x=752.5)

# Or, with less error in rounding:
ci.sc.ancova(adj.means=c(7.54, 11.98, 13.98), s.ancova=5.393, c.weights=c(-1,0,1), 
n=10, cov.means=c(17, 17.7, 17.4), SSwithin.x=752.5)

# Now, using the standard deviation from ANOVA (and not ANCOVA as above), we have:
ci.sc.ancova(adj.means=c(7.54, 11.98, 13.98), s.anova=6.294, s.ancova=5.393, c.weights=c(-1,0,1),
n=10, cov.means=c(17, 17.7, 17.4), SSwithin.x=752.5, standardizer= "s.anova", conf.level=.95)

Confidence Interval for the Standardized Mean

Description

Function to obtain the exact confidence interval for the standardized mean.

Usage

ci.sm(sm = NULL, Mean = NULL, SD = NULL, ncp = NULL, N = NULL, 
conf.level = 0.95, alpha.lower = NULL, alpha.upper = NULL, ...)

Arguments

Details

The user must specify the standardized mean in one and only one of the three ways: a) mean and standard deviation (Mean and SD), b) standardized mean (sm), and c) noncentral parameter (ncp). The confidence level must be specified in one of following two ways: using confidence interval coverage (conf.level), or lower and upper confidence limits (alpha.lower and alpha.upper).

This function uses the exact confidence interval method based on noncentral t-distributions. The confidence interval for noncentral t-parameter can be obtained from the conf.limits.nct function in MBESS.

Value

Note

The standardized mean is the mean divided by the standard deviation.

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

References

Kelley, K. (2007). Constructing confidence intervals for standardized effect sizes: Theory, application, and implementation. Journal of Statistical Software, 20 (8), 1–24.

Steiger, J. H., & Fouladi, R. T. (1997). Noncentrality interval estimation and the evaluation of statistical methods. In L. L. Harlow, S. A. Mulaik, & J.H. Steiger (Eds.), What if there were no significance tests? (pp. 221–257). Mahwah, NJ: Lawrence Erlbaum.

See Also

conf.limits.nct

Examples

ci.sm(sm=2.037905, N=13, conf.level=.95)
ci.sm(Mean=30, SD=14.721, N=13, conf.level=.95)
ci.sm(ncp=7.347771, N=13, conf.level=.95)
ci.sm(sm=2.037905, N=13, alpha.lower=.05, alpha.upper=0)
ci.sm(Mean=50, SD=10, N=25, conf.level=.95)

Confidence limits for the standardized mean difference.

Description

Function to calculate the confidence limits for the population standardized mean difference using the square root of the pooled variance as the divisor. This function is thus used to determine the confidence bounds for the population quantity of what is generally referred to as Cohen's d (delta being that population quantity).

Usage

ci.smd(ncp=NULL, smd=NULL, n.1=NULL, n.2=NULL, conf.level=.95, 
alpha.lower=NULL, alpha.upper=NULL, tol=1e-9, ...)

Arguments

Value

Warning

This function uses conf.limits.nct, which has as one of its arguments tol (and can be modified with tol of the present function). If the present function fails to converge (i.e., if it runs but does not report a solution), it is likely that the tol value is too restrictive and should be increased by a factor of 10, but probably by no more than 100. Running the function conf.limits.nct directly will report the actual probability values of the limits found. This should be done if any modification to tol is necessary in order to ensure acceptable confidence limits for the noncentral-t parameter have been achieved.

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

References

Cohen, J. (1988) Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum.

Cumming, G. & Finch, S. (2001). A primer on the understanding, use, and calculation of confidence intervals that are based on central and noncentral distributions, Educational and Psychological Measurement, 61, 532–574.

Hedges, L. V. (1981). Distribution theory for Glass's Estimator of effect size and related estimators. Journal of Educational Statistics, 2, 107–128.

Kelley, K. (2007). Constructing confidence intervals for standardized effect sizes: Theory, application, and implementation. Journal of Statistical Software, 20 (8), 1–24.

Kelley, K., Maxwell, S. E., & Rausch, J. R. (2003). Obtaining Power or Obtaining Precision: Delineating Methods of Sample-Size Planning, Evaluation and the Health Professions, 26, 258–287.

Steiger, J. H., & Fouladi, R. T. (1997). Noncentrality interval estimation and the evaluation of statistical methods. In L. L. Harlow, S. A. Mulaik,&J.H. Steiger (Eds.), What if there were no significance tests? (pp. 221–257). Mahwah, NJ: Lawrence Erlbaum.

See Also

smd, smd.c, ci.smd.c, conf.limits.nct

Examples

# Steiger and Fouladi (1997) example values.
ci.smd(ncp=2.6, n.1=10, n.2=10, conf.level=1-.05)
ci.smd(ncp=2.4, n.1=300, n.2=300, conf.level=1-.05)

Confidence limits for the standardized mean difference using the control group standard deviation as the divisor.

Description

Function to calculate the confidence limits for the standardized mean difference using the control group standard deviation as the divisor (Glass's g).

Usage

ci.smd.c(ncp = NULL, smd.c = NULL, n.C = NULL, n.E = NULL, 
conf.level = 0.95, alpha.lower = NULL, alpha.upper = NULL, 
tol = 1e-09, ...)

Arguments

Value

Warning

This function uses conf.limits.nct, which has as one of its arguments tol (and can be modified with tol of the present function). If the present function fails to converge (i.e., if it runs but does not report a solution), it is likely that the tol value is too restrictive and should be increased by a factor of 10, but probably by no more than 100. Running the function conf.limits.nct directly will report the actual probability values of the limits found. This should be done if any modification to tol is necessary in order to ensure acceptable confidence limits for the noncentral-t parameter have been achieved.

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

References

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum.

Cumming, G. & Finch, S. (2001). A primer on the understanding, use, and calculation of confidence intervals that are based on central and noncentral distributions, Educational and Psychological Measurement, 61, 532–574.

Glass, G. V. (1976). Primary, secondary, and meta-analysis of research. Educational Researcher, 5, 3–8.

Hedges, L. V. (1981). Distribution theory for Glass's Estimator of effect size and related estimators. Journal of Educational Statistics, 2, 107–128.

Kelley, K. (2007). Constructing confidence intervals for standardized effect sizes: Theory, application, and implementation. Journal of Statistical Software, 20 (8), 1–24.

Steiger, J. H., & Fouladi, R. T. (1997). Noncentrality interval estimation and the evaluation of statistical methods. In L. L. Harlow, S. A. Mulaik, & J. H. Steiger (Eds.), What if there were no significance tests? (pp. 221–257). Mahwah, NJ: Lawrence Erlbaum.

See Also

smd.c, smd, ci.smd, conf.limits.nct

Examples

ci.smd.c(smd.c=.5, n.C=100, n.E=100, conf.level=.95)

Confidence Interval for the Signal-To-Noise Ratio

Description

Function to obtain the exact confidence interval for the signal-to-noise ratio (i.e., the variance of the specific factor over the error variance).

Usage

ci.snr(F.value = NULL, df.1 = NULL, df.2 = NULL, N = NULL, conf.level = 0.95,
 alpha.lower = NULL, alpha.upper = NULL, ...)

Arguments

Details

The confidence level must be specified in one of following two ways: using confidence interval coverage (conf.level), or lower and upper confidence limits (alpha.lower and alpha.upper).

This function uses the confidence interval transformation principle (Steiger, 2004) to transform the confidence limits for the noncentality parameter to the confidence limits for the population's signal-to-noise ratio. The confidence interval for noncentral F-parameter can be obtained from the conf.limits.ncf function in MBESS, which is used internally within this function.

Value

Returns the confidence limits for the signal-to-noise ratio.

Note

The signal to noise ratio is defined as the variance due to the particular factor over the error variance (i.e., the mean square error).

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

References

Kelley, K. (2007). Constructing confidence intervals for standardized effect sizes: Theory, application, and implementation. Journal of Statistical Software, 20 (8), 1–24.

Fleishman, A. I. (1980). Confidence intervals for correlation ratios. Educational and Psychological Measurement, 40, 659–670.

Steiger, J. H. (2004). Beyond the F Test: Effect size confidence intervals and tests of close fit in the Analysis of Variance and Contrast Analysis. Psychological Methods, 9, 164–182.

See Also

ci.srsnr, ci.omega2 conf.limits.ncf

Examples

## Bargman (1970) gave an example in which a 5-group ANOVA with 11 subjects in each 
## group is conducted and the observed F value is 11.2213. This example was 
## used in Venables (1975),  Fleishman (1980), and Steiger (2004). If one wants to calculate 
## the exact confidence interval for the signal-to-noise ratio of that example, this
## function can be used.

ci.snr(F.value=11.221, df.1=4, df.2=50, N=55)

ci.snr(F.value=11.221, df.1=4, df.2=50, N=55, conf.level=.90)

ci.snr(F.value=11.221, df.1=4, df.2=50, N=55,  alpha.lower=.02, alpha.upper=.03)

Confidence Interval for a Standardized Regression Coefficient

Description

Function to obtain the confidence interval for a standardized regression coefficient.

Usage

ci.src(beta.k = NULL, SE.beta.k = NULL, N = NULL, K = NULL, R2.Y_X = NULL, 
R2.k_X.without.k = NULL, conf.level = 0.95, R2.Y_X.without.k = NULL, 
t.value = NULL, b.k = NULL, SE.b.k = NULL, s.Y = NULL, s.X = NULL, 
alpha.lower = NULL, alpha.upper = NULL, Suppress.Statement = FALSE, ...)

Arguments

Details

For standardized variables, do not specify the standard deviation of the variables and input the standardized regression coefficient for b.k.

Value

Returns the confidence limits specified for the regression coefficient of interest from the standard approach to confidence interval formation or from the noncentral approach to confidence interval formation using the noncentral t-distribution.

Note

This function calls upon ci.reg.coef in MBESS, but has a different naming scheme. See ci.reg.coef for more details.

To form a confidence interval for the unstandardized regression coefficient, use ci.rc. This function is used to form a confidence interval for the standardized regression coefficient.

Not all of the values need to be specified, only those that contain all of the necessary information in order to compute the confidence interval (options are thus given for the values that need to be specified).

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

References

Kelley, K. (2007). Constructing confidence intervals for standardized effect sizes: Theory, application, and implementation. Journal of Statistical Software, 20 (8), 1–24.

Kelley, K., & Maxwel, S. E. (2003). Sample size for Multiple Regression: Obtaining regression coefficients that are accurate, not simply significant. Psychological Methods, 8, 305–321.

Kelley, K., & Maxwell, S. E. (2008). Sample Size Planning with applications to multiple regression: Power and accuracy for omnibus and targeted effects. In P. Alasuuta, J. Brannen, & L. Bickman (Eds.), The Sage handbook of social research methods (pp. 166–192). Newbury Park, CA: Sage.

Smithson, M. (2003). Confidence intervals. New York, NY: Sage Publications.

Steiger, J. H. (2004). Beyond the F Test: Effect size confidence intervals and tests of close fit in the Analysis of Variance and Contrast Analysis. Psychological Methods, 9, 164–182.

See Also

ss.aipe.reg.coef, conf.limits.nct, ci.reg.coef, ci.rc

Confidence Interval for the Square Root of the Signal-To-Noise Ratio

Description

Function to calculate the exact confidence interval for the square root of the signal-to-noise ratio.

Usage

ci.srsnr(F.value = NULL, df.1 = NULL, df.2 = NULL, N = NULL, 
conf.level = 0.95, alpha.lower = NULL, alpha.upper = NULL, ...)

Arguments

Details

The confidence level must be specified in one of following two ways: using confidence interval coverage (conf.level), or lower and upper confidence limits (alpha.lower and alpha.upper).

The square root of the signal-to-noise ratio is defined as the standard deviation due to the particular factor over the standard deviation of the error (i.e., the square root of the mean square error). This function uses the confidence interval transformation principle (Steiger, 2004) to transform the confidence limits for the noncentality parameter to the confidence limits for square root of signal-to-noise ratio. The confidence interval for noncentral F parameter can be abtained from function conf.limits.ncf in MBESS.

Value

Returns the square root of the confidence limits for the signal to noise ratio.

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

References

Fleishman, A. I. (1980). Confidence intervals for correlation ratios. Educational and Psychological Measurement, 40, 659–670.

Kelley, K. (2007). Constructing confidence intervals for standardized effect sizes: Theory, application, and implementation. Journal of Statistical Software, 20 (8), 1–24.

Steiger, J. H. (2004). Beyond the F Test: Effect size confidence intervals and tests of close fit in the Analysis of Variance and Contrast Analysis. Psychological Methods, 9, 164–182.

See Also

ci.snr, conf.limits.ncf

Examples

## To illustrate the calculation of the confidence interval for noncentral 
## F parameter,Bargman (1970) gave an example in which a 5-group ANOVA with 
## 11 subjects in each group is conducted and the observed F value is 11.2213. 
## This exmaple continued to be used in Venables (1975),  Fleishman (1980), 
## and Steiger (2004). If one wants to calculate the exact confidence interval 
## for square root of the signal-to-noise ratio of that example, this 
## function can be used.

ci.srsnr(F.value=11.221, df.1=4, df.2=50, N=55)

ci.srsnr(F.value=11.221, df.1=4, df.2=50, N=55, conf.level=.90)

ci.srsnr(F.value=11.221, df.1=4, df.2=50, N=55,  alpha.lower=.02, alpha.upper=.03)

Confidence limits for noncentral chi square parameters

Description

Function to determine the noncentral parameter that leads to the observed Chi.Square-value, so that a confidence interval for the population noncentral chi-squrae value can be formed.

Usage

conf.limits.nc.chisq(Chi.Square=NULL, conf.level=.95, df=NULL, 
alpha.lower=NULL, alpha.upper=NULL, tol=1e-9, Jumping.Prop=.10)

Arguments

Details

If the function fails (or if a function relying upon this function fails), adjust the Jumping.Prop (to a smaller value).

Value

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu); Keke Lai (University of California–Merced)

See Also

conf.limits.nct, conf.limits.ncf

Examples

# A typical call to the function.
conf.limits.nc.chisq(Chi.Square=30, conf.level=.95, df=15)

# A one sided (upper) confidence interval.
 conf.limits.nc.chisq(Chi.Square=30, alpha.lower=0, alpha.upper=.05, 
 conf.level=NULL, df=15)

Confidence limits for noncentral F parameters

Description

Function to determine the noncentral parameter that leads to the observed F-value, so that a confidence interval around the population F-value can be conducted. Used for forming confidence intervals around noncentral parameters (given the monotonic relationship between the F-value and the noncentral value).

Usage

conf.limits.ncf(F.value = NULL, conf.level = .95, df.1 = NULL, 
df.2 = NULL, alpha.lower = NULL, alpha.upper = NULL, tol = 1e-09,
Jumping.Prop = 0.1)

Arguments

Details

This function is the relied upon by the ci.R2 and ss.aipe.R2. If the function fails (or if a function relying upon this function fails), adjust the Jumping.Prop (to a smaller value).

Value

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu); Keke Lai (University of Califonia-Merced)

See Also

ss.aipe.R2, ci.R2, conf.limits.nct

Examples

conf.limits.ncf(F.value = 5, conf.level = .95, df.1 = 5, 
df.2 = 100)

# A one sided confidence interval.
conf.limits.ncf(F.value = 5, conf.level = NULL, df.1 = 5, 
df.2 = 100, alpha.lower = .05, alpha.upper = 0, tol = 1e-09,
Jumping.Prop = 0.1)

Confidence limits for a noncentrality parameter from a t-distribution

Description

Function to determine the noncentrality parameters necessary to form a confidence interval around the population noncentrality parameter and related parameters.

Usage

conf.limits.nct(ncp, df, conf.level = 0.95, alpha.lower = NULL, 
alpha.upper = NULL, t.value, tol = 1e-09, sup.int.warns = TRUE, 
...)

Arguments

Details

Function for finding the upper and lower confidence limits for a noncentral parameter from a noncentral t-distribution with df degrees of freedom. This function is especially helpful when forming confidence intervals around standardized mean differences (i.e., Cohen's d; Glass's g; Hedges' g), standardized regression coefficients, and coefficients of variations. The Lower.Limit and the Upper.Limit values correspond to the noncentral parameters of a t-distribution with df degrees of freedom whose upper and lower tails contain the desired proportion of the respective noncentral t-distribution. When ncp is zero, the Lower.Limit and Upper.Limit are simply the desired quantiles of the central t-distribution with df degrees of freedom.

Note that the confidence interval limit(s) are found twice, using two different methods. The first method uses the optimize function, whereas the second method uses the nlm function. The best of the two methods, if not equal and numerically exact, is taken. This does not concern the user.

Value

Warning

At the present time, the largest ncp that R can accurately handle is 37.62.

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

References

Cumming, G. & Finch, S. (2001). A primer on the understanding, use, and calculation of confidence intervals that are based on central and noncentral distributions, Educational and Psychological Measurement, 61, 532–574.

Kelley, K. (2005). The effects of nonnormal distributions on confidence intervals around the standardized mean difference: Bootstrap and parametric confidence intervals, Educational and Psychological Measurement, 65, 51–69.

Kelley, K. (2007). Constructing confidence intervals for standardized effect sizes: Theory, application, and implementation. Journal of Statistical Software, 20 (8), 1–24.

Steiger, J. & Fouladi, T. (1997). Noncentrality interval estimation and the evaluation of statistical models. In L. Harlow, S. Muliak, & J. Steiger (Eds.), What if there were no significance tests?. Mahwah, NJ: Lawrence Erlbaum.

See Also

pt, qt, ci.smd, ci.smd.c, ss.aipe, conf.limits.ncf, conf.limits.nc.chisq

Examples

# Suppose observed t-value based on 'df'=126 is 2.83. Finding the lower 
# and upper critical values for the population noncentrality parameter 
# with a symmetric confidence interval with 95% confidence is given as:
conf.limits.nct(ncp=2.83, df=126, conf.level=.95)

# Modifying the above example so that a nonsymmetric 95% confidence interval
# can be formed:
conf.limits.nct(ncp=2.83, df=126, alpha.lower=.01, alpha.upper=.04,
conf.level=NULL)

# Modifying the above example so that a single-sided 95% confidence interval
# can be formed:
conf.limits.nct(ncp=2.83, df=126, alpha.lower=0, alpha.upper=.05,
conf.level=NULL)

Correlation Matrix to Covariance Matrix Conversion

Description

Function to convert a correlation matrix to a covariance matrix.

Usage

cor2cov(cor.mat, sd, discrepancy=1e-5)

Arguments

Details

The correlation matrix to convert can be either symmetric or triangular. The covariance matrix returned is always a symmetric matrix.

Note

The correlation matrix input should be a square matrix, and the length of sd should be equal to the number of variables in the correlation matrix (i.e., the number of rows/columns). Sometimes the correlation matrix input may not have exactly 1's on the main diagonal, due to, eg, rounding; discrepancy specifies the allowable discrepancy so that the function still considers the input as a correlation matrix and can proceed (but the function does not change the numbers on the main diagonal).

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu), Keke Lai

Covariance matrix from confirmatory (single) factor model.

Description

Function calculates a covariance matrix using the specified Lambda and Psi.Square values from a confirmatory factor model approach (McDonald, 1999).

Usage

covmat.from.cfm(Lambda, Psi.Square, tol.det = 1e-05)

Arguments

Value

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu); Leann Terry (Indiana University; ljterry@Indiana.Edu)

References

McDonald, R. P. (1999). Test theory: A unified approach. Mahwah, NJ: Erlbaum.

See Also

CFA.1;sem

Examples

# General Congeneric
# covmat.from.cfm(Lambda=c(.8, .9, .6, .8), Psi.Square=c(.6, .2, .1, .3), tol.det=.00001)


# True-score equivalent
# covmat.from.cfm(Lambda=c(.8, .8, .8, .8), Psi.Square=c(.6, .2, .1, .3), tol.det=.00001)


# Parallel 
# covmat.from.cfm(Lambda=c(.8, .8, .8, .8), Psi.Square=c(.2, .2, .2, .2), tol.det=.00001)

Function to calculate the regular (which is also biased) estimate of the coefficient of variation or the unbiased estimate of the coefficient of variation.

Description

Returns the estimated coefficient of variation or the unbiased estimate of the coefficient of variation.

Usage

cv(C.of.V=NULL, mean=NULL, sd=NULL, N=NULL, unbiased=FALSE)

Arguments

Details

A function to calculate the usual estimate of the coefficient of variation or its unbiased estimate.

Value

Returns the estimated coefficient of variation (regular but biased estimate or unbiased estimate.

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

See Also

ci.cv

Examples

cv(mean=100, sd=15)
cv(mean=100, sd=15, N=50, unbiased=TRUE)
cv(C.of.V=.15, N=2, unbiased=TRUE)

Regression Surface Containing Interaction

Description

To plot a three dimentional figure of a multiple regression surface containing one two-way interaction.

Usage

intr.plot(b.0, b.x, b.z, b.xz, x.min = NULL, x.max = NULL, z.min = NULL, 
z.max = NULL, n.x = 50, n.z = 50, x = NULL, z = NULL, col = "lightblue", 
hor.angle = -60, vert.angle = 15, xlab = "Value of X", zlab = "Value of Z", 
ylab = "Dependent Variable", expand = 0.5, lines.plot=TRUE, col.line = "red", 
line.wd = 2, gray.scale = FALSE, ticktype="detailed", ...)

Arguments

Details

The user can input either the limits of x and z, or specific x and z vectors, to draw the regression surface. If the user inputs simply the limits of the predictors, the function would generate predictor vectors for plotting. If the user inputs specific predictor vectors, the function would plot the regression surface based on those vectors.

Note

If the user enters specific vectors instead of the ranges of predictors, please make sure elements in those vectors are in ascending order. This is required by function persp, which is used within this function.

Author(s)

Keke Lai (University of California – Merced) and Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

References

Cohen, J., Cohen, P., West, S. G. and Aiken, L. S. (2003). Applied multiple regression/correlation analysis for the behavioral sciences (3rd ed.). Mahwah, NJ: Erlbaum.

See Also

intr.plot.2d, persp

Examples

##  A way to replicate the example given by Cohen et al. (2003) (pp. 258--263):
##  The regression equation with interaction is y=.2X+.6Z+.4XZ+2
##  To plot a regression surface and regression lines of Y on X holding Z 
##  at -1, 0, and 1 standard deviation above the mean

x<- c(0,2,4,6,8,10)
z<-c(0,2,4,6,8,10)
intr.plot(b.0=2, b.x=.2, b.z=.6, b.xz=.4, x=x, z=z)

## input limits of the predictors instead of specific x and z predictor vectors
intr.plot(b.0=2, b.x=.2, b.z=.6, b.xz=.4, x.min=5, x.max=10, z.min=0, z.max=20)

intr.plot(b.0=2, b.x=.2, b.z=.6, b.xz=.4, x.min=0, x.max=10, z.min=0, z.max=10, 
col="gray", hor.angle=-65, vert.angle=10)

##  To plot a black-and-white figure
intr.plot(b.0=2, b.x=.2, b.z=.6, b.xz=.4, x.min=0, x.max=10, z.min=0, z.max=10, 
gray.scale=TRUE)

## to adjust the tick marks on the axes
intr.plot(b.0=2, b.x=.2, b.z=.6, b.xz=.4, x.min=0, x.max=10, z.min=0, z.max=10, 
ticktype="detailed", nticks=8)

Plotting Conditional Regression Lines with Interactions in Two Dimensions

Description

To plot regression lines for one two-way interactions, holding one of the predictors (in this function, z) at values -2, -1, 0, 1, and 2 standard deviations above the mean.

Usage

intr.plot.2d(b.0, b.x, b.z, b.xz,x.min=NULL, x.max=NULL, x=NULL, 
n.x=50, mean.z=NULL, sd.z=NULL, z=NULL,xlab="Value of X",  
ylab="Dependent Variable", sd.plot=TRUE, sd2.plot=TRUE, sd_1.plot=TRUE, 
sd_2.plot=TRUE, type.sd=2, type.sd2=3, type.sd_1=4, type.sd_2=5, 
legend.pos="bottomright", legend.on=TRUE, ... )

Arguments

Details

To input the predictor x, one can use either the limits of x (x.max and x.min) , or a specific vector x (x). To input the predictor z, one can use either the mean and standard deviation of z (mean.z and sd.z ), or a specific vector z (z).

Note

Sometimes some of the regression lines are outside the default scope of the coordinates and thus cannot be seen; in such situations, one needs to, by entering additional arguments, adjust the scope to let proper sections of regression lines be seen. Refer to examples below for more details.

Author(s)

Keke Lai, Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

References

Cohen, J., Cohen, P., West, S. G. and Aiken, L. S. (2003). Applied multiple regression/correlation analysis for the behavioral sciences (3rd ed.). Mahwah, NJ: Erlbaum.

See Also

intr.plot

Examples

## A situation where one regression line is outside the default scope of the coordinates
intr.plot.2d(b.0=16, b.x=2.2, b.z=2.6, b.xz=.4, x.min=0, x.max=20, mean.z=0, sd.z=3)

## Adjust the scope of x and y axes so that proper sections of regression lines can be seen 
intr.plot.2d(b.0=16, b.x=2.2, b.z=2.6, b.xz=.4, x.min=0, x.max=50, mean.z=0, 
sd.z=3, xlim=c(0,50), ylim=c(-20,100) )

## Use specific vector(s) to define the predictor(s) 
intr.plot.2d(b.0=16, b.x=2.2, b.z=2.6, b.xz=.4, x=c(1:10), z=c(0,2,4,6,8,10))

intr.plot.2d(b.0=16, b.x=2.2, b.z=2.6, b.xz=.4, x.min=0, x.max=20, 
z=c(1,3,6,7,9,13,16,20), ylim=c(0,100))

## Change the position of the legend so that it does not block regression lines
intr.plot.2d(b.0=10, b.x=-.3, b.z=1, b.xz=.5, x.min=0, x.max=40, mean.z=-5, sd.z=3, 
ylim=c(-100,100),legend.pos="topright" )

Effect sizes and confidence intervals in a mediation model

Description

Automate the process of simple mediation analysis (one independent variable and one mediator) and effect size estimation for mediation models, as discussed in Preacher and Kelley (2011).

Usage

mediation(x, mediator, dv, S = NULL, N = NULL, x.location.S = NULL, 
mediator.location.S = NULL, dv.location.S = NULL, mean.x = NULL, 
mean.m = NULL, mean.dv = NULL, conf.level = 0.95, 
bootstrap = FALSE, B = 10000, which.boot="both", save.bs.replicates=FALSE,
complete.set=FALSE)

Arguments

Details

Based on the work of Preacher and Kelley (2010) and works cited therein, this function implements (simple) mediation analysis in a way that automates much of the results that are generally of interest, where "simple" means one independent variable, one mediator, and one dependent variable. More specifically, three regression outputs are automated as is the calculation of effect sizes that are thought to be useful or potentially useful in the context of mediation. Much work on mediation models exists in the literature, which should be consulted for proper interpretation of the effect sizes, models, and meaning of results. The usefulness of effect size \kappa^2 was called into question by Wen and Fan (2015). Further, another paper by Lachowicz, Preacher, and Kelley (submitted) offers a better was of quantifying the effect size and it is developed for more complex models. Users are encouraged to use, instead of or in addition to this function, the upsilon function.

Value

Author(s)

Ken Kelley (University of Notre Dame; KKelley@nd.edu)

References

Buyse, M., & Molenberghs, G. (1998). Criteria for the validation of surrogate endpoints in randomized experiments. Biometrics, 54, 1014–1029.

Ditlevsen, S., Christensen, U., Lynch, J., Damsgaard, M. T., & Keiding, N. (2005). The mediation proportion: A structural equation approach for estimating the proportion of exposure effect on outcome explained by an intermediate variable. Epidemiology, 16, 114–120.

Freedman, L. S. (2001). Confidence intervals and statistical power of the 'Validation' ratio for surrogate or intermediate endpoints. Journal of Statistical Planning and Inference, 96, 143–153.

Hansen, W. B., & McNeal, R. B. (1996). The law of maximum expected potential effect: Constraints placed on program effectiveness by mediator relationships. Health Education Research, 11, 501–507.

Huang, B., Sivaganesan, S., Succop, P., & Goodman, E. (2004). Statistical assessment of mediational effects for logistic mediational models. Statistics in Medicine, 23, 2713–2728.

Lachowicz, M. J., Preacher, K. J., & Kelley, K. (submitted). A novel measure of effect size for mediation analysis. Submited for publication.

Lindenberger, U., & Potter, U. (1998). The complex nature of unique and shared effects in hierarchical linear regression: Implications for developmental psychology. Psychological Methods, 3, 218–230.

MacKinnon, D. P. (2008). Introduction to statistical mediation analysis. Mahwah, NJ: Erlbaum.

Preacher, K. J., & Hayes, A. F. (2008b). Asymptotic and resampling strategies for assessing and comparing indirect effects in multiple mediator models. Behavior Research Methods, 40, 879–891.

Preacher, K. J., & Kelley, K. (2011). Effect size measures for mediation models: Quantitative and graphical strategies for communicating indirect effects. Psychological Methods, 16, 93–115.

Sobel, M. E. (1982). Asymptotic confidence intervals for indirect effects in structural equation models. In S. Leinhardt (Ed.), Sociological Methodology 1982 (pp. 290–312). Washington DC: American Sociological Association.

Wen, Z., & Fan, X. (2015). Monotonicity of effect sizes: Questioning kappa-squared as mediation effect size measure. Psychological Methods, 20, 193–203.

See Also

mediation.effect.plot, mediation.effect.bar.plot

Examples

## Not run: 
############################################
# EXAMPLE 1
# Using the Jessor data discussed in Preacher and Kelley (2011), to illustrate
# the methods based on summary statistics. 
 
mediation(S=rbind(c(2.26831107,  0.6615415, -0.08691755), 
c(0.66154147,  2.2763549, -0.22593820), c(-0.08691755, -0.2259382,  0.09218055)), 
N=432, x.location.S=1, mediator.location.S=2, dv.location.S=3, mean.x=7.157645, 
mean.m=5.892785, mean.dv=1.649316, conf.level=.95)

############################################
# EXAMPLE 2
# Clear the workspace:
rm(list=ls(all=TRUE))

# An (unrealistic) example data (from Hayes) 
Data <- rbind(
  c(-5.00, 25.00, -1.00),
  c(-4.00, 16.00, 2.00),
  c(-3.00, 9.00, 3.00),
  c(-2.00, 4.00, 4.00),
  c(-1.00, 1.00, 5.00),
  c(.00, .00, 6.00),
  c(1.00, 1.00, 7.00),
  c(2.00, 4.00, 8.00),
  c(3.00, 9.00, 9.00),
  c(4.00, 16.00, 10.00),
  c(5.00, 25.00, 13.00),
  c(-5.00, 25.00, -1.00),
  c(-4.00, 16.00, 2.00),
  c(-3.00, 9.00, 3.00),
  c(-2.00, 4.00, 4.00),
  c(-1.00, 1.00, 5.00),
  c(.00, .00, 6.00),
  c(1.00, 1.00, 7.00),
  c(2.00, 4.00, 8.00),
  c(3.00, 9.00, 9.00),
  c(4.00, 16.00, 10.00),
  c(5.00, 25.00, 13.00))


# Raw data example of the Hayes data.
mediation(x=Data[,1], mediator=Data[,2], dv=Data[,3], conf.level=.95)

# Sufficient statistics example of the Hayes data.
mediation(S=var(Data), N=22, x.location.S=1, mediator.location.S=2, dv.location.S=3, 
mean.x=mean(Data[,1]), mean.m=mean(Data[,2]), mean.dv=mean(Data[,3]), conf.level=.95)

# Example had there been two groups. 
gp.size <- length(Data[,1])/2 # adjust if using an odd number of observations.
grouping.variable <- c(rep(0, gp.size), rep(1, gp.size))
mediation(x=grouping.variable, mediator=Data[,2], dv=Data[,3])

############################################
# EXAMPLE 3
# Bootstrap of continuous data. 
set.seed(12414) # Seed used for repeatability (there is nothing special about this seed)
bs.Results <- mediation(x=Data[,1], mediator=Data[,2], dv=Data[,3], 
bootstrap=TRUE, B=5000, save.bs.replicates=TRUE)

ls() # Notice that Bootstrap.Replicates is available in the 
workspace (if save.bs.replicates=TRUE in the above call). 

#Now, given the Bootstrap.Replicates object, one can do whatever they want with them. 

# See the names of the effect sizes (and their ordering)
colnames(Bootstrap.Replicates)

# Define IE as the indirect effect from the Bootstrap.Replicates object. 
IE <- Bootstrap.Replicates$Indirect.Effect

# Summary statistics
mean(IE)
median(IE)
sqrt(var(IE))

# CIs from percentile perspective
quantile(IE, probs=c(.025, .975))

# Two-sided p-value. 
## First, calculate obseved value of the indirect effect and extract it here. 
IE.Observed <- mediation(x=Data[,1], mediator=Data[,2], dv=Data[,3], 
conf.level=.95)$Effect.Sizes[1,]

## Now, find those values of the bootstrap indirect effects that are more extreme (in an absolute 
## sense) than the indirect effect observed. Note that the p-value is 1 here because the observed
## indirect effect is exactly 0. 
mean(abs(IE) >= abs(IE.Observed))

## End(Not run)

Bar plots of mediation effects

Description

Provides an effect bar plot in the context of simple mediation.

Usage

mediation.effect.bar.plot(x, mediator, dv, 
main = "Mediation Effect Bar Plot", width = 1, left.text.adj = 0, 
right.text.adj = 0, rounding = 3, file = "", save.pdf = FALSE, 
save.eps = FALSE, save.jpg = FALSE, ...)

Arguments

Details

Provides an effect bar for mediation (Bauer, Preacher, & Gil, 2006) may be used to plot the results of a mediation analysis compactly. Effect bars represent, in a single metric, the relative magnitudes of several values that are important for interpreting indirect effects. Preacher and Kelley (2011) discuss this plotting method also.

Value

Only a figure is returned

Author(s)

Ken Kelley (University of Notre Dame; KKelley@nd.edu)

References

Bauer, D. J., Preacher, K. J., & Gil, K. M. (2006). Conceptualizing and testing random indirect effects and moderated mediation in multilevel models: New procedures and recommendations. Psychological Methods, 11, 142–163.

Preacher, K. J., & Kelley, K. (2011). Effect size measures for mediation models: Quantitative and graphical strategies for communicating indirect effects. Psychological Methods, 16, 93–115.

See Also

mediation, mediation.effect.bar.plot

Visualizing mediation effects

Description

Create a mediation effect plot

Usage

mediation.effect.plot(x, mediator, dv, ylab = "Dependent Variable", 
xlab = "Mediator", main = "Mediation Effect Plot", 
pct.from.top.a = 0.05, pct.from.left.c = 0.05, arrow.length.a = 0.05, 
arrow.length.c = 0.05, legend.loc = "topleft", file = "", pch = 20, 
xlim = NULL, ylim = NULL, save.pdf = FALSE, save.eps = FALSE, 
save.jpg = FALSE, ...)

Arguments

Details

Merrill (1994; see also MacKinnon, 2008; MacKinnon et al., 2007; Sy, 2004) presents a method that involves plotting the indirect effect as the vertical distance between two lines. Fritz and MacKinnon (2008) present a detailed exposition of this method too. Preacher and Kelley (2011) discuss this plotting method and implement their own code, which was also independently done as part of Fritz and MacKinnon (2008).

In this type of plot, the two horizontal lines correspond to the predicted values of Y regressed on X at the mean of X and at one unit above the mean of X. The distance between these two lines is thus \hat{c}. The two vertical lines correspond to predicted values of M regressed on X at the same two values of X. The distance between these lines is \hat{a}. The lines corresponding to the regression of Y on M (controlling for X) are plotted for the same two values of X.

Value

A figure is returned.

Note

Requires raw data.

Author(s)

Ken Kelley (University of Notre Dame; KKelley@nd.edu)

References

Fritz, M. S., & MacKinnon, D. P. (2008). A graphical representation of the mediated effect. Behavior Research Methods, 40, 55–60.

MacKinnon, D. P. (2008). Introduction to statistical mediation analysis. Mahwah, NJ: Erlbaum.

MacKinnon, D. P., Fairchild, A. J., & Fritz, M. S. (2007). Mediation analysis. Annual Review of Psychology, 58, 593–614.

Merrill, R. M. (1994). Treatment effect evaluation in non-additive mediation models. Unpublished dissertation, Arizona State University.

Preacher, K. J., & Kelley, K. (2011). Effect size measures for mediation models: Quantitative and graphical strategies for communicating indirect effects. Psychological Methods, 16, 93–115.

Sy, O. S. (2004). Multilevel mediation analysis: Estimation and applications. Unpublished dissertation, Kansas State University.

See Also

mediation.effect.plot, mediation.effect.bar.plot

Minimum risk point estimation of the population coefficient of variation

Description

A function for the sequential estimation of the coefficient of variations with minimum risk. The function implements the ideas of Chattopadhyay and Kelley (in press), which considers study cost and accuracy of the estimated coefficient of variation simultaneously.

Usage

mr.cv(data, A, structural.cost, epsilon, sampling.cost, pilot=FALSE, m0=4, gamma=.49, 
verbose=FALSE)

Arguments

Details

The value of epsilon is context specific; the smaller the value the closer the estimated value will tend to be to the population value.

Value

Note

When a study's aim is to estimate a parameter accurately, such as the coefficient of variation, the structural costs and the maximum probable error of the estimate (i.e., \epsilon) are combined to form A. When we say “what the researcher is willing to pay," we literally mean the structural cost (c) the researcher is willing to invest in a study in order to estimate the parameter of interest with the desired degree of accuracy. This value is implicitly included (along with anticipated sampling cost) in grant applications for empirical studies when a certain amount of money is requested to conduct a study. If a researcher is willing to pay more and/or desire a smaller value of \epsilon, A is larger than it would have been. A larger A value will translate into a more expensive study, holding everything else constant. Notice that A is a fixed value in any investigation, as the researcher specifies A directly or by specifying its two components (structural cost and \epsilon) individually. However, what is not fixed but rather evaluated in multiple steps throughout the process is the sampling cost, as it is unknown the necessary sample size in order to accomplish the study's goal of achieving a sufficiently accurate estimate of the coefficient of variation. This is the core of our contributions: minimizing sampling cost, and thereby study cost, by using a sequential procedure that evaluates a stopping rule using the risk function to determine if the optimation criterion has been satisfied (based on the goals of the researcher and current information available). This function implements the ideas of sampling error and the study costs are considered simultaneously, so that the cost is not higher than necessary for the tolerable sampling error.

Author(s)

Ken Kelley (University of Notre Dame; kkelley@nd.edu) and Bhargab Chattopadhyay (University of Texas - Dallas; bhargab@utdallas.edu)

References

Chattopadhyay, B., & Kelley, K. (in press). Estimation of the Coefficient of Variation with Minimum Risk: A Sequential Method for Minimizing Sampling Error and Study Cost. Multivariate Behavioral Research, X, X–X.

Kelley, K. (2007). Sample size planning for the coefficient of variation from the accuracy in parameter estimation approach. Behavior Research Methods, 39 4, 755–766.

See Also

ci.cv, cv, mr.smd

Examples

# Determine pilot sample size:
mr.cv(pilot=TRUE, A=400000, sampling.cost=75, gamma=.49)

# Collect data (the size of which is the pilot sample size)
Data <- c(36, 53, 19, 11, 10, 24, 14, 65, 18, 48, 25, 35, 13, 18, 3, 41, 5, 3)

# Use mr.cv() to assess if the criterion for stopping the sequential study has been satisfied:
mr.cv(data=Data, A=400000, sampling.cost=75, gamma=.49)

# Collect another data (m=1 here) and perform another check:
Data <- c(Data, 44)
mr.cv(data=Data, A=400000, sampling.cost=75, gamma=.49)

# Continue adding obervations, checking each time if m=1, until the minimum risk criteria 
# are satisfied:
Data <- c(Data, 26, 13, 39, 2, 3, 26, 22, 8, 15, 12, 22, 5, 21, 23, 40, 18)
mr.cv(data=Data, A=400000, sampling.cost=75, gamma=.49)

Minimum risk point estimation of the population standardized mean difference

Description

A function for the sequential estimation of the standardized mean difference with minimum risk. The function implements the ideas of Chattopadhyay and Kelley (submitted, Psychological Methods), which considers study cost and accuracy of the estimated standardized mean difference simultaniously. This is important to specify that mr.smd.R was developed under the assumption of normally distributed data with equal sample size and equal cost of sampling per observation for each group.

Usage

mr.smd(A, structural.cost, epsilon, d, n, sampling.cost, pilot = FALSE, m0 = 4, 
gamma = 0.49)

Arguments

Details

The standardized mean difference is a widely used measure effect size. In this article, we developed a general theory for estimating the population standardized mean difference by minimizing both the mean square error of the estimator and the total sampling cost. This function implements our ideas discussed in Chattopadhyay and Kelley (submitted). See also Kelley and Rausch (2006) for additional information on the standardized mean difference.

Value

Note

When pilot=TRUE the function returns the size of the pilot sample size, per group, that should be used (thus, the total sample size is twice the pilot sample size).

Author(s)

Ken Kelley (University of Notre Dame; kkelley@nd.edu) and Bhargab Chattopadhyay (University of Texas - Dallas; bhargab@utdallas.edu)

References

Chattopadhyay, B., & Kelley, K. (submitted, minor revision requested). Estimating the standardized mean difference with minimum risk: Maximizing accuracy and minimizing cost with sequential estimation. Psychological Methods, X, X–X.

Chattopadhyay, B., & Kelley, K. (in press). Estimation of the Coefficient of Variation with Minimum Risk: A Sequential Method for Minimizing Sampling Error and Study Cost. Multivariate Behavioral Research, X, X–X.

Kelley, K., & Rausch, J. R. (2006). Sample size planning for the standardized mean difference: Accuracy in Parameter Estimation via narrow confidence intervals. Psychological Methods, 11, 363–385.

See Also

ci.smd, mr.cv

Examples

# To obtain pilot sample size in a situation in which A=10000. Note that 'A' is 
# 'structural.cost' divided by the square of 'epsilon'.

# From Chattopadhyay and Kelley (submitted, minor revision requested)
mr.smd(pilot=TRUE, A=10000, sampling.cost=2.4, gamma=.49)

High.SLS <- c(11, 7, 22, 13, 6, 9, 11, 16, 12, 17, 14, 8, 16)
Low.SLS  <- c(3, 6, 10, 8, 14, 5, 12, 10, 6, 8, 13, 5, 9)

mr.smd(d=1.021484, n=13, A=10000, sampling.cost=2.40, gamma=.49)

# Or, using the smd() function:
mr.smd(d=smd(Group.1=High.SLS, Group.2=Low.SLS), n=13, A=10000, sampling.cost=2.40, gamma=.49)

# Here, for this situation, the stopping rule is satisfied:
mr.smd(d=1.00, n=75, A=10000, sampling.cost=2.40, gamma=.49)

Density for power of two one-sided tests procedure (TOST) for equivalence

Description

A function to calculate density for the power of the two one-sided tests prodedure (TOST). (See package equivalence, function tost.)

Usage

power.density.equivalence.md(power_sigma, alpha = alpha, theta1 = theta1, 
theta2 = theta2, diff = diff, sigma = sigma, n = n, nu = nu)

Arguments

Value

Author(s)

Kem Phillips; kemphillips@comcast.net

References

Diletti, E., Hauschke D. & Steinijans, V.W. (1991). Sample size determination of bioequivalence assessment by means of confidence intervals, International Journal of Clinical Pharmacology, Therapy and Toxicology, 29, No. 1, 1–8.

Phillips, K.F. (1990). Power of the Two One-Sided Tests Procedure in Bioquivalence, Journal of Pharmacokinetics and Biopharmaceutics, 18, No. 2, 139–144.

Schuirmann, D.J. (1987). A comparison of the two one-sided tests procedure and the power approach for assessing the equivalence of average bioavailability, Journal of Pharmacokinetics and Biopharmaceutics, 15. 657–680.

See Also

power.equivalence.md.plot, power.density.equivalence.md

Examples

## Not run: 
# This function is called by power.equivalence.md within 
# the integrate function. It is integrated over 
# appropriate limits to compute the power. Use

power.density.equivalence.md(.1, alpha=.05, theta1=-.2, theta2=.2, diff=.05, 
	sigma= .20, n=24, nu=22)

# The usage for the logarithmic scale is the same, except that 
# theta1, theta2, and diff must be on that scale. That is, use log(.8), etc.

## End(Not run)

Power of Two One-Sided Tests Procedure (TOST) for Equivalence

Description

A function to calculate the power of the two one-sided tests prodedure (TOST). This is the probability that a confidence interval lies within a specified equivalence interval. (See also package equivalence, function tost.)

Usage

power.equivalence.md(alpha, logscale, ltheta1, ltheta2, ldiff, sigma, n, nu)

Arguments

Value

Author(s)

Kem Phillips; kemphillips@comcast.net

References

Diletti, E., Hauschke D. & Steinijans, V.W. (1991). Sample size determination of bioequivalence assessment by means of confidence intervals, International Journal of Clinical Pharmacology, Therapy and Toxicology, 29, No. 1, 1–8.

Phillips, K.F. (1990). Power of the Two One-Sided Tests Procedure in Bioquivalence, Journal of Pharmacokinetics and Biopharmaceutics, 18, No. 2, 139–144.

Schuirmann, D.J. (1987). A comparison of the two one-sided tests procedure and the power approach for assessing the equivalence of average bioavailability, Journal of Pharmacokinetics and Biopharmaceutics, 15. 657–680.

Examples

 # Suppose that two formulations of a drug are to be compared on 
 # the regular scale using a two-period crossover design, with 
 # theta1 = -0.20, theta2 = 0.20, rm{CV} = 0.20, the 
 # difference in the mean bioavailability is 0.05 (5 percent), and we choose 
 # n=24, corresponding to 22 degrees of freedom.  We need to test 
 # bioequivalence at the 5 percent significance level, which corresponds to 
 # having a 90 percent confidence interval lying within (-0.20, 0.20). Then 
 # the power will be 0.8029678.  This corresponds to Phillips (1990), 
 # Table 1, 5th row, 5th column, and Figure 3.  Use
 
power.equivalence.md(.05, FALSE, -.2, .2, .05, .20, 24, 22)


# If the formulations are compared on the logarithmic scale with 
# theta1 = 0.80, theta2 = 1.25, n=18 (16 degrees of freedom), and 
# a ratio of test to reference of 1.05. Then the power will be 0.7922796.
# This corresponds to Diletti, Table 1, power=.80, CV=.20, ratio=1.05, and Figure 1c. Use

power.equivalence.md(.05, TRUE, .8, 1.25, 1.05, .20, 18, 16)

Plot power of Two One-Sided Tests Procedure (TOST) for Equivalence

Description

A function to plot the power of the two one-sided tests prodedure (TOST) for various alternatives. (See also package equivalence, function tost.)

Usage

power.equivalence.md.plot(alpha, logscale, theta1, theta2, sigma, n, nu, title2)

Arguments

Value

Author(s)

Kem Phillips; kemphillips@comcast.net

References

Diletti, E., Hauschke D. & Steinijans, V.W. (1991) Sample size determination of bioequivalence assessment by means of confidence intervals, International Journal of Clinical Pharmacology, Therapy and Toxicology, 29, No. 1, 1-8.

Phillips, K.F. (1990) Power of the Two One-Sided Tests Procedure in Bioquivalence, Journal of Pharmacokinetics and Biopharmaceutics, 18, No. 2, 139-144.

Schuirmann, D.J. (1987) A comparison of the two one-sided tests procedure and the power approach for assessing the equivalence of average bioavailability, Journal of Pharmacokinetics and Biopharmaceutics, 15. 657-680.

Examples

## Not run: 
# Suppose that two formulations of a drug are to be compared 
# on the regular scale using a two-period crossover design, 
# with theta1 = -0.20, theta2 = 0.20, rm(CV) = 0.20, and 
# we choose 
n<-c(9,12,18,24,30,40,60)

# corresponding to 
nu<-c(7,10,16,22,28,38,58)

# degrees of freedom.  We need to test bioequivalence at the 
# .05 significance level, which corresponds to having a .90 confidence
# interval lying within (-0.20, 0.20). This corresponds to 
# Phillips (1990),  Figure 3.  Use

power.equivalence.md.plot(.05, FALSE, -.2, .2, .20, n, nu, 'Phillips Figure 3')

# If the formulations are compared on the logarithmic scale with 
# theta1 = 0.80, theta2 = 1.25, and 

n<-c(8,12,18,24,30,40,60)

# corresponding  to 
nu<-c(6,10,16,22,28,38,58)

# degrees of freedom. This corresponds to Diletti, Figure 1c. Use

power.equivalence.md.plot(.05, TRUE, .8, 1.25, .20, n, nu, 'Diletti, Figure 1c')

## End(Not run)

Cohen et. al. (2003)'s professor salary data set

Description

The data set of the salaries and other information of 62 some professors in Cohen et. al. (2003, pp. 81-82).

Usage

data(prof.salary)

Format

A data frame with 62 observations on the following 6 variables.

id

the identification number

time

the time since getting the Ph.D. degree

pub

the number of publications

sex

the gender, 1 for female and 0 for male

citation

the citation count

salary

the professor's current salary

Source

Cohen, J., Cohen, P., West, S. G., & Aiken, L. S. (2003). Applied multiple regression/correlation analysis for the behavioral sciences (3rd ed.). Mahwah, NJ: Erlbaum.

References

Cohen, J., Cohen, P., West, S. G., & Aiken, L. S. (2003). Applied multiple regression/correlation analysis for the behavioral sciences (3rd ed.). Mahwah, NJ: Erlbaum.

Examples

data(prof.salary)

Unbiased estimate of the population standard deviation

Description

Transforms the usual (and biased) estimate of the standard deviation into an unbiased estimator.

Usage

s.u(s=NULL, N=NULL, X=NULL)

Arguments

Details

Returns the unbiased estimate for the standard deviation.

Value

The unbiased estimate for the standard deviation.

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

References

Holtzman, W. H. (1950). The unbiased estimate of the population variance and standard deviation. American Journal of Psychology, 63, 615–617.

Examples

set.seed(113)
X <- rnorm(10, 100, 15)

# Square root of the unbiased estimate of the variance (not unbiased)
var(X)^.5

# One way to implement the function.
s.u(s=var(X)^.5, N=length(X))

# Another way to implement the function.
s.u(X=X)

Signal to noise using squared multiple correlation coefficient

Description

Function that calculates five different signal-to-noise ratios using the squared multiple correlation coefficient.

Usage

signal.to.noise.R2(R.Square, N, p)

Arguments

Details

The method of choice is phi2.UMVUE.NL, but it requires p of 5 or more. In situations where p < 5, it is suggested that phi2.UMVUE.L be used.

Value

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

References

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum.

Muirhead, R. J. (1985). Estimating a particular function of the multiple correlation coefficient. Journal of the American Statistical Association, 80, 923–925.

See Also

ci.R2, ss.aipe.R2

Examples

signal.to.noise.R2(R.Square=.5, N=50, p=2)
signal.to.noise.R2(R.Square=.5, N=50, p=5)
signal.to.noise.R2(R.Square=.5, N=100, p=2)
signal.to.noise.R2(R.Square=.5, N=100, p=5)

Standardized mean difference

Description

Function to calculate the standardized mean difference (regular or unbiased) using either raw data or summary measures.

Usage

smd(Group.1 = NULL, Group.2 = NULL, Mean.1 = NULL, Mean.2 = NULL, 
s.1 = NULL, s.2 = NULL, s = NULL, n.1 = NULL, n.2 = NULL,
Unbiased=FALSE)

Arguments

Details

When Unbiased=TRUE, the unbiased estimate of the standardized mean difference is returned (Hedges, 1981).

Value

Returns the estimated standardized mean difference.

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

References

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum.

Cumming, G. & Finch, S. (2001). A primer on the understanding, use, and calculation of confidence intervals that are based on central and noncentral distributions, Educational and Psychological Measurement, 61, 532–574.

Hedges, L. V. (1981). Distribution theory for Glass's Estimator of effect size and related estimators. Journal of Educational Statistics, 2, 107–128.

Kelley, K. (2005) The effects of nonnormal distributions on confidence intervals around the standardized mean difference: Bootstrap and parametric confidence intervals, Educational and Psychological Measurement, 65, 51–69.

Kelley, K. (2007). Constructing confidence intervals for standardized effect sizes: Theory, application, and implementation. Journal of Statistical Software, 20 (8), 1–24.

Steiger, J. H., & Fouladi, R. T. (1997). Noncentrality interval estimation and the evaluation of statistical methods. In L. L. Harlow, S. A. Mulaik, & J. H. Steiger (Eds.), What if there were no significance tests? (pp. 221–257). Mahwah, NJ: Lawrence Erlbaum.

See Also

smd.c, conf.limits.nct, ss.aipe

Examples

# Generate sample data.
set.seed(113)
g.1 <- rnorm(n=25, mean=.5, sd=1)
g.2 <- rnorm(n=25, mean=0, sd=1)
smd(Group.1=g.1, Group.2=g.2)

M.x <- .66745
M.y <- .24878
sd <- 1.048
smd(Mean.1=M.x, Mean.2=M.y, s=sd)

M.x <- .66745
M.y <- .24878
n1 <- 25
n2 <- 25
sd.1 <- .95817
sd.2 <- 1.1311
smd(Mean.1=M.x, Mean.2=M.y, s.1=sd.1, s.2=sd.2, n.1=n1, n.2=n2)

smd(Mean.1=M.x, Mean.2=M.y, s.1=sd.1, s.2=sd.2, n.1=n1, n.2=n2, 
Unbiased=TRUE)

Standardized mean difference using the control group as the basis of standardization

Description

Function to calculate the standardized mean difference (regular or unbiased) using the control group standard deviation as the basis of standardization (for either raw data or summary measures).

Usage

smd.c(Group.T = NULL, Group.C = NULL, Mean.T = NULL, Mean.C = NULL, 
s.C = NULL, n.C = NULL, Unbiased=FALSE)

Arguments

Details

When Unbiased=TRUE, the unbiased estimate of the standardized mean difference (using the control group as the basis of standardization) is returned (Hedges, 1981). Although the unbiased estimate of the standardized mean difference is not often reported, at least at the present time, it is nevertheless made available to those who are interested in calculating this quantity.

Value

Returns the estimated standardized mean difference using the control group standard deviation as the basis of standardization.

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

References

Hedges, L. V. (1981). Distribution theory for Glass's Estimator of effect size and related estimators. Journal of Educational Statistics, 2, 107–128.

Glass, G. (1976). Primary, secondary, and meta-analysis of research. Educational Researcher, 5, 3–8.

See Also

smd, conf.limits.nct

Examples

# Generate sample data.
set.seed(113)
g.T <- rnorm(n=25, mean=.5, sd=1)
g.C <- rnorm(n=25, mean=0, sd=1)
smd.c(Group.T=g.T, Group.C=g.C)

M.T <- .66745
M.C <- .24878
sd.c <- 1.1311
n.c <- 25
smd.c(Mean.T=M.T, Mean.C=M.C, s=sd.c)
smd.c(Mean.T=M.T, Mean.C=M.C, s=sd.c, n.C=n.c, Unbiased=TRUE)

Sample Size Planning for Accuracy in Parameter Estimation for the multiple correlation coefficient.

Description

Determines necessary sample size for the multiple correlation coefficient so that the confidence interval for the population multiple correlation coefficient is sufficiently narrow. Optionally, there is a certainty parameter that allows one to be a specified percent certain that the observed interval will be no wider than desired.

Usage

ss.aipe.R2(Population.R2 = NULL, conf.level = 0.95, width = NULL, 
Random.Predictors = TRUE, Random.Regressors, which.width = "Full", p = NULL, 
K, degree.of.certainty = NULL, assurance=NULL, certainty=NULL, 
verify.ss = FALSE, Tol = 1e-09, ...)

Arguments

Details

This function determines a necessary sample size so that the expected confidence interval width for the squared multiple correlation coefficient is sufficiently narrow (when degree.of.certainty=NULL) so that the obtained confidence interval is no larger than the value specified with some desired degree of certainty (i.e., a probability that the obtained width is less than the specified width). The method depends on whether or not the regressors are regarded as fixed or random. This is the case because the distribution theory for the two cases is different and thus the confidence interval procedure is conditional on the type of regressors. The default methods are approximate but can be made exact with the specification of verify.ss=TRUE, which performs an a priori Monte Carlo simulation study. Kelley (2007) and Kelley & Maxwell (2008) detail the methods used in the function, with the former focusing on random regressors and the latter on fixed regressors.

It is recommended that the option verify.ss should always be used! Doing so uses the method implied sample size as an estimate and then evaluates with an internal Monte Carlo simulation (i.e., via "brute-force" methods) the exact sample size given the goals specified. When verify.ss=TRUE, the default number of iterations is 10,000 but this can be changed by specifying G=5000 (or some other value; 10000 is the recommended) When verify.ss=TRUE is specified, an internal function verify.ss.aipe.R2 calls upon the ss.aipe.R2.sensitivity function for purposes of the internal Monte Carlo simulation study. See the verify.ss.aipe.R2 function for arguments that can be passed from ss.aipe.R2 to verify.ss.aipe.R2.

Value

Note

This function without verify.SS=FALSE can be slow to converge when verify.SS=TRUE, the function can take some time to converge (e.g., 15 minutes). Most times this will not be the case, but it is possible in some situations.

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

References

Algina, J. & Olejnik, S. (2000). Determining sample size for accurate estimation of the squared multiple correlation coefficient. Multivariate Behavioral Research, 35, 119–136.

Steiger, J. H. & Fouladi, R. T. (1992). R2: A computer program for interval estimation, power calculation, and hypothesis testing for the squared multiple correlation. Behavior research methods, instruments and computers, 4, 581–582.

Kelley, K. (2007). Sample size planning for the squared multiple correlation coefficient: Accuracy in parameter estimation via narrow confidence intervals, manuscripted submitted for publication.

Kelley, K. & Maxwell, S. E. (2008). Power and accuracy for omnibus and targeted effects: Issues of sample size planning with applications to multiple regression. In P. Alasuuta, J. Brannen, & L. Bickman (Eds.), Handbook of Social Research Methods (pp. 166–192). Newbury Park, CA: Sage.

See Also

ci.R2, conf.limits.nct, ss.aipe.R2.sensitivity

Examples

## Not run: 
# Returned sample size should be considered approximate; exact sample
# size is obtained by specifying the argument 'verify.ss=TRUE' (see below).
# ss.aipe.R2(Population.R2=.50, conf.level=.95, width=.10, which.width="Full",
# p=5, Random.Predictors=TRUE)
# Uncomment to run in order to get exact sample size.
# ss.aipe.R2(Population.R2=.50, conf.level=.95, width=.10, which.width="Full",
# p=5, Random.Predictors=TRUE, verify.ss=TRUE)


# Same as above, except the predictor variables are considered fixed.
# Returned sample size should be considered approximate; exact sample
# size is obtained by specifying the argument 'verify.ss=TRUE'.
# ss.aipe.R2(Population.R2=.50, conf.level=.95, width=.10, which.width="Full",
# p=5, Random.Predictors=FALSE)
# Uncomment to run in order to get exact sample size.
#ss.aipe.R2(Population.R2=.50, conf.level=.95, width=.10, which.width="Full",
#p=5, Random.Predictors=FALSE, verify.ss=TRUE)


# Returned sample size should be considered approximate; exact sample
# size is obtained by specifying the argument 'verify.ss=TRUE'.
# ss.aipe.R2(Population.R2=.50, conf.level=.95, width=.10, which.width="Full",
# p=5, degree.of.certainty=.85, Random.Predictors=TRUE)
# Uncomment to run in order to get exact sample size.
#ss.aipe.R2(Population.R2=.50, conf.level=.95, width=.10, which.width="Full",
#p=5, degree.of.certainty=.85, Random.Predictors=TRUE, verify.ss=TRUE)


# Same as above, except the predictor variables are considered fixed.
# Returned sample size should be considered approximate; exact sample
# size is obtained by specifying the argument 'verify.ss=TRUE'.
# ss.aipe.R2(Population.R2=.50, conf.level=.95, width=.10, which.width="Full",
# p=5, degree.of.certainty=.85, Random.Predictors=FALSE)
# Uncomment to run in order to get exact sample size.
#ss.aipe.R2(Population.R2=.50, conf.level=.95, width=.10, which.width="Full",
#p=5, degree.of.certainty=.85, Random.Predictors=FALSE, verify.ss=TRUE)

## End(Not run)

Sensitivity analysis for sample size planning with the goal of Accuracy in Parameter Estimation (i.e., a narrow observed confidence interval)

Description

Given Estimated.R2 and True.R2, one can perform a sensitivity analysis to determine the effect of a misspecified population squared multiple correlation coefficient using the Accuracy in Parameter Estimation (AIPE) approach to sample size planning. The function evaluates the effect of a misspecified True.R2 on the width of obtained confidence intervals.

Usage

ss.aipe.R2.sensitivity(True.R2 = NULL, Estimated.R2 = NULL, w = NULL, 
p = NULL, Random.Predictors=TRUE, Selected.N=NULL, 
degree.of.certainty = NULL, assurance=NULL, certainty=NULL, 
conf.level = 0.95, Generate.Random.Predictors=TRUE, rho.yx = 0.3, 
rho.xx = 0.3,  G = 10000, print.iter = TRUE, ...)

Arguments

Details

When Estimated.R2=True.R2, the results are that of a simulation study when all assumptions are satisfied. Rather than specifying Estimated.R2, one can specify Selected.N to determine the results of a particular sample size (when doing this Estimated.R2 cannot be specified).

The sample size estimation procedure technically assumes multivariate normal variables (p+1) with fixed predictors (x/indepdent variables), yet the function assumes random multivariate normal predictors (having a p+1 multivariate distribution). As Gatsonis and Sampson (1989) note in the context of statistical power analysis (recall this function is used in the context of precision), there is little difference in the outcome.

In the behavioral, educational, and social sciences, predictor variables are almost always random, and thus Random.Predictors should generally be used. Random.Predictors=TRUE specifies how both the sample size planning procedure and the confidence intervals are calculated based on the random predictors/regressors. The internal simulation generates random or fixed predictors/regressors based on whether variables predictor variables are random or fixed. However, when Random.Predictors=FALSE, only the sample size planning procedure and the confidence intervals are calculated based on the parameter. The parameter Generate.Random.Predictors (where the default is TRUE so that random predictors/regressors are generated) allows random or fixed predictor variables to be generated. Because the sample size planning procedure and the internal simulation are both specified, for purposes of sensitivity analysis random/fixed can be crossed to examine the effects of specifying sample size based on one but using it on data based on the other.

Value

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

References

Algina, J. & Olejnik, S. (2000). Determining Sample Size for Accurate Estimation of the Squared Multiple Correlation Coefficient. Multivariate Behavioral Research, 35, 119–136.

Gatsonis, C. & Sampson, A. R. (1989). Multiple Correlation: Exact power and sample size calculations. Psychological Bulletin, 106, 516–524.

Steiger, J. H. & Fouladi, R. T. (1992). R2: A computer program for interval estimation, power calculation, and hypothesis testing for the squared multiple correlation. Behavior research methods, instruments and computers, 4, 581–582.

Kelley, K. (2008). Sample size planning for the squared multiple correlation coefficient: Accuracy in parameter estimation via narrow confidence intervals, Multivariate Behavioral Research, 43, 524–555.

Kelley, K. & Maxwell, S. E. (2008). Sample Size Planning with applications to multiple regression: Power and accuracy for omnibus and targeted effects. In P. Alasuuta, J. Brannen, & L. Bickman (Eds.), The Sage handbook of social research methods (pp. 166–192). Newbury Park, CA: Sage.

See Also

ci.R2, conf.limits.nct, ss.aipe.R2

Examples

## Not run: 
# Change 'G' to some large number (e.g., G=10,000)
# ss.aipe.R2.sensitivity(True.R2=.5, Estimated.R2=.4, w=.10, p=5, conf.level=0.95,
# G=25)

## End(Not run)

Sample size planning for an ANOVA contrast from the Accuracy in Parameter Estimation (AIPE) perspective

Description

A function to calculate the appropriate sample size per group for the (unstandardized) ANOVA contrast so that the width of the confidence interval is sufficiently narrow.

Usage

ss.aipe.c(error.variance = NULL, c.weights, width, conf.level = 0.95, 
assurance = NULL, certainty = NULL, MSwithin = NULL, SD = NULL, ...)

Arguments

Value

Note

Be sure to use the error varaince and not its square root (i.e., the standard deviation of the errors).

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu), Keke Lai

References

Kelley, K., Maxwell, S. E., & Rausch, J. R. (2003). Obtaining power or obtaining precesion: Delineating methods of sample size planning. Evaluation and the Health Professions, 26, 258–287.

Maxwell, S. E., & Delaney, H. D. (2004). Designing experiments and analyzing data: A model comparison perspective. Mahwah, NJ: Erlbaum.

See Also

ss.aipe.sc, ss.aipe.c.ancova, ci.c

Examples

# Suppose the population error variance of some three-group ANOVA model
# is believed to be 40. The researcher is interested in the difference 
# between the mean of group 1 and the average of means of group 2 and 3. 
# To plan the sample size so that, with 90 percent certainty, the 
# obtained 95 percent full confidence interval width is no wider than 3:

ss.aipe.c(error.variance=40, c.weights=c(1, -0.5, -0.5), width=3, assurance=.90)

Sample size planning for a contrast in randomized ANCOVA from the Accuracy in Parameter Estimation (AIPE) perspective

Description

A function to calculate the appropriate sample size per group for the (unstandardized) contrast, in one-covariate randomized ANCOVA, so that the width of the confidence interval is sufficiently narrow.

Usage

ss.aipe.c.ancova(error.var.ancova = NULL, error.var.anova = NULL, 
rho = NULL, c.weights, width, conf.level = 0.95, 
assurance = NULL, certainty = NULL)

Arguments

Details

Either the error variance of the ANCOVA model or of the ANOVA model can be used to plan the appropriate sample size per group. When using the error variance of the ANOVA model to plan sample size, the correlation coefficient of the response and the covariate is also needed.

Value

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu); Keke Lai <Lai.15@ND.Edu>

References

Kelley, K., Maxwell, S. E., & Rausch, J. R. (2003). Obtaining power or obtaining precision: Delineating methods of sample size planning. Evaluation and the Health Professions, 26, 258-287.

Maxwell, S. E., & Delaney, H. D. (2004). Designing experiments and analyzing data: A model comparison perspective. Mahwah, NJ: Erlbaum.

See Also

ci.c.ancova, ci.sc.ancova, ss.aipe.c

Examples

# Suppose the population error variance of some three-group ANOVA model
# is believed to be 40, and the population correlation coefficient 
# of the response and the covariate is 0.22. The researcher is 
# interested in the difference between the mean of group 1 and 
# the average of means of group 2 and 3. To plan the sample size so 
# that, with 90 percent certainty, the obtained 95 percent full 
# confidence interval width is no wider than 3:

ss.aipe.c.ancova(error.var.anova=40, rho=.22, 
c.weights=c(1, -0.5, -0.5), width=3, assurance=.90)

Sensitivity analysis for sample size planning for the (unstandardized) contrast in randomized ANCOVA from the Accuracy in Parameter Estimation (AIPE) Perspective

Description

Performs a sensitivity analysis when planning sample size from the Accuracy in Parameter Estimation (AIPE) Perspective for the (unstandardized) contrast in randomized ANCOVA design.

Usage

ss.aipe.c.ancova.sensitivity(true.error.var.ancova = NULL, 
est.error.var.ancova = NULL, true.error.var.anova = NULL, 
est.error.var.anova = NULL, rho, est.rho = NULL, G = 10000, 
mu.y, sigma.y, mu.x, sigma.x, c.weights, width,
conf.level = 0.95, assurance = NULL, certainty=NULL)

Arguments

Details

The arguments mu.y, mu.x, sigma.y, and sigma.x are used to generate random data in the simulations for the sensitivity analysis. The value of mu.y should be the same as the square root of true.error.var.anova

So far this function is based on one-covariate randomized ANCOVA design only. The argument mu.x should be a single number, because it is assumed that the population mean of the covariate is equal across groups in randomized ANCOVA.

Value

Author(s)

Keke Lai (University of Notre Dame; Lai.15@ND.Edu)

Examples

## Not run: 
ss.aipe.c.ancova.sensitivity(true.error.var.ancova=30, 
est.error.var.ancova=30, rho=.2, mu.y=c(10,12,15,13), mu.x=2, 
G=1000, sigma.x=1.3, sigma.y=2, c.weights=c(1,0,-1,0), width=3)

ss.aipe.c.ancova.sensitivity(true.error.var.anova=36, 
est.error.var.anova=36, rho=.2, est.rho=.2, G=1000, 
mu.y=c(10,12,15,13), mu.x=2, sigma.x=1.3, sigma.y=6, 
c.weights=c(1,0,-1,0), width=3, assurance=NULL)

## End(Not run)

Find target sample sizes for the accuracy in unstandardized conditions means estimation in CRD

Description

Find target sample sizes (the number of clusters, cluster size, or both) for the accuracy in unstandardized conditions means estimation in CRD. If users wish to seek for both types of sample sizes simultaneously, an additional constraint is required, such as a desired width or a desired budget.

Usage

ss.aipe.crd.nclus.fixedwidth(width, nindiv, prtreat, tauy=NULL, sigma2y=NULL, 
	totalvar=NULL, iccy=NULL, r2between = 0, r2within = 0, numpredictor = 0, 
	assurance=NULL, conf.level = 0.95, cluscost=NULL, indivcost=NULL, diffsize=NULL)
ss.aipe.crd.nindiv.fixedwidth(width, nclus, prtreat, tauy=NULL, sigma2y=NULL, 
	totalvar=NULL, iccy=NULL, r2between = 0, r2within = 0, numpredictor = 0, 
	assurance=NULL, conf.level = 0.95, cluscost=NULL, indivcost=NULL, diffsize=NULL)
ss.aipe.crd.nclus.fixedbudget(budget, nindiv, cluscost = 0, indivcost = 1, 
	prtreat = NULL, tauy=NULL, sigma2y=NULL, totalvar=NULL, iccy=NULL, r2between = 0, 
	r2within = 0, numpredictor = 0, assurance=NULL, conf.level = 0.95, diffsize=NULL)
ss.aipe.crd.nindiv.fixedbudget(budget, nclus, cluscost = 0, indivcost = 1, 
	prtreat = NULL, tauy=NULL, sigma2y=NULL, totalvar=NULL, iccy=NULL, r2between = 0, 
	r2within = 0, numpredictor = 0, assurance=NULL, conf.level = 0.95, diffsize=NULL)
ss.aipe.crd.both.fixedbudget(budget, cluscost=0, indivcost=1, prtreat, tauy=NULL, 
	sigma2y=NULL, totalvar=NULL, iccy=NULL, r2between = 0, r2within = 0, 
	numpredictor = 0, assurance=NULL, conf.level = 0.95, diffsize=NULL)
ss.aipe.crd.both.fixedwidth(width, cluscost=0, indivcost=1, prtreat, tauy=NULL, 
	sigma2y=NULL, totalvar=NULL, iccy=NULL, r2between = 0, r2within = 0, 
	numpredictor = 0, assurance=NULL, conf.level = 0.95, diffsize=NULL)

Arguments

Details

Here are the functions' descriptions:

• ss.aipe.crd.nclus.fixedwidth Find the number of clusters given a specified width of the confidence interval and the cluster size

• ss.aipe.crd.nindiv.fixedwidth Find the cluster size given a specified width of the confidence interval and the number of clusters

• ss.aipe.crd.nclus.fixedbudget Find the number of clusters given a budget and the cluster size

• ss.aipe.crd.nindiv.fixedbudget Find the cluster size given a budget and the number of clusters

• ss.aipe.crd.both.fixedbudget Find the sample size combinations (the number of clusters and that cluster size) providing the narrowest confidence interval given the fixed budget

• ss.aipe.crd.both.fixedwidth Find the sample size combinations (the number of clusters and that cluster size) providing the lowest cost given the specified width of the confidence interval

Value

The ss.aipe.crd.nclus.fixedwidth and ss.aipe.crd.nclus.fixedbudget functions provide the number of clusters. The ss.aipe.crd.nindiv.fixedwidth and ss.aipe.crd.nindiv.fixedbudget functions provide the cluster size. The ss.aipe.crd.both.fixedbudget and ss.aipe.crd.both.fixedwidth provide the number of clusters and the cluster size, respectively.

Author(s)

Sunthud Pornprasertmanit (psunthud@gmail.com)

References

Pornprasertmanic, S., & Schneider, W. J. (2014). Accuracy in parameter estimation in cluster randomized designs. Psychological Methods, 19, 356–379.

Examples

## Not run: 
# Examples for each function
ss.aipe.crd.nclus.fixedwidth(width=0.3, nindiv=30, prtreat=0.5, tauy=0.25, sigma2y=0.75)

ss.aipe.crd.nindiv.fixedwidth(width=0.3, nclus=250, prtreat=0.5, tauy=0.25, sigma2y=0.75)

ss.aipe.crd.nclus.fixedbudget(budget=10000, nindiv=20, cluscost=20, indivcost=1)

ss.aipe.crd.nindiv.fixedbudget(budget=10000, nclus=30, cluscost=20, indivcost=1, 
prtreat=0.5, tauy=0.05, sigma2y=0.95, assurance=0.8)

ss.aipe.crd.both.fixedbudget(budget=10000, cluscost=30, indivcost=1, prtreat=0.5, tauy=0.25, 
	sigma2y=0.75)

ss.aipe.crd.both.fixedwidth(width=0.3, cluscost=0, indivcost=1, prtreat=0.5, tauy=0.25, 
	sigma2y=0.75)

# Examples for different cluster size
ss.aipe.crd.nclus.fixedwidth(width=0.3, nindiv=30, prtreat=0.5, tauy=0.25, sigma2y=0.75, 
diffsize = c(-2, 1, 0, 2, -1, 3, -3, 0, 0))

ss.aipe.crd.nclus.fixedwidth(width=0.3, nindiv=30, prtreat=0.5, tauy=0.25, sigma2y=0.75, 
diffsize = c(0.6, 1.2, 0.8, 1.4, 1, 1, 1.1, 0.9))

## End(Not run)

Find target sample sizes for the accuracy in standardized conditions means estimation in CRD

Description

Find target sample sizes (the number of clusters, cluster size, or both) for the accuracy in standardized conditions means estimation in CRD. If users wish to seek for both types of sample sizes simultaneously, an additional constraint is required, such as a desired width or a desired budget. This function uses the likelihood-based confidence interval (Cheung, 2009) by the OpenMx package (Boker et al., 2011). See further details at Pornprasertmanit and Schneider (2010, submitted).

Usage

ss.aipe.crd.es.nclus.fixedwidth(width, nindiv, es, estype = 1, iccy, prtreat, 
	r2between = 0, r2within = 0, numpredictor = 0, assurance=NULL, 
	conf.level = 0.95, nrep = 1000, iccz = NULL, seed = 123321, multicore = FALSE, 
	numProc=NULL, cluscost=NULL, indivcost=NULL, diffsize=NULL)
ss.aipe.crd.es.nindiv.fixedwidth(width, nclus, es, estype = 1, iccy, prtreat, 
	r2between = 0, r2within = 0, numpredictor = 0, assurance=NULL, 
	conf.level = 0.95, nrep = 1000, iccz = NULL, seed = 123321, multicore = FALSE, 
	numProc=NULL, cluscost=NULL, indivcost=NULL, diffsize=NULL) 
ss.aipe.crd.es.nclus.fixedbudget(budget, nindiv, cluscost, indivcost, nrep=NULL, 
	prtreat=NULL, iccy=NULL, es=NULL, estype = 1, numpredictor = 0, 
	iccz=NULL, r2within=NULL, r2between=NULL, assurance=NULL, 
	seed=123321, multicore=FALSE, numProc=NULL, conf.level=0.95, diffsize=NULL)
ss.aipe.crd.es.nindiv.fixedbudget(budget, nclus, cluscost, indivcost, nrep=NULL, 
	prtreat=NULL, iccy=NULL, es=NULL, estype = 1, numpredictor = 0, 
	iccz=NULL, r2within=NULL, r2between=NULL, assurance=NULL, 
	seed=123321, multicore=FALSE, numProc=NULL, conf.level=0.95, diffsize=NULL) 
ss.aipe.crd.es.both.fixedbudget(budget, cluscost=0, indivcost=1, es, estype = 1, 
	iccy, prtreat, r2between = 0, r2within = 0, numpredictor = 0, assurance=NULL, 
	conf.level = 0.95, nrep = 1000, iccz = NULL, seed = 123321, multicore = FALSE, 
	numProc=NULL, diffsize=NULL)
ss.aipe.crd.es.both.fixedwidth(width, cluscost=0, indivcost=1, es, estype = 1, 
	iccy, prtreat, r2between = 0, r2within = 0, numpredictor = 0, assurance=NULL, 
	conf.level = 0.95, nrep = 1000, iccz = NULL, seed = 123321, multicore = FALSE, 
	numProc=NULL, diffsize=NULL)

Arguments

Details

Here are the functions' descriptions:

• ss.aipe.crd.es.nclus.fixedwidth Find the number of clusters given a specified width of the confidence interval and the cluster size

• ss.aipe.crd.es.nindiv.fixedwidth Find the cluster size given a specified width of the confidence interval and the number of clusters

• ss.aipe.crd.es.nclus.fixedbudget Find the number of clusters given a budget and the cluster size

• ss.aipe.crd.es.nindiv.fixedbudget Find the cluster size given a budget and the number of clusters

• ss.aipe.crd.es.both.fixedbudget Find the sample size combinations (the number of clusters and that cluster size) providing the narrowest confidence interval given the fixed budget

• ss.aipe.crd.es.both.fixedwidth Find the sample size combinations (the number of clusters and that cluster size) providing the lowest cost given the specified width of the confidence interval

Value

The ss.aipe.crd.es.nclus.fixedwidth and ss.aipe.crd.es.nclus.fixedbudget functions provide the number of clusters. The ss.aipe.crd.es.nindiv.fixedwidth and ss.aipe.crd.es.nindiv.fixedbudget functions provide the cluster size. The ss.aipe.crd.es.both.fixedbudget and ss.aipe.crd.es.both.fixedwidth provide the number of clusters and the cluster size, respectively.

Author(s)

Sunthud Pornprasertmanit (psunthud@gmail.com)

References

Boker, S., M., N., Maes, H., Wilde, M., Spiegel, M., Brick, T., et al. (2011). OpenMx: An open source extended structural equation modeling framework. Psychometrika, 76, 306-317.

Cheung, M. W.-L. (2009). Constructing approximate confidence intervals for parameters with structural constructing approximate confidence intervals for parameters with structural equation models. Structural Equation Modeling, 16, 267-294.

Pornprasertmanit, S., & Schneider, W. J. (2010). Efficient sample size for power and desired accuracy in Cohen's d estimation in two-group cluster randomized design (Master Thesis). Illinois State University, Normal, IL.

Pornprasertmanic, S., & Schneider, W. J. (2014). Accuracy in parameter estimation in cluster randomized designs. Psychological Methods, 19, 356–379.

Examples

## Not run: 
# Examples for each function
ss.aipe.crd.es.nclus.fixedwidth(width=0.3, nindiv=20, es=0.5, estype=1, iccy=0.25, prtreat=0.5, 
	nrep=20)

ss.aipe.crd.es.nindiv.fixedwidth(width=0.3, 250, es=0.5, estype=1, iccy=0.25, prtreat=0.5, 
	nrep=20)

ss.aipe.crd.es.nclus.fixedbudget(budget=1000, nindiv=20, cluscost=0, indivcost=1, nrep=20, 
	prtreat=0.5, iccy=0.25, es=0.5)

ss.aipe.crd.es.nindiv.fixedbudget(budget=1000, nclus=200, cluscost=0, indivcost=1, nrep=20, 
	prtreat=0.5, iccy=0.25, es=0.5)

ss.aipe.crd.es.both.fixedbudget(budget=1000, cluscost=5, indivcost=1, es=0.5, estype=1, 
	iccy=0.25, prtreat=0.5, nrep=20)

ss.aipe.crd.es.both.fixedwidth(width=0.5, cluscost=5, indivcost=1, es=0.5, estype=1, iccy=0.25, 
	prtreat=0.5, nrep=20)

# Examples for different cluster size
ss.aipe.crd.es.nclus.fixedwidth(width=0.3, nindiv=20, es=0.5, estype=1, iccy=0.25, prtreat=0.5, 
nrep=20, diffsize = c(-2, 1, 0, 2, -1, 3, -3, 0, 0))

ss.aipe.crd.es.nclus.fixedwidth(width=0.3, nindiv=20, es=0.5, estype=1, iccy=0.25, prtreat=0.5, 
nrep=20, diffsize = c(0.6, 1.2, 0.8, 1.4, 1, 1, 1.1, 0.9))

## End(Not run)

Sample size planning for the coefficient of variation given the goal of Accuracy in Parameter Estimation approach to sample size planning

Description

Determines the necessary sample size so that the expected confidence interval width for the coefficient of variation will be sufficiently narrow, optionally with a desired degree of certainty that the interval will not be wider than desired. The value of C.of.V should be positive.

Usage

ss.aipe.cv(C.of.V = NULL, width = NULL, conf.level = 0.95, 
degree.of.certainty = NULL, assurance=NULL, certainty=NULL, 
mu = NULL, sigma = NULL, alpha.lower = NULL, alpha.upper = NULL, 
Suppress.Statement = TRUE, sup.int.warns = TRUE, ...)

Arguments

Value

Returns the necessary sample size given the input specifications.

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

See Also

ss.aipe.cv.sensitivity, cv

Examples

# Suppose one wishes to have a confidence interval with an expected width of .10 
# for a 99% confidence interval when the population coefficient of variation is .25.
ss.aipe.cv(C.of.V=.1, width=.1, conf.level=.99)

# Ensuring that the confidence interval will be sufficiently narrow with a 99% 
# certainty for the situation above.
ss.aipe.cv(C.of.V=.1, width=.1, conf.level=.99, degree.of.certainty=.99)

Sensitivity analysis for sample size planning given the Accuracy in Parameter Estimation approach for the coefficient of variation.

Description

Performs sensitivity analysis for sample size determination for the coefficient of variation given a population coefficient of variation (or population mean and standard deviation) and goals for the sample size procedure. Allows one to determine the effect of being wrong when estimating the population coefficient of variation in terms of the width of the obtained (two-sided) confidence intervals. The values of True.C.of.V and Estimated.C.of.V should be positive.

Usage

ss.aipe.cv.sensitivity(True.C.of.V = NULL, Estimated.C.of.V = NULL,
width = NULL, degree.of.certainty = NULL, assurance=NULL, certainty=NULL, 
mean = 100, Specified.N = NULL, conf.level = 0.95, 
G = 1000, print.iter = TRUE)

Arguments

Details

For sensitivity analysis when planning sample size given the desire to obtain narrow confidence intervals for the population coefficient of variation. Given a population value and an estimated value, one can determine the effects of incorrectly specifying the population coefficient of variation (True.C.of.V) on the obtained widths of the confidence intervals. Also, one can evaluate the percent of the confidence intervals that are less than the desired width (especially when modifying the degree.of.certainty parameter); see ss.aipe.cv)

Alternatively, one can specify Specified.N to determine the results at a particular sample size (when doing this Estimated.C.of.V cannot be specified).

Value

Note

Returns three lists, where each list has multiple components.

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

See Also

cv, ss.aipe.cv

Sample size planning for polynomial change models in longitudinal study

Description

This function plans sample size with respect to the group-by-time interaction in the context of a longitudinal design with two groups. It plans sample size from the accuracy in parameter estimation (AIPE) perspective, where the goal is to obtain a sufficiently narrow confidence interval for the fixed effect polynomial change coefficient parameter (e.g., linear, quadratic, etc.). The sample size returned can be one such that (a) the expected confidence interval width is sufficiently narrow, or (b) the observed confidence interval will be sufficiently narrow with a specified high degree of assurance (e.g., .99, .95, .90, etc.). This function accompanies Kelley and Rausch (2011).

Usage

ss.aipe.pcm(true.variance.trend, error.variance, 
variance.true.minus.estimated.trend = NULL, duration, frequency, 
width, conf.level = 0.95, trend = "linear", assurance = NULL)

Arguments

Value

Returns the necessary sample size for the combination of the desired goals and values of the population parameters for a specific design.

Note

Like in all formal sample size planning methods that require the value of one or more population parameter(s), if the population parameters are incorrectly specified, there is no guarantee that the sample size this function returns will be accurate. Of course, the further away from the true values, the further away the true sample size will tend to be.

The number of timepoints in a study (say M) is defined by f \times D + 1, where f is the frequency and D is the duration.

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

References

Kelley, K., & Rausch, J. R. (2011). Accuracy in parameter estimation for polynomial change models. Psychological Methods.

Examples

 
## Not run: 
# An example used in Kelley and Rausch for the expected confidence interval 
# width (returns 278). Thus, a necessary sample size of 278 is required when 
# the duration of the study will be 4 units and the frequency of measurement 
# occasions is 1 year in order for the expected confidence interval 
# width to be 0.025 units.  

ss.aipe.pcm(true.variance.trend=0.003, error.variance=0.0262, duration=4, 
frequency=1, width=0.025, conf.level=.95)

# Now, when incorporating an assurance parameter (returns 316). 
# Thus, a necessary sample size of 316 will ensure that the 95% confidence 
# interval will be sufficiently narrow (i.e., have a width less than .025 units) 
# at least 99% of the time.

ss.aipe.pcm(true.variance.trend=.003, error.variance=.0262, duration=4, 
frequency=1, width=.025, conf.level=.95, assurance=.99)

## End(Not run)

Sample size necessary for the accuracy in parameter estimation approach for an unstandardized regression coefficient of interest

Description

A function used to plan sample size from the accuracy in parameter estimation perspective for an unstandardized regression coefficient of interest given the input specification.

Usage

ss.aipe.rc(Rho2.Y_X = NULL, Rho2.k_X.without.k = NULL, 
K = NULL, b.k = NULL, width, which.width = "Full", sigma.Y = 1, 
sigma.X.k = 1, RHO.XX = NULL, Rho.YX = NULL, which.predictor = NULL, 
alpha.lower = NULL, alpha.upper = NULL, conf.level = .95, 
degree.of.certainty = NULL, assurance=NULL,  certainty=NULL, 
Suppress.Statement = FALSE)

Arguments

Details

Not all of the arguments need to be specified, only those that provide all of the necessary information so that the sample size can be determined for the conditions specified.

Value

Returns the necessary sample size in order for the goals of accuracy in parameter estimation to be satisfied for the confidence interval for a particular regression coefficient given the input specifications.

Note

This function calls upon ss.aipe.reg.coef in MBESS but has a different naming scheme. See ss.aipe.reg.coef for more details.

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

References

Kelley, K. & Maxwel, S. E. (2003). Sample size for Multiple Regression: Obtaining regression coefficients that are accuracy, not simply significant. Psychological Methods, 8, 305–321.

See Also

ss.aipe.reg.coef.sensitivity, conf.limits.nct,

ss.aipe.reg.coef, ss.aipe.src

Examples

## Not run: 
# Exchangable correlation structure
# Rho.YX <- c(.3, .3, .3, .3, .3)
# RHO.XX <- rbind(c(1, .5, .5, .5, .5), c(.5, 1, .5, .5, .5), c(.5, .5, 1, .5, .5),
# c(.5, .5, .5, 1, .5), c(.5, .5, .5, .5, 1))

# ss.aipe.rc(width=.1, which.width="Full", sigma.Y=1, sigma.X=1, RHO.XX=RHO.XX,
# Rho.YX=Rho.YX, which.predictor=1, conf.level=1-.05)

# ss.aipe.rc(width=.1, which.width="Full", sigma.Y=1, sigma.X=1, RHO.XX=RHO.XX,
# Rho.YX=Rho.YX, which.predictor=1,  conf.level=1-.05, degree.of.certainty=.85)

## End(Not run)

Sensitivity analysis for sample size planing from the Accuracy in Parameter Estimation Perspective for the unstandardized regression coefficient

Description

Performs a sensitivity analysis when planning sample size from the Accuracy in Parameter Estimation Perspective for the unstandardized regression coefficient.

Usage

ss.aipe.rc.sensitivity(True.Var.Y = NULL, True.Cov.YX = NULL, 
True.Cov.XX = NULL, Estimated.Var.Y = NULL, Estimated.Cov.YX = NULL, 
Estimated.Cov.XX = NULL, Specified.N = NULL, which.predictor = 1, 
w = NULL, Noncentral = FALSE, Standardize = FALSE, conf.level = 0.95, 
degree.of.certainty = NULL, assurance=NULL, certainty=NULL, 
G = 1000, print.iter = TRUE)

Arguments

Details

Direct specification of True.Rho.YX and True.RHO.XX is necessary, even if one is interested in a single regression coefficient, so that the covariance/correlation structure can be specified when when the simulation student within the function runs.

Value

Note

Note that when True.Rho.YX=Estimated.Rho.YX and True.RHO.XX=Estimated.RHO.XX, the results are not literally from a sensitivity analysis, rather the function performs a standard simulation study. A simulation study can be helpful in order to determine if the sample size procedure under or overestimates necessary sample size.

See ss.aipe.reg.coef.sensitivity in MBESS for more details.

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

References

Kelley, K. & Maxwell, S. E. (2003). Sample size for Multiple Regression: Obtaining regression coefficients that are accuracy, not simply significant. Psychological Methods, 8, 305–321.

See Also

ss.aipe.reg.coef.sensitivity, ss.aipe.src.sensitivity,

ss.aipe.reg.coef, ci.reg.coef

Sample size necessary for the accuracy in parameter estimation approach for a regression coefficient of interest

Description

A function used to plan sample size from the accuracy in parameter estimation approach for a regression coefficient of interest given the input specification.

Usage

ss.aipe.reg.coef(Rho2.Y_X=NULL, Rho2.j_X.without.j=NULL, p=NULL, 
b.j=NULL, width, which.width="Full", sigma.Y=1, sigma.X=1, RHO.XX=NULL, 
Rho.YX=NULL, which.predictor=NULL, Noncentral=FALSE, alpha.lower=NULL,
alpha.upper=NULL, conf.level=.95, degree.of.certainty=NULL, assurance=NULL, 
certainty=NULL, Suppress.Statement=FALSE)

Arguments

Details

Not all of the arguments need to be specified, only those that provide all of the necessary information so that the sample size can be determined for the conditions specified.

Value

Returns the necessary sample size in order for the goals of accuracy in parameter estimation to be satisfied for the confidence interval for a particular regression coefficient given the input specifications.

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

References

Kelley, K. & Maxwel, S. E. (2003). Sample size for Multiple Regression: Obtaining regression coefficients that are accurate, not simply significant. Psychological Methods, 8, 305–321.

See Also

ss.aipe.reg.coef.sensitivity, conf.limits.nct

Examples

## Not run: 
# Exchangable correlation structure
# Rho.YX <- c(.3, .3, .3, .3, .3)
# RHO.XX <- rbind(c(1, .5, .5, .5, .5), c(.5, 1, .5, .5, .5), c(.5, .5, 1, .5, .5), 
# c(.5, .5, .5, 1, .5), c(.5, .5, .5, .5, 1))
# ss.aipe.reg.coef(width=.1, which.width="Full", sigma.Y=1, sigma.X=1, RHO.XX=RHO.XX,
# Rho.YX=Rho.YX, which.predictor=1, Noncentral=FALSE, conf.level=1-.05, 
# degree.of.certainty=NULL, Suppress.Statement=FALSE)

# ss.aipe.reg.coef(width=.1, which.width="Full", sigma.Y=1, sigma.X=1, RHO.XX=RHO.XX,
# Rho.YX=Rho.YX, which.predictor=1, Noncentral=FALSE, conf.level=1-.05, 
# degree.of.certainty=.85, Suppress.Statement=FALSE)

# ss.aipe.reg.coef(width=.1, which.width="Full", sigma.Y=1, sigma.X=1, RHO.XX=RHO.XX,
# Rho.YX=Rho.YX, which.predictor=1, Noncentral=TRUE, conf.level=1-.05,
# degree.of.certainty=NULL, Suppress.Statement=FALSE)

# ss.aipe.reg.coef(width=.1, which.width="Full", sigma.Y=1, sigma.X=1, RHO.XX=RHO.XX, 
# Rho.YX=Rho.YX, which.predictor=1, Noncentral=TRUE, conf.level=1-.05, 
# degree.of.certainty=.85, Suppress.Statement=FALSE)
## End(Not run)

Sensitivity analysis for sample size planning from the Accuracy in Parameter Estimation Perspective for the (standardized and unstandardized) regression coefficient

Description

Performs a sensitivity analysis when planning sample size from the Accuracy in Parameter Estimation Perspective for the standardized or unstandardized regression coefficient.

Usage

ss.aipe.reg.coef.sensitivity(True.Var.Y = NULL, True.Cov.YX = NULL,
True.Cov.XX = NULL, Estimated.Var.Y = NULL, Estimated.Cov.YX = NULL,
Estimated.Cov.XX = NULL, Specified.N = NULL, which.predictor = 1, 
w = NULL, Noncentral = FALSE, Standardize = FALSE, conf.level = 0.95, 
degree.of.certainty = NULL, assurance=NULL, certainty=NULL,
 G = 1000, print.iter = TRUE)

Arguments

Details

Direct specification of True.Rho.YX and True.RHO.XX is necessary, even if one is interested in a single regression coefficient, so that the covariance/correlation structure can be specified when the simulation student within the function runs.

Value

Note

Note that when True.Rho.YX=Estimated.Rho.YX and True.RHO.XX=Estimated.RHO.XX, the results are not literally from a sensitivity analysis, rather the function performs a standard simulation study. A simulation study can be helpful in order to determine if the sample size procedure under or overestimates necessary sample size.

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

References

Kelley, K. & Maxwell, S. E. (2003). Sample size for Multiple Regression: Obtaining regression coefficients that are accuracy, not simply significant. Psychological Methods, 8, 305–321.

See Also

ss.aipe.reg.coef, ci.reg.coef

Sample Size Planning for Accuracy in Parameter Estimation for Reliability Coefficients.

Description

This function determines a necessary sample size so that the expected confidence interval width for the alpha coefficient or omega coefficient is sufficiently narrow (when assurance=NULL) or so that the obtained confidence interval is no larger than the value specified with some desired degree of certainty (i.e., a probability that the obtained width is less than the specified width; assurance=.85). This function calculates coefficient alpha based on McDonald's (1999) formula for coefficient alpha, also known as Guttman-Cronbach alpha. It also uses coefficient omega from McDonald (1999). When the 'Parallel' or 'True Score' model is used, coefficient alpha is calculated. When the 'Congeneric' model is used, coefficient omega is calculated.

Usage

ss.aipe.reliability(model = NULL, type = NULL, width = NULL, S = NULL, 
conf.level = 0.95, assurance = NULL, data = NULL, i = NULL, cor.est = NULL, 
lambda = NULL, psi.square = NULL, initial.iter = 500, 
final.iter = 5000, start.ss = NULL, verbose=FALSE)

Arguments

Details

Use verbose=TRUE if the function is taking a very long time to provide an answer.

Value

Warning

In some conditions, you may receive a warning, such as "In sem.default(ram = ram, S = S, N = N, param.names = pars, var.names = vars,; Could not compute QR decomposition of Hessian. Optimization probably did not converge." This indicates that the model likely did not converge. In certain conditions this may occur because the model is not being fit well due to small sample size, a low number of iterations, or a poorly behaved covariance matrix.

Note

Not all of the items can be entered into the function to represent the population values. For example, either 'data' can be used, or S, or i, cor.est, and psi.square, or i, lambda, and psi.square. With a large number of iterations (final.iter) this function may take considerable time.

Author(s)

Leann J. Terry (Indiana University; ljterry@Indiana.Edu); Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

References

McDonald, R. P. (1999). Test theory: A unified approach. Mahwah, New Jersey: Lawrence Erlbaum Associates, Publishers.

van Zyl, J. M., Neudecker, H., & Nel, D. G. (2000). On the distribution of the maximum likelihood estimator of Cronbach's alpha. Psychometrika, 65 (3), 271–280.

See Also

CFA.1; sem; ci.reliability;

Examples

## Not run: 
ss.aipe.reliability (model='Parallel', type='Normal Theory', width=.1, i=6, 
                     cor.est=.3, psi.square=.2, conf.level=.95, assurance=NULL, initial.iter=500, 
                     final.iter=5000)

# Same as above but now 'assurance' is used. 
ss.aipe.reliability (model='Parallel', type='Normal Theory', width=.1, i=6, 
cor.est=.3, psi.square=.2, conf.level=.95, assurance=.85, initial.iter=500, 
final.iter=5000)


# Similar to the above but now the "True Score" model is used. Note how the psi.square changes 
# from a scalar to a vector of length i (number of items). 
# Also note, however, that cor.est is a single value (due to the true-score model specified)
ss.aipe.reliability (model='True Score', type='Normal Theory', width=.1, i=5, 
                     cor.est=.3, psi.square=c(.2, .3, .3, .2, .3), conf.level=.95, 
                     assurance=.85, initial.iter=500, final.iter=5000)
                     
ss.aipe.reliability (model='True Score', type='Normal Theory', width=.1, i=5, 
                     cor.est=.3, psi.square=c(.2, .3, .3, .2, .3), conf.level=.95, 
                     assurance=.85, initial.iter=500, final.iter=5000)                 

# Now, a congeneric model is used with the factor analytic appraoch. This is likely the 
# most realistic scenario (and maps onto the ideas of Coefficient Omega). 
ss.aipe.reliability (model='Congeneric', type='Factor Analytic', width=.1, i=5, 
lambda=c(.4, .4, .3, .3, .5), psi.square=c(.2, .4, .3, .3, .2), conf.level=.95, 
assurance=.85, initial.iter=1000, final.iter=5000)

# Now, the presumed population matrix among the items is used. 
Pop.Mat<-rbind(c(1.0000000, 0.3813850, 0.4216370, 0.3651484, 0.4472136), 
c(0.3813850, 1.0000000, 0.4020151, 0.3481553, 0.4264014), c(0.4216370, 
0.4020151, 1.0000000, 0.3849002, 0.4714045), c(0.3651484, 0.3481553, 
0.3849002, 1.0000000, 0.4082483), c(0.4472136, 0.4264014, 0.4714045, 
0.4082483, 1.0000000))

ss.aipe.reliability (model='True Score', type='Normal Theory', width=.15, 
S=Pop.Mat, conf.level=.95, assurance=.85, initial.iter=1000, final.iter=5000) 


## End(Not run)

Sample size planning for RMSEA in SEM

Description

Sample size planning for the population root mean square error of approximation (RMSEA) from the accuracy in parameter estimation (AIPE) perspective. The sample size is planned so that the expected width of a confidence interval for the population RMSEA is no larger than desired.

Usage

ss.aipe.rmsea(RMSEA, df, width, conf.level = 0.95)

Arguments

Value

Returns the necessary total sample size in order to achieve the desired degree of accuracy (i.e., the sufficiently narrow confidence interval).

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu) and Keke Lai

See Also

ci.rmsea

Examples

## Not run: 
# ss.aipe.rmsea(RMSEA=.035, df=50, width=.05, conf.level=.95)

## End(Not run)

a priori Monte Carlo simulation for sample size planning for RMSEA in SEM

Description

Conduct a priori Monte Carlo simulation to empirically study the effects of (mis)specifications of input information on the calculated sample size. The sample size is planned so that the expected width of a confidence interval for the population RMSEA is no larger than desired. Random data are generated from the true covariance matrix but fit to the proposed model, whereas sample size is calculated based on the input covariance matrix and proposed model.

Usage

ss.aipe.rmsea.sensitivity(width, model, Sigma, N=NULL, 
conf.level=0.95, G=200, save.file="sim.results.txt", ...)

Arguments

Details

This function implements the sample size planning methods proposed in Kelley and Lai (2010). It depends on the function sem in the sem package to fit the proposed model to random data, and uses the same notation to specify SEM models as does sem. Please refer to sem for more detailed documentation about model specifications, the RAM notation, and model fitting techniques. For technical discussion on how to obtain the model implied covariance matrix in the RAM notation given model parameters, see McArdle and McDonald (1984)

Value

Note

Sometimes this function jumps out of the loop before it finishes the simulation. The reason is because the sem function that this function calls to fit the model fails to converge when searching for maximum likelihood estimates of model parameters. Since the results in previous replications are saved, the user can start this function again, and specify the number of replications (i.e., G) to be the desired total number of replications minus the number of previous successful replications.

Author(s)

Keke Lai (University of California – Merced) and Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

References

Cudeck, R., & Browne, M. W. (1992). Constructing a covariance matrix that yields a specified minimizer and a specified minimum discrepancy function value. Psychometrika, 57, 357–369.

Fox, J. (2006). Structural equation modeling with the sem package in R. Structural Equation Modeling, 13, 465–486.

Kelley, K., & Lai, K. (2010). Accuracy in parameter estimation for the root mean square of approximation: Sample size planning for narrow confidence intervals. Manuscript under review.

McArdle, J. J., & McDonald, R. P. (1984). Some algebraic properties of the reticular action model. British Journal of Mathematical and Statistical Psychology, 37, 234–251.

See Also

sem; specify.model; ss.aipe.rmsea; theta.2.Sigma.theta; Sigma.2.SigmaStar

Examples

## Not run: 
#########################
EXAMPLE 1 
######################### 
# To replicate the simulation in the first panel, second column of 
# Table 2 (i.e., population RMSEA=0.0268, df=23, desired width=0.02) 
# in Lai and Kelley (2010), the following steps can be used.

## STEP 1: Obtain the (correct) population covariance matrix implied by Model 2
# This requires the model and its population model parameter values.
library(MASS)
library(sem)

# Specify Model 2 in the RAM notation
model.2<-specifyModel()
xi1 -> y1, lambda1, 1
xi1 -> y2, NA, 1
xi1 -> y3, lambda2, 1
xi1 -> y4, lambda3, 0.3
eta1 -> y4, lambda4, 1
eta1 -> y5, NA, 1
eta1 -> y6, lambda5, 1
eta1 -> y7, lambda6, 0.3
eta2 -> y6, lambda7, 0.3
eta2 -> y7, lambda8, 1
eta2 -> y8, NA, 1
eta2 -> y9, lambda9, 1
xi1 -> eta1, gamma11, 0.6
eta1 -> eta2, beta21, 0.6 
xi1 <-> xi1, phi11, 0.49
eta1 <-> eta1, psi11, 0.3136
eta2 <-> eta2, psi22, 0.3136
y1 <-> y1, delta1, 0.51
y2 <-> y2, delta2, 0.51
y3 <-> y3, delta3, 0.51
y4 <-> y4, delta4, 0.2895
y5 <-> y5, delta5, 0.51
y6 <-> y6, delta6, 0.2895
y7 <-> y7, delta7, 0.2895
y8 <-> y8, delta8, 0.51
y9 <-> y9, delta9, 0.51


# To inspect the specified model
model.2

# Specify model parameter values
theta <- c(1, 1, 0.3, 1,1, 0.3, 0.3, 1, 1, 0.6, 0.6,
0.49, 0.3136, 0.3136, 0.51, 0.51, 0.51, 0.2895, 0.51, 0.2895, 0.2895, 0.51, 0.51)

names(theta) <- c("lambda1","lambda2","lambda3",
"lambda4","lambda5","lambda6","lambda7","lambda8","lambda9",
"gamma11", "beta21",
"phi11", "psi11", "psi22", 
"delta1","delta2","delta3","delta4","delta5","delta6","delta7",
"delta8","delta9")

res<-theta.2.Sigma.theta(model=model.2, theta=theta, 
latent.vars=c("xi1", "eta1","eta2"))

Sigma.theta <- res$Sigma.theta
# Then 'Sigma.theta' is the (true) population covariance matrix

## STEP 2: Create a misspecified model
# The following model is misspecified in the same way as did Lai and Kelley (2010)
# with the goal to obtain a relatively small population RMSEA

model.2.mis<-specifyModel()
xi1 -> y1, lambda1, 1
xi1 -> y2, NA, 1
xi1 -> y3, lambda2, 1
xi1 -> y4, lambda3, 0.3
eta1 -> y4, lambda4, 1
eta1 -> y5, NA, 1
eta1 -> y6, lambda5, 0.96
eta2 -> y6, lambda7, 0.33
eta2 -> y7, lambda8, 1.33
eta2 -> y8, NA, 1
eta2 -> y9, lambda9, 1
xi1 -> eta1, gamma11, 0.6
eta1 -> eta2, beta21, 0.65 
xi1 <-> xi1, phi11, 0.49
eta1 <-> eta1, psi11, 0.3136
eta2 <-> eta2, psi22, 0.23
y1 <-> y1, delta1, 0.51
y2 <-> y2, delta2, 0.51
y3 <-> y3, delta3, 0.51
y4 <-> y4, delta4, 0.2895
y5 <-> y5, delta5, 0.51
y6 <-> y6, delta6, 0.29
y7 <-> y7, delta7, 0.22
y8 <-> y8, delta8, 0.56
y9 <-> y9, delta9, 0.56


# To verify the population RMSEA of this misspecified model
fit<-sem(ram=model.2.mis, S=Sigma.theta, N=1000000)
summary(fit)$RMSEA

## STEP 3: Conduct the simulation
# The number of replications is set to a very small value just to demonstrate
# and save time. Real simulation studies require a larger number (e.g., 500, 1,000)

ss.aipe.rmsea.sensitivity(width=0.02, model=model.2.mis, Sigma=Sigma.theta, G=10)

## STEP 3+: In cases where this function stops before it finishes the simulation
# Suppose it stops at the 7th replication. The text 
# file "results_ss.aipe.rmsea.sensitivity.txt" saves the results in all 
# previous replications; in this case it contains 6 replications since
# the simulation stopped at the 7th. The user can start this function again and specify
# 'G' to 4 (i.e., 10-6). New results will be appended to previous ones in the same file.

ss.aipe.rmsea.sensitivity(width=0.02, model=model.2.mis, Sigma=Sigma.theta, G=4)

########################################
EXAMPLE 2
########################################
# In addition to create a misspecified model by changing the model
# parameters in the true model as does Example 1, a misspecified
# model can also be created with the Cudeck-Browne (1992) procedure. 
# This procedure is implemented in the 'Sigma.2.SigmaStar( )' function in
# the MBESS package. Please refer to the help file of 'Sigma.2.SigmaStar( )'
# for detailed documentation.

## STEP 1: Specify the model
# This model is the same as the model in the first step of Example 1, but the
# model-implied population covariance matrix is no longer the true population 
# covariance matrix. The true population covariance matrix will be generated
# in Step 2 with the Cudeck-Browne procedure.
library(MASS)
library(sem)

model.2<-specifyModel()
xi1 -> y1, lambda1, 1
xi1 -> y2, NA, 1
xi1 -> y3, lambda2, 1
xi1 -> y4, lambda3, 0.3
eta1 -> y4, lambda4, 1
eta1 -> y5, NA, 1
eta1 -> y6, lambda5, 1
eta1 -> y7, lambda6, 0.3
eta2 -> y6, lambda7, 0.3
eta2 -> y7, lambda8, 1
eta2 -> y8, NA, 1
eta2 -> y9, lambda9, 1
xi1 -> eta1, gamma11, 0.6
eta1 -> eta2, beta21, 0.6 
xi1 <-> xi1, phi11, 0.49
eta1 <-> eta1, psi11, 0.3136
eta2 <-> eta2, psi22, 0.3136
y1 <-> y1, delta1, 0.51
y2 <-> y2, delta2, 0.51
y3 <-> y3, delta3, 0.51
y4 <-> y4, delta4, 0.2895
y5 <-> y5, delta5, 0.51
y6 <-> y6, delta6, 0.2895
y7 <-> y7, delta7, 0.2895
y8 <-> y8, delta8, 0.51
y9 <-> y9, delta9, 0.51



theta <- c(1, 1, 0.3, 1,1, 0.3, 0.3, 1, 1, 0.6, 0.6,
0.49, 0.3136, 0.3136, 0.51, 0.51, 0.51, 0.2895, 0.51, 0.2895, 0.2895, 0.51, 0.51)

names(theta) <- c("lambda1","lambda2","lambda3",
"lambda4","lambda5","lambda6","lambda7","lambda8","lambda9",
"gamma11", "beta21",
"phi11", "psi11", "psi22", 
"delta1","delta2","delta3","delta4","delta5","delta6","delta7",
"delta8","delta9")

## STEP 2: Create the true population covariance matrix, so that (a) the model fits
# to this covariance matrix with specified discrepancy; (b) the population model
# parameters (the object 'theta') is the minimizer in fitting the model to the true 
# population covariance matrix.

# Since the desired RMSEA is 0.0268 and the df is 22, the MLE discrepancy value
# is specified to be 22*0.0268*0.0268, given the definition of RMSEA.

res <- Sigma.2.SigmaStar(model=model.2, model.par=theta, 
latent.var=c("xi1", "eta1", "eta2"), discrep=22*0.0268*0.0268)

Sigma.theta.star <- res$Sigma.star

# To verify that the population RMSEA is 0.0268
res2 <- sem(ram=model.2, S=Sigma.theta.star, N=1000000)
summary(res2)$RMSEA

## STEP 3: Conduct the simulation
# Note although Examples 1 and 2 have the same population RMSEA, the
# model df and true population covariance matrix are different. Example 1
# uses 'model.2.mis' and 'Sigma.theta', whereas Example 2 uses 'model.2'
# and 'Sigma.theta.star'. Since the df is different, it requires a different sample
# size to achieve the same desired confidence interval width. 
ss.aipe.rmsea.sensitivity(width=0.02, model=model.2, Sigma=Sigma.theta.star, G=10)

## End(Not run)

Sample size planning for Accuracy in Parameter Estimation (AIPE) of the standardized contrast in ANOVA

Description

A function to calculate the appropriate sample size per group for the standardized contrast in ANOVA such that the width of the confidence interval is sufficiently narrow.

Usage

ss.aipe.sc(psi, c.weights, width, conf.level = 0.95, 
assurance = NULL, certainty = NULL, ...)

Arguments

Value

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu); Keke Lai

References

Cumming, G. & Finch, S. (2001). A primer on the understanding, use, and calculation of confidence intervals that are based on central and noncentral distributions, Educational and Psychological Measurement, 61, 532–574.

Hedges, L. V. (1981). Distribution theory for Glass's Estimator of effect size and related estimators. Journal of Educational Statistics, 2, 107–128.

Kelley, K. (2005). The effects of nonnormal distributions on confidence intervals around the standardized mean difference: Bootstrap and parametric confidence intervals, Educational and Psychological Measurement, 65, 51–69.

Kelley, K. (2007). Constructing confidence intervals for standardized effect sizes: Theory, application, and implementation. Journal of Statistical Software, 20 (8), 1–24.

Kelley, K., & Rausch, J. R. (2006). Sample size planning for the standardized mean difference: Accuracy in Parameter Estimation via narrow confidence intervals. Psychological Methods, 11(4), 363–385.

Lai, K., & Kelley, K. (2007). Sample size planning for standardized ANCOVA and ANOVA contrasts: Obtaining narrow confidence intervals. Manuscript submitted for publication.

Steiger, J. H., & Fouladi, R. T. (1997). Noncentrality interval estimation and the evaluation of statistical methods. In L. L. Harlow, S. A. Mulaik, & J.H. Steiger (Eds.), What if there were no significance tests? (pp. 221–257). Mahwah, NJ: Lawrence Erlbaum.

See Also

ci.sc, conf.limits.nct, ss.aipe.c

Examples

# Suppose the population standardized contrast is believed to be .6 
# in some 5-group ANOVA model. The researcher is interested in comparing 
# the average of means of group 1 and 2 with the average of group 3 and 4.

# To calculate the necessary sample size per gorup such that the width 
# of 95 percent confidence interval of the standardized 
# contrast is, with 90 percent assurance, no wider than .4:

# ss.aipe.sc(psi=.6, c.weights=c(.5, .5, -.5, -.5, 0), width=.4, assurance=.90) 

Sample size planning from the AIPE perspective for standardized ANCOVA contrasts

Description

Sample size planning from the accuracy in parameter estimation (AIPE) perspective for standardized ANCOVA contrasts.

Usage

ss.aipe.sc.ancova(Psi = NULL, sigma.anova = NULL, sigma.ancova = NULL,
psi = NULL, ratio = NULL, rho = NULL, divisor = "s.ancova", 
c.weights, width, conf.level = 0.95, assurance = NULL, ...)

Arguments

Details

The sample size planning method this function is based on is developed in the context of simple (i.e., one-response-one-covariate) ANCOVA model and randomized design (i.e., same population covariate mean across groups).

An ANCOVA contrast can be standardized in at least two ways: (a) divided by the error standard deviation of the ANOVA model, (b) divided by the error standard deviation of the ANCOVA model. This function can be used to analyze both types of standardized ANCOVA contrasts.

Not all of the arguments about the effect sizes need to be specified. If divisor="s.ancova" is used in the argument, then input either (a) psi, or (b) Psi and s.ancova. If divisor="s.anova" is used in the argument, possible specifications are (a) Psi, s.ancova, and s.anova; (b) psi, and ratio; (c) psi, and rho.

Value

This function returns the sample size per group.

Note

When divisor="s.anova" and the argument assurance is specified, the necessary sample size per group returned by the function with assurance specified is slightly underestimated. The method to obtain exact sample size in the above situation has not been developed yet. A practical solution is to use the sample size returned as the starting value to conduct a priori Montre Carlo simulations with function ss.aipe.sc.ancova.sensitivity, as discussed in Lai & Kelley (under review).

Author(s)

Keke Lai (University of California–Merced)

References

Kelley, K. (2007). Constructing confidence intervals for standardized effect sizes: Theory, application, and implementation. Journal of Statistical Software, 20 (8), 1–24.

Kelley, K., & Rausch, J. R. (2006). Sample size planning for the standardized mean difference: Accuracy in Parameter Estimation via narrow confidence intervals. Psychological Methods, 11 (4), 363–385.

Lai, K., & Kelley, K. (2012). Accuracy in parameter estimation for ANCOVA and ANOVA contrasts: Sample size planning via narrow confidence intervals. British Journal of Mathematical and Statistical Psychology, 65, 350–370.

Steiger, J. H., & Fouladi, R. T. (1997). Noncentrality interval estimation and the evaluation of statistical methods. In L. L. Harlow, S. A. Mulaik, & J.H. Steiger (Eds.), What if there were no significance tests? (pp. 221–257). Mahwah, NJ: Lawrence Erlbaum.

See Also

ss.aipe.sc, ss.aipe.sc.ancova.sensitivity

Examples

## Not run: 
ss.aipe.sc.ancova(psi=.8, width=.5, c.weights=c(.5, .5, 0, -1))

ss.aipe.sc.ancova(psi=.8, ratio=.6, width=.5, 
c.weights=c(.5, .5, 0, -1), divisor="s.anova")

ss.aipe.sc.ancova(psi=.5, rho=.4, width=.3, 
c.weights=c(.5, .5, 0, -1), divisor="s.anova")

## End(Not run)

Sensitivity analysis for the sample size planning method for standardized ANCOVA contrast

Description