Quantitative Psychology Tools

Description

Contains tools useful in screening univariate and multivariate data, testing simple moderating relationships (Cohen et. al. 2003)<doi:10.4324/9780203774441>, extimating indirect effects based on simple (proximal) and complex (distal) mediating relationships(MacKinnon et. al. 2002)<doi:10.1037/1082-989x.7.1.83> . A tool for computing power in a given F distribution is also included (Murphy & Myors 2014)<ISBN:9781315773155>. These are basic operations covered in a multivariate course in most Doctoral level Psychology programs. THese functions will also likely be useful in other domains (e.g., sociology, business management, medicine).

Details

These functions can be grouped into 4 sets.
Data Screening
Moderation
Mediation
Power Calculation

Data Screening tools include both graphical and statistical methods for assessing the shape of the distributions as well as look for any outliers. Key functions include norm and mult.norm. Moderation functions are based on Aiken & West (1991) and Cohen, Cohen, West & Aiken (2003). Currently, only simple models are permissible (Y ~ X + Z + XZ). Key functions include moderate.lm, sim.slopes, and graph.mod. Mediation functions are largely based on MacKinnon et al (2002) and Fletcher (2006). Both simple (one mediator) and complex (chain of two mediators) relationships can be estimated. Key functions include proximal.med, and distal.med. Finally, powerF will calculate power based on an F distribution given percent variance accounted for (e.g. effect size) and degrees of freedom (e.g., model parameters and sample size). Some of the functions are original and others borrowed from numerous sources. I have taken care to reference appropriately.

Author(s)

Thomas D. Fletcher

Maintainer: Thomas D. Fletcher <t.d.fletcher05@gmail.com>

References

Aiken, L. S., & West, S. G. (1991). Multiple regression: Testing and interpreting interactions. Newbury Park: Sage Publications.

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

Fletcher, T. D. (2006, August). Methods and approaches to assessing distal mediation. Paper presented at the 66th annual meeting of the Academy of Management, Atlanta, GA.

Khattree, R. & Naik, D. N. (1999). Applied multivariate statistics with SAS software (2nd ed.). Cary, NC: SAS Institute Inc.

MacKinnon, D. P., Lockwood, C. M., Hoffman, J. M., West, S. G., & Sheets, V. (2002). A comparison of methods to test mediation and other intervening variable effects. Psychological Methods, 7, 83-104.

Murphy, K. R., & Myors, B. (2004). Statistical power analysis: A simple and general model for traditional and modern hypothesis tests (2nd ed.). Mahwah, NJ: Lawrence Erlbaum Associates.

Examples

# Data Screening

data(USJudgeRatings)
norm(USJudgeRatings$CONT)
mult.norm(USJudgeRatings[,1:4])

# Moderation
data(tra)
lm.mod1 <- moderate.lm(beliefs, values, attitudes, tra)
ss.mod1 <- sim.slopes(lm.mod1,meanCenter(tra$values))
summary(lm.mod1)
ss.mod1
# use mouse click to place legend in graph.mod
# graph.mod(ss.mod1,beliefs,attitudes,tra,"Interaction Example")

# Mediation
# create object with names x, m, y
# data(tra)
temp.tra <- tra
names(temp.tra) <- c("x", "z", "m", "y")
proximal.med(temp.tra)

Classification for Logistic Regression

Description

Provides a Classification analysis for a logistic regression model. Also provides McFadden's Rsq.

Usage

ClassLog(MOD, resp, cut=.5)

Arguments

Value

A list containing: /CR

• rawtab two-way table of classifications as frequencies

• classtab two-way table of classifications as percentages

• overall Overall percent classifications that are correct

• mcFadden McFaddens pseudoRsq; 1 - ModelDeviance / NullDeviance

Warning

This is a primative function. I have a long to do list. For example, it is not yet written to handle missing observations.

Author(s)

Thomas D. Fletcher t.d.fletcher05@gmail.com

See Also

glm

Examples

# create some data
x <- rnorm(100)
y <- as.numeric(cut(.5*x + rnorm(100), breaks=2))-1
tdf <- data.frame(x=x, y=y)

# run a logistic regression	
glm1 <- glm(y ~ x, data=tdf, family=binomial)

# Get typical summary of results
summary(glm1)

# Classification Analysis
ClassLog(glm1, tdf$y)

Standardize Data

Description

Converts data to standard normal (mean = 0; SD = 1) - i.e., z-scores.

Usage

Make.Z(x)

Arguments

Details

Takes the data (by columns if necessary) and subtracts out the mean and then divides by the standard deviation. The result is a standard normal z score.

Value

A numeric or matrix containing standardized data (i.e., z scores)

Warning

The result is a matrix. One may wish to convert to data.frame or use as.data.frame(Make.Z(x))

Author(s)

Thomas D. Fletcher t.d.fletcher05@gmail.com

Examples

zUSJR <- Make.Z(USJudgeRatings)	# creates new object containg z scores
dim(zUSJR)			# shows that there are 43 observed z scores for 12 variables
zUSJR[,1]			# to look at only the first column of z scores

Normalize Data

Description

Convert data to Normal Scores with the same Mean and SD. This reshapes data to conform to a Normal Distribution. It is not converting to z-scores (i.e., it is not standardizing data)

Usage

Normalize(x) 

Arguments

Value

A numeric with the same Mean and SD as x, but without skew or kurtosis

Warning

This is a primative function. I have a long to do list. For example, it is not yet written to handle missing observations.

Author(s)

Thomas D. Fletcher t.d.fletcher05@gmail.com

References

Joreskog, K., Sorbom, D., du Toit, S., & du Toit, M. (2000). LISREL 8: New Statistical Features. SSI.

Snippets of code were borrowed and modified from: http://zoonek2.free.fr/UNIX/48_R/03.html

See Also

norm

Examples

summary(USJudgeRatings$CONT)
plot(density(USJudgeRatings$CONT))

ContN <- Normalize(USJudgeRatings$CONT)
summary(ContN)
lines(density(ContN), col=2)

Complex Mediation for use in Bootstrapping

Description

Computes the 'total indirect effect' from distal.med for use in boot

Usage

distInd.ef(data, i)

Arguments

Details

This function is not useful of itself. It is specifically created as an intermediate step in bootstrapping the indirect effect.

Value

indirect effect that is passed to boot for each bootstrap sample

Author(s)

Thomas D. Fletcher t.d.fletcher05@gmail.com

References

Davison, A. C. & Hinkley, D. V. (1997). Bootstrap methods and their application. Cambridge, UK: Cambridge University Press.

Fletcher, T. D. (2006, August). Methods and approaches to assessing distal mediation. Paper presented at the 66th annual meeting of the Academy of Management, Atlanta, GA.

MacKinnon, D. P., Lockwood, C. M., Hoffman, J. M., West, S. G., & Sheets, V. (2002). A comparison of methods to test mediation and other intervening variable effects. Psychological Methods, 7, 83-104.

MacKinnon, D. P., Lockwood, C. M., & Williams, J. (2004). Confidence limit for indirect effect: distribution of the product and resampling methods. Multivariate Behavioral Research, 39, 99-128.

See Also

distal.med

Examples

cormat <- matrix (c(1,.3,.15,.075,.3,1,.3,.15,.15,.3,1,.3,.075,.15,.3,1),ncol=4)
require(MASS)
d200 <- data.frame(mvrnorm(200, mu=c(0,0,0,0), cormat))
names(d200) <- c("x","m1","m2","y")

require(boot)
distmed.boot <- boot(d200, distInd.ef, R=999)
sort(distmed.boot$t)[c(25,975)]	#95% CI
plot(density(distmed.boot$t))	# Distribution of bootstapped indirect effect
summary(distmed.boot$t)

Distal Indirect Effect

Description

Computes the indirect effect (and all paths) in a 4 variable system, assuming all paths estimated.

Usage

distal.med(data)

Arguments

Details

Computes the paths in the model system: /cr Y = t'X + fM1 + cM2
M2 = eX + bM1
M1 = aX
and the indirect effect a*b*c + a*f + e*c

Value

Returns a table with all the effects and decomposition of effects in the above 4 variable system inclucing the standard errors and t-values.

Warning

This function is primative in that it is based on a simplistic model AND forces the user to name the variables in the dataset x, m1, m2, and y.

Note

This function uses the following undocumented functions: se.indirect3

Author(s)

Thomas D. Fletcher t.d.fletcher05@gmail.com

References

Fletcher, T. D. (2006, August). Methods and approaches to assessing distal mediation. Paper presented at the 66th annual meeting of the Academy of Management, Atlanta, GA.

MacKinnon, D. P., Lockwood, C. M., Hoffman, J. M., West, S. G., & Sheets, V. (2002). A comparison of methods to test mediation and other intervening variable effects. Psychological Methods, 7, 83-104.

See Also

proximal.med, distInd.ef

Examples

cormat <- matrix (c(1,.3,.15,.075,.3,1,.3,.15,.15,.3,1,.3,.075,.15,.3,1),ncol=4)
require(MASS)
d200 <- data.frame(mvrnorm(200, mu=c(0,0,0,0), cormat))
names(d200) <- c("x","m1","m2","y")
distal.med(d200)

Plots for Exploratory Data Analysis

Description

This function is a modified version of eda.shape found in the S+ Guide to Statistics, v1, p. 124. It is based on work by Tukey (1977) and each plot is described in more detail in Ch. 4 of Cohen et al. Creates 4 plots useful in assesing univariate distributions of data.

Usage

eda.uni(x, title = "")

Arguments

Details

Simply provides a histogram, smoothed histogram, qq-plot, and boxplot for x.

Value

A single graph object with 4 basic plots.

Author(s)

Thomas D. Fletcher t.d.fletcher05@gmail.com

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: Lawrence Erlbaum Associates. Chapter 4, Data Visualization, exploration, and assumption checking: Diagnosing and solving regression problems.

Insightful (2001). S-Plus 6 for Windows Guide to Statistics, Volume 1. Seattle: Insightful.

Tukey (1977). Exploratory Data Analysis. Reading, MA: Addison-Wesley.

See Also

hist, plot.density, qqnorm, boxplot

Examples

# create negatively skewed dat with 100 observations
xc <-  -rchisq(100,3)	
eda.uni(xc)

Moderation Graph

Description

x-y plot containing the simple slopes conditioned on z.

Usage

graph.mod(ssmod, x, y, data, title = "", xlab = "Centered X", 
  ylab = "Raw Y", ylimit = 1.5, ...)

Arguments

Details

Given moderate.lm and sim.slopes, this function plots x,y and adds the simple slopes corresponding to arbitrary values of z defined in sim.slopes. Users must 'click' on an area to add the legend.

Value

A plot object.

Warning

This is based on a simple 3 variable moderation model with continuous variables. Users must modify the functions to accomodate other models (e.g., categorical moderators, covariates)

Author(s)

Thomas D. Fletcher t.d.fletcher05@gmail.com

References

Aiken, L. S., & West, S. G. (1991). Multiple regression: Testing and interpreting interactions. Newbury Park: Sage Publications.

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

See Also

moderate.lm, sim.slopes

Examples

data(tra)
lm.mod1 <- moderate.lm(beliefs, values, attitudes, tra)
ss.mod1 <- sim.slopes(lm.mod1, tra$values)
## requires user interaction 
## graph.mod(ss.mod1,beliefs,attitudes,tra,"Interaction Example")

Standardized Regression Coefficients

Description

Computes the standardized regression coeffients (beta) from an object of class (lm)

Usage

lm.beta(MOD)

Arguments

Value

A "numeric" representing each standardized coefficient from lm() model

Warning

This function does not produce 'correct' standardized coefficients when interaction terms are present

Author(s)

Thomas D. Fletcher t.d.fletcher05@gmail.com

See Also

Make.Z, lm

Examples

us <- USJudgeRatings
names(us)
lm1 <- lm ( CONT ~ INTG + DMNR + DILG, us)
lm.beta(lm1)

# Standardized data (using Make.Z())

usz <- data.frame (Make.Z (us))
lm1.z <- lm ( CONT ~ INTG + DMNR + DILG, usz)

# compare standardized data versus lm.beta

summary(lm1.z)

Mean Center Variables

Description

This simple function subtracts the mean from a variable rendering mean-centered variables.

Usage

meanCenter(x)

Arguments

Details

This is particularly useful in lm() with higher order terms as in moderation.

Value

a numeric; mean centered 'x'

Author(s)

Thomas D. Fletcher t.d.fletcher05@gmail.com

Examples

data(USJudgeRatings)
usCONT <- meanCenter(USJudgeRatings$CONT)
summary(usCONT)

Simple Moderated Regression Model

Description

This function creates an object of class lm() specific to a moderated multiple regression involving 3 variables.

Usage

moderate.lm(x, z, y, data, mc = FALSE)

Arguments

Details

This model takes x and z and creates the interaction term x*z. If the data are not already mean centered, then x and z are mean centered by subtracting out the means. This is necessary for interpretation and to reduce multicolinearity. The lm() is then computed thusly: Y ~ X + Z + XZ.

Value

An object of class lm(). One can use summary(), coef() or any other function useful to lm(). This model is used by other moderator tools - see below.

Warning

This is a very simplistic model. If x or z are categorical, the results will not be accurate. The function can be modified by the user to deal with complications such as covariates, non-continuous variables, etc.

Author(s)

Thomas D. Fletcher t.d.fletcher05@gmail.com

References

Aiken, L. S., & West, S. G. (1991). Multiple regression: Testing and interpreting interactions. Newbury Park: Sage Publications.

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

See Also

sim.slopes, graph.mod

Examples

data(tra)
lm.mod1 <- moderate.lm(beliefs, values, attitudes, tra)
summary(lm.mod1)

Tests for Multivariate Normality

Description

Returns tests for multivariate Skewness and kurtosis as well as Mahalanobis' D-squared.

Usage

mult.norm(x, s = var(x), chicrit = 0.005)

Arguments

Details

Tests for multivariate skewness and kurtosis were adapted from SAS macros in Khatree & Naik (1999). They attribute the formula to Mardia (1970; 1974). Mahalanobis' Dsq is based on Mahalanobis (1936). Dsq is multivariate analogue to z scores, but based on the chi-sq distribution rather than normal distribution. Once can specify at what level one wishes to define multivariate outliers (e.g., .005, .001)

Value

A list containing the following:

Note

Mahalanobis is returned without regard to NAs (missing observations) and is useful only in detecting IF multivariate outliers are present. If one wishes to determine which cases are multivariate outliers and if one has missing observations, mahalanobis is perhaps a better choice.

These statistics are known to be susceptible to sample size (as in their univariate counterparts). One should always use graphical methods such as qqplot in addition to statistical.

Author(s)

Thomas D. Fletcher t.d.fletcher05@gmail.com

References

Khattree, R. & Naik, D. N. (1999). Applied multivariate statistics with SAS software (2nd ed.). Cary, NC: SAS Institute Inc.

See Also

mahalanobis, qqplot

Examples

# assess the multivariate normality of variables 4,5,6 in USJudgeRatings
data(USJudgeRatings)	
mn <- mult.norm(USJudgeRatings[,4:6],chicrit=.001)
mn

mn$Dsq > mn$CriticalDsq

Skewness and Kurtosis

Description

Computes Skewness and Kurtosis of data.

Usage

norm(x)
Skew(x)
Kurt(x)

Arguments

Value

norm returns a table containing Skew & Kurt. Each contain the following elements:

Warning

These statistics should be used with caution as they are influenced by sample size!

Author(s)

Thomas D. Fletcher t.d.fletcher05@gmail.com

Examples

# create negatively skewed dat with 100 observations
xc <-  -rchisq(100,3)	
norm(xc)

Normal Density Plot

Description

Plots the density distribution of 'x' (e.g., smoothed histogram) with an overlaying normal density plot with the same mean and SD. This is useful for 'seeing' the degree of deviance from normality.

Usage

plotNormX(x)
plotNormXm(x, im)

Arguments

Details

plotNormX is useful for single use (univariate) objects, but plotNormXm is more useful for creating multiple graphs (i.e., multivariate) as in sending graphs to a postcript or pdf device. See examples below.

Value

A graph of density of x.

Author(s)

Thomas D. Fletcher t.d.fletcher05@gmail.com

See Also

eda.uni

Examples

#	plot.normX	
data(USJudgeRatings)	# data packaged with R
plotNormX(USJudgeRatings$CONT) 

# plot all 12 variables in USJudgeRatings, using function plotNormXm

data(USJudgeRatings)	
plotNormXm(USJudgeRatings, 12) 

Power in F distribution

Description

Computes power (1 - beta) to detect an effect with a given effect size, sample size (df) and specified alpha (significance) level.

Usage

powerF(PV, df2, df1 = 1, alpha = 0.05)

Arguments

Details

Murphy & Myors (2004) detail the use of a similar function and the notion that most distributions can be converted to F. Therefore, they argue that the F distribution is the most versatile in computing power. Typically, alpha is set at .05 (default). Users will likely find conversions of various distributions to F corresponding to a df1=1 (default). Therefore, users can manipulate df2 based on their model to estiamte sample size needs. Likewise, one may begin with a given sample size (i.e., df2) and manipulate PV (effect size) to iteratively determine what power their study is likely to detect. Conventions maintain that .80 is a sufficient target, and that no study shold be designed with power = .5 or less.

Value

A numeric value representing the power to detect the effect

Warning

It is critical that the user correctly specify the model for which the effect is obtained. For instance, if a single coeficient from a regression model is the object of inquiry (e.g., interaction effect in moderation model), the DF should reflect that effect and not the overall model, which also contains the 'main effects'.

Author(s)

Thomas D. Fletcher t.d.fletcher05@gmail.com

References

Murphy, K. R., & Myors, B. (2004). Statistical power analysis: A simple and general model for traditional and modern hypothesis tests (2nd ed.). Mahwah, NJ: Lawrence Erlbaum Associates.

Examples

# Simulated TRA example
data(tra)
lm1 <- lm (attitudes ~ beliefs*values, tra)
summary(lm1) 
# power to detect the interaction effect, where df1 = 1 and df2 = n-k-1 = 996
# PV = t^2/(t^2+df2) = .1863
powerF(.1863, 996)

# Estimate sample size needed to detect interaction effect with PV = .01 and power = .8
powerF(.01, 200)  # too low
powerF(.01, 1000) # too high
powerF(.01, (800-3-1))  # just right: n=800 - k=3 - 1

Simple Mediation for use in Bootstrapping

Description

Calculates the indirect effect from proximal.med in a form useful to send to boot

Usage

proxInd.ef(data, i)

Arguments

Details

This function is not useful of itself. It is specifically created as an intermediate step in bootstrapping the indirect effect.

Value

indirect effect that is passed to boot for each bootstrap sample

Author(s)

Thomas D. Fletcher t.d.fletcher05@gmail.com

References

Davison, A. C. & Hinkley, D. V. (1997). Bootstrap methods and their application. Cambridge, UK: Cambridge University Press.

MacKinnon, D. P., Lockwood, C. M., Hoffman, J. M., West, S. G., & Sheets, V. (2002). A comparison of methods to test mediation and other intervening variable effects. Psychological Methods, 7, 83-104.

MacKinnon, D. P., Lockwood, C. M., & Williams, J. (2004). Confidence limit for indirect effect: distribution of the product and resampling methods. Multivariate Behavioral Research, 39, 99-128.

See Also

proximal.med

Examples

require(boot)
data(tra)
tmp.tra <- tra
names(tmp.tra) <- c('x','z','m','y')
med.boot <- boot(tmp.tra, proxInd.ef, R=999)
sort(med.boot$t)[c(25,975)]	#95% CI
plot(density(med.boot$t))	# Distribution of bootstapped indirect effect
summary(med.boot$t)

Simple Mediation Models

Description

Computes the Indirect Effect for a simple 3 variable mediation model: X -> M -> Y assuming direct effect X -> Y

Usage

proximal.med(data)

Arguments

Details

This function computes all paths in the simple 3 variable system involving the following regressions:
Y = t'X + bM, and
M = aX
where t' + ab = t

The indirect effect is computed as the product of a*b. Several formula are used for the computation of the standard error for the indirect effect (see MacKinnon et al for a comprehensive review).
As noted below, one can use this function to create the indirect effect and then utilize bootstrapping for a more accurate estimate of the standard error and model the distribution of the direct effect.

Value

Creates a table containing the following effects, their standard errors, and t-values :

Warning

This function is primative in that it is based on a simplistic model AND forces the user to name the variables in the dataset x, m, and y.

Note

This function uses the following undocumented functions: aroian.se.indirect2, goodman.se.indirect2, se.indirect2

Author(s)

Thomas D. Fletcher t.d.fletcher05@gmail.com

References

MacKinnon, D. P., Lockwood, C. M., Hoffman, J. M., West, S. G., & Sheets, V. (2002). A comparison of methods to test mediation and other intervening variable effects. Psychological Methods, 7, 83-104.

See Also

distal.med, proxInd.ef

Examples

data(tra)
tmp.tra <- tra
names(tmp.tra) <- c('x','z','m','y')
data.frame(proximal.med(tmp.tra))  ## data.frame() simple makes the table 'pretty'

Moderated Simple Slopes

Description

Computes the simple slopes for a moderated regression model.

Usage

sim.slopes(mod, z, zsd = 1, mcz = FALSE)

Arguments

Details

Constructs the simple slopes for arbitrary values of z (e.g., +/- 1, 2, 3 standard deviations) involved in a moderated multiple regression equation.

Value

A table with the following values for zHigh (Meanz + zsd*SDz), Mean(Meanz), and zLow (Meanz - zsd*SDz):

Author(s)

Thomas D. Fletcher t.d.fletcher05@gmail.com

References

Aiken, L. S., & West, S. G. (1991). Multiple regression: Testing and interpreting interactions. Newbury Park: Sage Publications.

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

See Also

moderate.lm, graph.mod

Examples

data(tra)
lm.mod1 <- moderate.lm(beliefs, values, attitudes, tra)
ss.mod1 <- sim.slopes(lm.mod1, tra$values)
ss.mod1

Simulated Theory of Reasoned Action Data

Description

Simulated data loosely based on the Theory of Reasoned Action from Social Psychology. Similar data are frequently used in publications involving moderation and mediation. These data were created for illustrative purposes.

Usage

data(tra)

Format

A data frame with 1000 observations on the following 4 variables.

beliefs

a numeric vector

values

a numeric vector

attitudes

a numeric vector

intentions

a numeric vector

Details

The data were constructed with the following model(s) in mind. Attitudes are a function of beliefs as moderated by values. Beliefs lead to intentions though their association with attitudes.

Examples

data(tra)
str(tra)
eda.uni(tra$intentions)