Bollen's Data on Industrialization and Political Democracy

Description

This data set includes four measures of democracy at two points in time, 1960 and 1965, and three measures of industrialization in 1960, for 75 developing countries.

Usage

Bollen

Format

A data frame with 75 observations on the following 11 variables.

y1

freedom of the press, 1960

y2

freedom of political opposition, 1960

y3

fairness of elections, 1960

y4

effectivness of elected legislature, 1960

y5

freedom of the press, 1965

y6

freedom of political opposition, 1965

y7

fairness of elections, 1965

y8

effectivness of elected legislature, 1965

x1

GNP per capita, 1960

x2

energy consumption per capita, 1960

x3

percentage of labor force in industry, 1960

Details

Variables y1 through y4 are intended to be indicators of the latent variable political democracy in 1960; y5 through y8 indicators of political democracy in 1965; and x1 through x3 indicators of industrialization in 1960.

Source

personal communication from Ken Bollen.

References

Bollen, K. A. (1989) Structural Equations With Latent Variables. Wiley.

Variables from the 1997 Canadian National Election Study

Description

These variables are from the mailback questionnaire to the 1997 Canadian National Election Study, and are intended to tap attitude towards “traditional values.”

Usage

CNES

Format

A data frame with 1529 observations on the following 4 variables.

MBSA2

an ordered factor with levels StronglyDisagree, Disagree, Agree, and StronglyAgree, in response to the statement, “We should be more tolerant of people who choose to live according to their own standards, even if they are very different from our own.”

MBSA7

an ordered factor with levels StronglyDisagree, Disagree, Agree, and StronglyAgree, in response to the statement, “Newer lifestyles are contributing to the breakdown of our society.”

MBSA8

an ordered factor with levels StronglyDisagree, Disagree, Agree, and StronglyAgree, in response to the statement, “The world is always changing and we should adapt our view of moral behaviour to these changes.”

MBSA9

an ordered factor with levels StronglyDisagree, Disagree, Agree, and StronglyAgree, in response to the statement, “This country would have many fewer problems if there were more emphasis on traditional family values.”

Source

York University Institute for Social Research.

Holizinger and Swineford's Data

Description

This data set, for scores on a variety of tests, was originally in the MBESS package. A new version of the data set in that package doesn't appear to be identical to this one.

Usage

HS.data

Format

A data frame with 301 observations on the following 32 variables.

id

a numeric vector

Gender

a factor with levels Female Male

grade

a numeric vector

agey

a numeric vector

agem

a numeric vector

school

a factor with levels Grant-White Pasteur

visual

a numeric vector

cubes

a numeric vector

paper

a numeric vector

flags

a numeric vector

general

a numeric vector

paragrap

a numeric vector

sentence

a numeric vector

wordc

a numeric vector

wordm

a numeric vector

addition

a numeric vector

code

a numeric vector

counting

a numeric vector

straight

a numeric vector

wordr

a numeric vector

numberr

a numeric vector

figurer

a numeric vector

object

a numeric vector

numberf

a numeric vector

figurew

a numeric vector

deduct

a numeric vector

numeric

a numeric vector

problemr

a numeric vector

series

a numeric vector

arithmet

a numeric vector

paperrev

a numeric vector

flagssub

a numeric vector

Source

Originally from Holzinger and Swineford (1939). This copy is originally from version 4.6.0 of the MBESS package.

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

summary(HS.data)

Klein's Data on the U. S. Economy

Description

Data for Klein's (1950) simple econometric model of the U. S. economy.

The Klein data frame has 22 rows and 10 columns.

Usage

Klein

Format

This data frame contains the following columns:

Year

1921–1941

C

consumption.

P

private profits.

Wp

private wages.

I

investment.

K.lag

capital stock, lagged one year.

X

equilibrium demand.

Wg

government wages.

G

government non-wage spending.

T

indirect business taxes and net exports.

Source

Greene, W. H. (1993) Econometric Analysis, Second Edition. Macmillan.

References

Klein, L. (1950) Economic Fluctuations in the United States 1921–1941. Wiley.

Examples

Klein$P.lag <- c(NA, Klein$P[-22])
Klein$X.lag <- c(NA, Klein$X[-22])

summary(tsls(C ~ P + P.lag + I(Wp + Wg), 
    instruments=~1 + G + T + Wg + I(Year - 1931) + K.lag + P.lag + X.lag,
    data=Klein))
    
summary(tsls(I ~ P + P.lag + K.lag,
    instruments=~1 + G + T + Wg + I(Year - 1931) + K.lag + P.lag + X.lag,
    data=Klein))
    
summary(tsls(Wp ~ X + X.lag + I(Year - 1931),
    instruments=~1 + G + T + Wg + I(Year - 1931) + K.lag + P.lag + X.lag,
    data=Klein))

Partly Artificial Data on the U. S. Economy

Description

These are partly contrived data from Kmenta (1986), constructed to illustrate estimation of a simultaneous-equation model.

The Kmenta data frame has 20 rows and 5 columns.

Usage

Kmenta

Format

This data frame contains the following columns:

Q

food consumption per capita.

P

ratio of food prices to general consumer prices.

D

disposable income in constant dollars.

F

ratio of preceding year's prices received by farmers to general consumer prices.

A

time in years.

Details

The exogenous variables D, F, and A are based on real data; the endogenous variables P and Q were generated by simulation.

Source

Kmenta, J. (1986) Elements of Econometrics, Second Edition, Macmillan.

Methods for sem Objects Fit Using the objectiveML, objectiveGLS, objectiveFIML, msemObjectiveML, and msemObjectiveGLS Objective Functions

Description

These functions are for objects fit by sem using the objectiveML (multivariate-normal full-information maximum-likelihood), link{objectiveFIML} (multivariate-normal full-information maximum-likihood in the presence of missing data), objectiveGLS (generalized least squares), and msemObjectiveML (multigroup multivariate-normal FIML) objective functions.

Usage

## S3 method for class 'objectiveML'
anova(object, model.2, robust=FALSE, ...)
## S3 method for class 'objectiveFIML'
anova(object, model.2, ...)

## S3 method for class 'objectiveML'
logLik(object, ...)
## S3 method for class 'objectiveFIML'
logLik(object, saturated=FALSE, 
    intercept="Intercept", iterlim=1000, ...)
## S3 method for class 'objectiveML'
deviance(object, ...)
## S3 method for class 'objectiveFIML'
deviance(object, saturated.logLik, ...)
## S3 method for class 'msemObjectiveML'
deviance(object, ...) 
## S3 method for class 'objectiveML'
AIC(object, ..., k)
## S3 method for class 'objectiveFIML'
AIC(object, saturated.logLik, ..., k)
## S3 method for class 'msemObjectiveML'
AIC(object, ..., k)
## S3 method for class 'objectiveML'
AICc(object, ...)
## S3 method for class 'objectiveFIML'
AICc(object, saturated.logLik, ...)
## S3 method for class 'msemObjectiveML'
AICc(object, ...)
## S3 method for class 'objectiveML'
BIC(object, ...)
## S3 method for class 'objectiveFIML'
BIC(object, saturated.logLik, ...)
## S3 method for class 'msemObjectiveML'
BIC(object, ...)
## S3 method for class 'objectiveML'
CAIC(object, ...)
## S3 method for class 'objectiveFIML'
CAIC(object, saturated.logLik, ...)

## S3 method for class 'objectiveML'
print(x, ...)
## S3 method for class 'objectiveGLS'
print(x, ...)
## S3 method for class 'objectiveFIML'
print(x, saturated=FALSE, ...)
## S3 method for class 'msemObjectiveML'
print(x, ...)
## S3 method for class 'msemObjectiveGLS'
print(x, ...)

## S3 method for class 'objectiveML'
summary(object, digits=getOption("digits"), 
    conf.level=.90, robust=FALSE, analytic.se=object$t <= 500, 
    fit.indices=c("GFI", "AGFI", "RMSEA", "NFI", "NNFI", "CFI", "RNI",
      "IFI", "SRMR", "AIC", "AICc", "BIC", "CAIC"), ...)
## S3 method for class 'objectiveFIML'
summary(object, digits=getOption("digits"), conf.level=.90,
    fit.indices=c("AIC", "AICc", "BIC", "CAIC"),
    saturated=FALSE, intercept="Intercept", saturated.logLik, ...)
## S3 method for class 'objectiveGLS'
summary(object, digits=getOption("digits"), conf.level=.90, 
    fit.indices=c("GFI", "AGFI", "RMSEA", "NFI", "NNFI", "CFI", "RNI", "IFI", "SRMR"),
    ...)
## S3 method for class 'msemObjectiveML'
summary(object, digits=getOption("digits"), 
    conf.level=.90, robust=FALSE, 
    analytic.se=object$t <= 500,
    fit.indices=c("GFI", "AGFI", "RMSEA", "NFI", "NNFI", "CFI", "RNI", 
      "IFI", "SRMR", "AIC", "AICc", "BIC"), ...)
## S3 method for class 'msemObjectiveGLS'
summary(object, digits=getOption("digits"), 
    conf.level=.90,
    fit.indices=c("GFI", "AGFI", "RMSEA", "NFI", "NNFI", 
      "CFI", "RNI", "IFI", "SRMR"), ...)

Arguments

Author(s)

John Fox jfox@mcmaster.ca and Jarrett Byrnes

References

See sem.

See Also

sem, objective.functions, modIndices.objectiveML

Six Mental Tests

Description

These data are from the SAS manual and consist of six mental tests for 32 students, with some missing data. The three x variables are intended to load on a verbal factor, and the three y variables on a math factor. The data can be used to illustrate the estimation of a confirmatory factor analysis model by multinormal full-information maximum-likelihood in the presence of missing data.

Usage

Tests

Format

A data frame with 32 observations on the following 6 variables.

x1

score on verbal test 1.

x2

score on verbal test 2.

x3

score on verbal test 3.

y1

score on math test 1.

y2

score on math test 2.

y3

score on math test 3.

Source

Example 25.13 from SAS/STAT 9.22 User's Guide, SAS Institute, 2010.

Bootstrap a Structural Equation Model

Description

Bootstraps a structural equation model in an sem object (as returned by the sem function).

Usage

bootSem(model, ...)

## S3 method for class 'sem'
bootSem(model, R=100, Cov=cov, data=model$data, 
    max.failures=10, show.progress=TRUE, ...)

## S3 method for class 'msem'
bootSem(model, R=100, Cov=cov, data=model$data, 
    max.failures=10, show.progress=TRUE, ...)

## S3 method for class 'bootsem'
print(x, digits=getOption("digits"), ...)

## S3 method for class 'bootsem'
summary(object,
    type=c("perc", "bca", "norm", "basic", "none"), level=0.95, ...)

Arguments

Details

bootSem implements the nonparametric bootstrap, assuming an independent random sample. Convergence failures in the bootstrap resamples are discarded (and a warning printed); more than max.failures consecutive convergence failures (default, 10) result in an error. You can use the boot function in the boot package for more complex sampling schemes and additional options.

Bootstrapping is implemented by resampling the observations in data, recalculating the input covariance matrix with Cov, and refitting the model with sem, using the parameter estimates from the original sample as start-values.

Warning: the bootstrapping process can be very time-consuming.

Value

bootSem returns an object of class bootsem, which inherits from class boot, supported by the boot package. The returned object contains the following components:

Author(s)

John Fox jfox@mcmaster.ca

References

Davison, A. C., and Hinkley, D. V. (1997) Bootstrap Methods and their Application. Cambridge.

Efron, B., and Tibshirani, R. J. (1993) An Introduction to the Bootstrap. Chapman and Hall.

See Also

boot, sem

Examples

    ## Not run:  # because of long execution time

# A simple confirmatory factor-analysis model using polychoric correlations.
#  The polycor package is required for the hetcor function.

if (require(polycor)){

# The following function returns correlations computed by hetcor,
#   and is used for the bootstrapping.

hcor <- function(data) hetcor(data, std.err=FALSE)$correlations

model.cnes <- specifyModel(text="
F -> MBSA2, lam1
F -> MBSA7, lam2
F -> MBSA8, lam3
F -> MBSA9, lam4
F <-> F, NA, 1
MBSA2 <-> MBSA2, the1
MBSA7 <-> MBSA7, the2
MBSA8 <-> MBSA8, the3
MBSA9 <-> MBSA9, the4
")

R.cnes <- hcor(CNES)

sem.cnes <- sem(model.cnes, R.cnes, N=1529)
summary(sem.cnes)
}

#  Note: this can take a minute:

set.seed(12345) # for reproducibility

system.time(boot.cnes <- bootSem(sem.cnes, R=100, Cov=hcor, data=CNES))
summary(boot.cnes, type="norm")  
# cf., standard errors to those computed by summary(sem.cnes)
    
## End(Not run)
    
    ## Not run:   # because of long execution time

# An example bootstrapping a multi-group model

mod.hs <- cfa(text="
spatial: visual, cubes, paper, flags
verbal: general, paragrap, sentence, wordc, wordm
memory: wordr, numberr, figurer, object, numberf, figurew
math: deduct, numeric, problemr, series, arithmet
")

mod.mg <- multigroupModel(mod.hs, groups=c("Female", "Male")) 

sem.mg <- sem(mod.mg, data=HS.data, group="Gender",
              formula = ~ visual + cubes + paper + flags +
              general + paragrap + sentence + wordc + wordm +
              wordr + numberr + figurer + object + numberf + figurew +
              deduct + numeric + problemr + series + arithmet
              )

# Note: this example can take several minutes or more;
#       you can decrease R if you just want to see how it works:

set.seed(12345) # for reproducibility

system.time(boot.mg <- bootSem(sem.mg, R=100))
summary(boot.mg, type="norm")
# cf., standard errors to those computed by summary(sem.mg)
    
## End(Not run)

Total, Direct, and Indirect Effects for Structural Equation Models

Description

The sem method for the standard generic function effects computes total, direct, and indirect effects for a fitted structural equation model according to the method described in Fox (1980).

Usage

## S3 method for class 'sem'
effects(object, ...)
## S3 method for class 'msem'
effects(object, ...)

## S3 method for class 'semeffects'
print(x, digits = getOption("digits"), ...)
## S3 method for class 'semeffectsList'
print(x, digits = getOption("digits"), ...)

Arguments

Value

effect.sem returns an object of class semeffects with Total, Direct, and Indirect elements.

Author(s)

John Fox jfox@mcmaster.ca

References

Fox, J. (1980) Effect analysis in structural equation models: Extensions and simplified methods of computation. Sociological Methods and Research 9, 3–28.

See Also

sem

Examples

    ## Not run: 

# These examples are from Fox (1980)

# In the first pair of examples, readMoments() and specifyModel() read from the
# input stream. These examples cannot be executed via example() but can be entered
# at the command prompt. The Blau and Duncan example is repeated using file input;
# this example can be executed via example(). 

# The recursive Blau and Duncan basic stratification model:
#  x1 is father's education, x2 father's SES, y3 respondent's education,
#  y4 SES of respondent's first job, y5 respondent's SES in 1962

R.bd <- readMoments(names=c("x1", "x2", "y3", "y4", "y5"))
1
.516 1
.453 .438 1
.332 .417 .538 1
.322 .405 .596 .541 1

mod.bd <- specifyModel()
y3 <- x1, gam31
y3 <- x2, gam32
y4 <- x2, gam42
y4 <- y3, beta43
y5 <- x2, gam52
y5 <- y3, beta53
y5 <- y4, beta54

sem.bd <- sem(mod.bd, R.bd, N=20700, fixed.x=c("x1", "x2"))
summary(sem.bd)
effects(sem.bd)


# The nonrecursive Duncan, Haller, and Portes peer-influences model for observed variables:

R.DHP <- readMoments(diag=FALSE, names=c("ROccAsp", "REdAsp", "FOccAsp", 
"FEdAsp", "RParAsp", "RIQ", "RSES", "FSES", "FIQ", "FParAsp"))
.6247     
.3269  .3669       
.4216  .3275  .6404
.2137  .2742  .1124  .0839
.4105  .4043  .2903  .2598  .1839
.3240  .4047  .3054  .2786  .0489  .2220
.2930  .2407  .4105  .3607  .0186  .1861  .2707
.2995  .2863  .5191  .5007  .0782  .3355  .2302  .2950
.0760  .0702  .2784  .1988  .1147  .1021  .0931 -.0438  .2087

model.dhp <- specifyModel()
RIQ      -> ROccAsp, gam51,  NA
RSES     -> ROccAsp, gam52,  NA
FSES     -> FOccAsp, gam63,  NA
FIQ      -> FOccAsp, gam64,  NA
FOccAsp  -> ROccAsp, beta56, NA
ROccAsp  -> FOccAsp, beta65, NA
ROccAsp <-> ROccAsp, ps55,   NA
FOccAsp <-> FOccAsp, ps66,   NA
ROccAsp <-> FOccAsp, ps56,   NA


# Note: The following generates a warning because not all of the variables
#       in the correlation matrix are used
sem.dhp <- sem(model.dhp, R.DHP, 329,
                fixed.x=c('RIQ', 'RSES', 'FSES', 'FIQ'))
summary(sem.dhp)
effects(sem.dhp)
    
## End(Not run)
    
## the following example may be executed via example()

etc <- system.file(package="sem", "etc") # path to data and model files

# The recursive Blau and Duncan basic stratification model:
#  x1 is father's education, x2 father's SES, y3 respondent's education,
#  y4 SES of respondent's first job, y5 respondent's SES in 1962

(R.bd <- readMoments(file=file.path(etc, "R-Blau-Duncan.txt"),
					names=c("x1", "x2", "y3", "y4", "y5")))
(mod.bd <- specifyModel(file=file.path(etc, "model-Blau-Duncan.txt")))
sem.bd <- sem(mod.bd, R.bd, N=20700, fixed.x=c("x1", "x2"))
summary(sem.bd)
effects(sem.bd)

Factor Scores for Latent Variables

Description

Calculate factor scores or factor-score coefficients for the latent variables in a structural-equation model.

Usage

## S3 method for class 'sem'
fscores(model, data=model$data, center=TRUE, scale=FALSE, ...)
## S3 method for class 'msem'
fscores(model, data=model$data, center=TRUE, scale=FALSE, ...)

Arguments

Details

Factor-score coefficients are computed by the “regression” method as B = C^{-1} C^{*}, where C is the model-implied covariance or moment matrix among the observed variables and C^{*} is the matrix of model-implied covariances or moments between the observed and latent variables.

Value

Either a matrix of estimated factor scores (if the data argument is supplied) or a matrix of factor-score coefficients (otherwise). In the case of an "msem" argument, a list of matrices is returned.

Author(s)

John Fox jfox@mcmaster.ca

References

Bollen, K. A. (1989) Structural Equations With Latent Variables. Wiley.

See Also

sem, scale

Examples

# In the first example, readMoments() and specifyModel() read from the
# input stream. This example cannot be executed via example() but can be entered
# at the command prompt. The example is repeated using file input;
# this example can be executed via example(). 
	   ## Not run: 

S.wh <- readMoments(names=c('Anomia67','Powerless67','Anomia71',
                                    'Powerless71','Education','SEI'))
   11.834                                    
    6.947    9.364                            
    6.819    5.091   12.532                    
    4.783    5.028    7.495    9.986            
   -3.839   -3.889   -3.841   -3.625   9.610     
  -21.899  -18.831  -21.748  -18.775  35.522  450.288

# This model in the SAS manual for PROC CALIS

model.wh.1 <- specifyModel()
    Alienation67   ->  Anomia67,      NA,     1
    Alienation67   ->  Powerless67,   NA,     0.833
    Alienation71   ->  Anomia71,      NA,     1
    Alienation71   ->  Powerless71,   NA,     0.833 
    SES            ->  Education,     NA,     1     
    SES            ->  SEI,           lamb,   NA
    SES            ->  Alienation67,  gam1,   NA
    Alienation67   ->  Alienation71,  beta,   NA
    SES            ->  Alienation71,  gam2,   NA
    Anomia67       <-> Anomia67,      the1,   NA
    Anomia71       <-> Anomia71,      the1,   NA
    Powerless67    <-> Powerless67,   the2,   NA
    Powerless71    <-> Powerless71,   the2,   NA
    Education      <-> Education,     the3,   NA
    SEI            <-> SEI,           the4,   NA
    Anomia67       <-> Anomia71,      the5,   NA
    Powerless67    <-> Powerless71,   the5,   NA
    Alienation67   <-> Alienation67,  psi1,   NA
    Alienation71   <-> Alienation71,  psi2,   NA
    SES            <-> SES,           phi,    NA
    
                        
sem.wh.1 <- sem(model.wh.1, S.wh, 932)

fscores(sem.wh.1)
   
## End(Not run)

# The following example can be executed via example():

etc <- system.file(package="sem", "etc") # path to data and model files
   
(S.wh <- readMoments(file=file.path(etc, "S-Wheaton.txt"),
					names=c('Anomia67','Powerless67','Anomia71',
                            'Powerless71','Education','SEI')))
(model.wh.1 <- specifyModel(file=file.path(etc, "model-Wheaton-1.txt")))        
(sem.wh.1 <- sem(model.wh.1, S.wh, 932))
fscores(sem.wh.1)

Additional Information Criteria

Description

These are generic functions for computing, respectively, the AICc (second-order corrected Akaike Information Criterion) and CAIC (consistent Akaike Information Criterion).

Usage

AICc(object, ...)

CAIC(object, ...)

Arguments

Author(s)

Jarrett Byrnes and John Fox jfox@mcmaster.ca

References

Burnham, K. P., and Anderson, D. R. (1998) Model Selection and Inference: A Practical Information-Theoretical Approach. Springer.

Bozdogan, H. (1987) Model selection and Akaike's information criterion (AIC). Psychometrika 52, 345–370.

See Also

AICc.objectiveML, CAIC.objectiveML

Estimate a Structural Equation Model By Multiple Imputation

Description

miSem uses the mi function in the mi package to generate multiple imputations of missing data, fitting the specified model to each completed data set.

Usage

miSem(model, ...)

## S3 method for class 'semmod'
miSem(model, ..., data, formula = ~., raw=FALSE, 
        fixed.x=NULL, objective=objectiveML,
        n.imp=5, n.chains=n.imp, n.iter=30, seed=sample(1e6, 1), mi.args=list(), 
        show.progress=TRUE)
    
## S3 method for class 'semmodList'
miSem(model, ..., data, formula = ~., group, raw=FALSE, 
        fixed.x=NULL, objective=msemObjectiveML,
        n.imp=5, n.chains=n.imp, n.iter=30, seed=sample(1e6, 1), mi.args=list(),
        show.progress=TRUE)

## S3 method for class 'miSem'
print(x, ...)

## S3 method for class 'miSem'
summary(object, digits=max(3, getOption("digits") - 2), ...)

Arguments

Value

miSem returns an object of class "miSem" with the following components:

Author(s)

John Fox jfox@mcmaster.ca

References

Yu-Sung Su, Andrew Gelman, Jennifer Hill, Masanao Yajima. (2011). “Multiple imputation with diagnostics (mi) in R: Opening windows into the black box.” Journal of Statistical Software 45(2).

See Also

sem, mi

Examples

    ## Not run:  # because of long execution time
mod.cfa.tests <- cfa(raw=TRUE, text="
verbal: x1, x2, x3
math: y1, y2, y3
")
imps <- miSem(mod.cfa.tests, data=Tests, fixed.x="Intercept", 
              raw=TRUE, seed=12345)
summary(imps, digits=3) 


# introduce some missing data to the HS.data data set:
HS <- HS.data[, c(2,7:10,11:15,20:25,26:30)]
set.seed(12345)
r <- sample(301, 100, replace=TRUE)
c <- sample(2:21, 100, replace=TRUE)
for (i in 1:100) HS[r[i], c[i]] <- NA

mod.hs <- cfa(text="
spatial: visual, cubes, paper, flags
verbal: general, paragrap, sentence, wordc, wordm
memory: wordr, numberr, figurer, object, numberf, figurew
math: deduct, numeric, problemr, series, arithmet
")

mod.mg <- multigroupModel(mod.hs, groups=c("Female", "Male")) 
system.time( # relatively time-consuming!
  imps.mg <- miSem(mod.mg, data=HS, group="Gender", seed=12345)
)
summary(imps.mg, digits=3)
    
## End(Not run)

Modification Indices for Structural Equation Models

Description

mod.indices calculates modification indices (score tests) and estimated parameter changes for the fixed and constrained parameters in a structural equation model fit by multinormal maximum likelihood.

Usage

## S3 method for class 'objectiveML'
modIndices(model, duplicated, deviance=NULL, ...)
## S3 method for class 'msemObjectiveML'
modIndices(model, ...)

## S3 method for class 'modIndices'
print(x, n.largest=5, ...)
## S3 method for class 'msemModIndices'
print(x, ...)

## S3 method for class 'modIndices'
summary(object, round=2, 
    print.matrices=c("both", "par.change", "mod.indices"), ...)
## S3 method for class 'msemModIndices'
summary(object, ...)

Arguments

Details

Modification indices are one-df chi-square score (“Lagrange-multiplier”) test statistics for the fixed and constrained parameters in a structural equation model. They may be regarded as an estimate of the improvement in the likelihood-ratio chi-square statistic for the model if the corresponding parameter is respecified as a free parameter. The modIndices function also estimates the change in the value of a fixed or constrained parameter if the parameter is respecified as free. When several parameters are set equal, modification indices and estimated changes are given for all but the first. Modification indices and estimated parameter changes for currently free parameters are given as NA.

The method employed is described in Saris, Satorra, and Sorbom (1987) and Sorbom (1989).

Value

modIndices returns an object of class modIndices with the following elements:

Author(s)

John Fox jfox@mcmaster.ca and Michael Culbertson

References

Sarris, W. E., Satorra, A., and Sorbom, D. (1987) The detection and correction of specification errors in structural equation models. Pp. 105–129 in Clogg, C. C. (ed.), Sociological Methodology 1987. American Sociological Association.

Sorbom, D. (1989) Model modification. Psychometrika 54, 371–384.

See Also

sem

Examples

# In the first example, readMoments() and specifyModel() read from the
# input stream. This example cannot be executed via example() but can be entered
# at the command prompt. The example is repeated using file input;
# this example can be executed via example(). 
	## Not run: 
# This example is adapted from the SAS manual

S.wh <- readMoments(names=c('Anomia67','Powerless67','Anomia71',
                                    'Powerless71','Education','SEI'))
   11.834                                    
    6.947    9.364                            
    6.819    5.091   12.532                    
    4.783    5.028    7.495    9.986            
   -3.839   -3.889   -3.841   -3.625   9.610     
  -21.899  -18.831  -21.748  -18.775  35.522  450.288

model.wh <- specifyModel()
    Alienation67   ->  Anomia67,      NA,   1
    Alienation67   ->  Powerless67,   NA,   0.833
    Alienation71   ->  Anomia71,      NA,   1
    Alienation71   ->  Powerless71,   NA,   0.833
    SES            ->  Education,     NA,   1     
    SES            ->  SEI,           lamb, NA
    SES            ->  Alienation67,  gam1, NA
    Alienation67   ->  Alienation71,  beta, NA
    SES            ->  Alienation71,  gam2, NA
    Anomia67       <-> Anomia67,      the1, NA
    Anomia71       <-> Anomia71,      the1, NA
    Powerless67    <-> Powerless67,   the2, NA
    Powerless71    <-> Powerless71,   the2, NA
    Education      <-> Education,     the3, NA
    SEI            <-> SEI,           the4, NA
    Anomia67       <-> Anomia71,      the5, NA
    Powerless67    <-> Powerless71,   the5, NA
    Alienation67   <-> Alienation67,  psi1, NA
    Alienation71   <-> Alienation71,  psi2, NA
    SES            <-> SES,           phi,  NA

sem.wh <- sem(model.wh, S.wh, 932)
modIndices(sem.wh)
	
## End(Not run)
	
# The following example can be executed via example():

etc <- system.file(package="sem", "etc") # path to data and model files

(S.wh <- readMoments(file=file.path(etc, "S-Wheaton.txt"),
					names=c('Anomia67','Powerless67','Anomia71',
                            'Powerless71','Education','SEI')))
(model.wh <- specifyModel(file=file.path(etc, "model-Wheaton-1.txt")))                    
(sem.wh <- sem(model.wh, S.wh, 932))
modIndices(sem.wh)

sem Objective-Function Builders

Description

These functions return objective functions suitable for use with optimizers called by sem. The user would not normally call these functions directly, but rather supply one of them in the objective argument to sem. Users may also write their own objective functions. objectiveML and objectiveML2 are for multinormal maximum-likelihood estimation; objectiveGLS and objectiveGLS2 are for generalized least squares; and objectiveFIML2 is for so-called “full-information maximum-likelihood” estimation in the presence of missing data. The FIML estimator provides the same estimates as the ML estimator when there is no missing data; it can be slow because it iterates over the unique patterns of missing data that occur in the data set. objectiveML and objectiveGLS use compiled code and are therefore substantially faster. objectiveML2 and objectiveGLS2 are provided primarily to illustrate how to write sem objective functions in R. msemObjectiveML uses compiled code is for fitting multi-group models by multinormal maximum likelihood; msemObjectiveML2 is similar but doesn't use compiled code. msemObjectiveGLS uses compiled code and is for fitting multi-group models by generalized least squares.

Usage

objectiveML(gradient=TRUE, hessian=FALSE)
objectiveML2(gradient=TRUE)

objectiveGLS(gradient=FALSE)
objectiveGLS2(gradient=FALSE)

objectiveFIML(gradient=TRUE, hessian=FALSE)
objectiveFIML2(gradient=TRUE, hessian=FALSE)

msemObjectiveML(gradient=TRUE)
msemObjectiveML2(gradient=TRUE)

msemObjectiveGLS(gradient=FALSE)

Arguments

Value

These functions return an object of class "semObjective", with up to two elements:

Author(s)

John Fox jfox@mcmaster.ca

References

See sem.

See Also

sem, optimizers

sem Optimizers

Description

The default optimizer used by sem is optimizerSem, which employs compiled code and is integrated with the objectiveML and objectiveGLS objective functions; optimizerSem, written by Zhenghua Nie, is a modified version of the standard R nlm optimizer, which was written by Saikat DebRoy, R-core, and Richard H. Jones. The other functions call optimizers (nlm, optim, or nlminb), to fit structural equation models, and are called by the sem function. The user would not normally call these functions directly, but rather supply one of them in the optimizer argument to sem. Users may also write them own optimizer functions. msemOptimizerNlm is for fitting multigroup models, and also adapts the nlm code.

Usage

optimizerSem(start, objective=objectiveML,  
	gradient=TRUE, maxiter, debug, par.size, model.description, warn, ...)
	
optimizerMsem(start, objective=msemObjectiveML, gradient=TRUE,
	maxiter, debug, par.size, model.description, warn=FALSE, ...)
	
optimizerNlm(start, objective=objectiveML, gradient=TRUE, 
	maxiter, debug, par.size, model.description, warn, ...)
	
optimizerOptim(start, objective=objectiveML, gradient=TRUE, 
	maxiter, debug, par.size, model.description, warn, method="CG", ...)

optimizerNlminb(start, objective=objectiveML, gradient=TRUE, maxiter, 
	debug, par.size, model.description, warn, ...)

msemOptimizerNlm(start, objective=msemObjectiveML, gradient=TRUE,
		maxiter, debug, par.size, model.description, warn=FALSE, ...)

Arguments

Value

An object of class "semResult", with elements:

Author(s)

John Fox jfox@mcmaster.ca, and Zhenghua Nie, in part adapting work by Saikat DebRoy, R-core, and Richard H. Jones.

See Also

sem, objective.functions, nlm, optim, nlminb

Draw Path Diagram

Description

pathDiagram creates a description of the path diagram for a structural-equation-model or SEM-specification object to be processed by the graph-drawing program dot.

Usage

pathDiagram(model, ...)

## S3 method for class 'sem'
pathDiagram(model, file = "pathDiagram", 
    style = c("ram", "traditional"),
    output.type = c("html", "graphics", "dot"), graphics.fmt = "pdf", 
    dot.options = NULL,
    size = c(8, 8), node.font = c("Helvetica", 14),
    edge.font = c("Helvetica", 10), digits = 2, 
    rank.direction = c("LR", "TB"),
    min.rank = NULL, max.rank = NULL, same.rank = NULL,
    variables = model$var.names, var.labels, parameters, par.labels,
    ignore.double = TRUE, ignore.self = FALSE, error.nodes = TRUE,
    edge.labels = c("names", "values", "both"),  
    edge.colors = c("black", "black"),
    edge.weight = c("fixed", "proportional"),
    node.colors = c("transparent", "transparent", "transparent"),
    standardize = FALSE, ...)

## S3 method for class 'semmod'
pathDiagram(model, obs.variables, ...)

math(text, html.only=FALSE, hat=FALSE)

Arguments

Details

pathDiagram creates a description of the path diagram for a structural-equation-model or SEM-specification object to be processed by the graph-drawing program dot, which can be called automatically; see Koutsofios and North (2002) and https://www.graphviz.org/. To obtain graphics output directly, the dot program must be on the system search path.

Alternatively, HTML output can be created in a web browser without an independent installation of dot using facilities in the DiagrammeR package.

The math function can be used to create node (variable) and edge (arrow) labels with symbols such as Greek letters, subscripts, and superscripts.

The semmod method of pathDiagram sets up a call to the sem method.

The various arguments to pathDiagram can be used to customize the diagram, but if there are too many constraints on node placement, dot may fail to produce a graph or may produce a distorted graph. pathDiagram can create both RAM-style diagrams, in which variances are represented as self-directed arrows, and traditional path diagrams, in which error variables appear explicitly as nodes. As is conventional, latent variables (including error variables) are represented as ellipses and observed variables as rectangles; double-headed arrows represent covariances (and in RAM diagrams, variances) and single-headed arrows represent structural coefficients.

Value

pathDiagram invisibly returns a character vector containing dot commands. math returns a character vector containing suitable HTML markup.

Author(s)

John Fox jfox@mcmaster.ca, Adam Kramer, and Michael Friendly

References

Koutsofios, E., and North, S. C. (2002) Drawing graphs with dot. https://graphviz.org/documentation/.

See Also

sem, specifyEquations, specifyModel, cfa

Examples

	if (interactive()) {
# The Duncan, Haller, and Portes Peer-Influences Model

R.DHP <- readMoments(diag=FALSE, names=c("ROccAsp", "REdAsp", "FOccAsp", 
                "FEdAsp", "RParAsp", "RIQ", "RSES", "FSES", "FIQ", "FParAsp"),
                text="
    .6247                                                              
    .3269  .3669                                                        
    .4216  .3275  .6404                                      
    .2137  .2742  .1124  .0839                                
    .4105  .4043  .2903  .2598  .1839                          
    .3240  .4047  .3054  .2786  .0489  .2220                    
    .2930  .2407  .4105  .3607  .0186  .1861  .2707              
    .2995  .2863  .5191  .5007  .0782  .3355  .2302  .2950        
    .0760  .0702  .2784  .1988  .1147  .1021  .0931 -.0438  .2087  
")

model.dhp <- specifyModel(text="
    RParAsp  -> RGenAsp, gam11,  NA
    RIQ      -> RGenAsp, gam12,  NA
    RSES     -> RGenAsp, gam13,  NA
    FSES     -> RGenAsp, gam14,  NA
    RSES     -> FGenAsp, gam23,  NA
    FSES     -> FGenAsp, gam24,  NA
    FIQ      -> FGenAsp, gam25,  NA
    FParAsp  -> FGenAsp, gam26,  NA
    FGenAsp  -> RGenAsp, beta12, NA
    RGenAsp  -> FGenAsp, beta21, NA
    RGenAsp  -> ROccAsp,  NA,       1
    RGenAsp  -> REdAsp,  lam21,  NA
    FGenAsp  -> FOccAsp,  NA,       1
    FGenAsp  -> FEdAsp,  lam42,  NA
    RGenAsp <-> RGenAsp, ps11,   NA
    FGenAsp <-> FGenAsp, ps22,   NA
    RGenAsp <-> FGenAsp, ps12,   NA
    ROccAsp <-> ROccAsp, theta1, NA
    REdAsp  <-> REdAsp,  theta2, NA
    FOccAsp <-> FOccAsp, theta3, NA
    FEdAsp  <-> FEdAsp,  theta4, NA
")

sem.dhp <- sem(model.dhp, R.DHP, 329,
    fixed.x=c("RParAsp", "RIQ", "RSES", "FSES", "FIQ", "FParAsp"))
    
pathDiagram(sem.dhp, min.rank="RIQ, RSES, RParAsp, FParAsp, FSES, FIQ", 
                     max.rank="ROccAsp, REdAsp, FEdAsp, FOccAsp",
                     same.rank="RGenAsp, FGenAsp",
                     edge.labels="values")
    
pathDiagram(model.dhp,
    obs.variables=c("RParAsp", "RIQ", "RSES", "FSES", "FIQ", 
        "FParAsp", "ROccAsp", "REdAsp", "FOccAsp", "FEdAsp"),
    style="traditional", 
    node.colors=c("pink", "lightblue", "lightgreen"),
    min.rank="RIQ, RSES, RParAsp, FParAsp, FSES, FIQ",
    max.rank="ROccAsp, REdAsp, FEdAsp, FOccAsp",
    same.rank="RGenAsp, FGenAsp",
    var.labels=c(RParAsp="Respondent Parental Aspiration", 
        RIQ="Respondent IQ",
        RSES="Respondent SES",
        FSES="Friend SES",
        FIQ="Friend IQ",
        FParAsp="Friend Parental Aspiration",
        ROccAsp="Respondent Occupational Aspiration",
        REdAsp="Respondent Educational Aspiration",
        RGenAsp="Respondent General Aspiration",
        FOccAsp="Friend Occupational Aspiration",
        FEdAsp="Friend Educational Aspiration",
        FGenAsp="Friend General Aspiration",
        math(c(RGenAsp.error="xi_{1}",
        FGenAsp.error="xi_{2}",
        ROccAsp.error="epsilon_{1}",
        REdAsp.error="epsilon_{2}",
        FOccAsp.error="epsilon_{3}",
        FEdAsp.error="epsilon_{4}"))),
    par.labels=math(c(gam11="gamma_{11}",
        gam12="gamma_{12}",
        gam13="gamma_{13}",
        gam14="gamma_{14}",
        gam23="gamma_{23}",
        gam24="gamma_{24}",
        gam25="gamma_{25}",
        gam26="gamma_{26}",
        beta12="beta_{12}",
        beta21="beta_{21}",
        lam21="lambda_{21}",
        lam42="lambda_{42}",
        ps11="psi_{11}",
        ps22="psi_{22}",
        ps12="psi_{12}",
        theta1="theta_{1}",
        theta2="theta_{2}",
        theta3="theta_{3}",
        theta4="theta_{4}")))
        
    # the following example contributed by Michael Friendly:
    
union <- readMoments(diag=TRUE,
    names=c('y1', 'y2', 'y3', 'x1', 'x2'), text="
14.610
-5.250  11.017
-8.057  11.087   31.971
-0.482   0.677    1.559   1.021
-18.857  17.861   28.250   7.139  215.662
")

union.mod <- specifyEquations(covs=c("x1, x2"), text="
y1 = gam12*x2
y2 = beta21*y1 + gam22*x2
y3 = beta31*y1 + beta32*y2 + gam31*x1
")

union.sem <- sem(union.mod, union, N=173)

dot <- pathDiagram(union.sem, style="traditional",  
    ignore.double=FALSE, error.nodes=FALSE,
    edge.labels="values", 
    min.rank=c("Years", "Age"), 
    max.rank=c("Sentiment", "Sentiment.error"),
    same.rank=c("Deference, Deference.error", "Activism, Activism.error"),
    variables=c("Deference", "Activism", "Sentiment", "Years", "Age"),
    edge.colors=c("black", "red"),
    node.colors = c("pink", "lightblue"))

cat(paste(dot, collapse="\n")) # dot commands

    }

RAM Matrix for a Structural-Equation Model

Description

Print the labelled RAM definition matrix for a structural-equation model fit by sem.

Usage

ram(object, digits=getOption("digits"), startvalues=FALSE)

Arguments

Value

A data frame containing the labelled RAM definition matrix, which is normally just printed.

Author(s)

John Fox jfox@mcmaster.ca

See Also

sem

Examples

# ------------- assumes that Duncan, Haller and Portes peer-influences model
# -------------     has been fit and is in sem.dhp

    ## Not run: 
ram(sem.dhp)
    
## End(Not run)

Compute Raw Moments Matrix

Description

Computes the “uncorrected” sum-of-squares-and-products matrix divided by the number of observations.

Usage

## S3 method for class 'formula'
rawMoments(formula, data, subset, na.action, 
    contrasts=NULL, ...)

## Default S3 method:
rawMoments(object, na.rm=FALSE, ...)

cov2raw(cov, mean, N, sd)

## S3 method for class 'rawmoments'
print(x, ...)

Arguments

.

Value

rawMoments and cov2raw return an object of class rawmoments, which is simply a matrix with an attribute "N" that contains the number of observations on which the moments are based.

Author(s)

John Fox jfox@mcmaster.ca

See Also

sem

Examples

# the following are all equivalent (with the exception of the name of the intercept):

rawMoments(cbind(1, Kmenta))

rawMoments(~ Q + P + D + F + A, data=Kmenta)

Cov <- with(Kmenta, cov(cbind(Q, P, D, F, A)))
cov2raw(Cov, colMeans(Kmenta), nrow(Kmenta))

Input a Covariance, Correlation, or Raw Moment Matrix

Description

This functions makes it simpler to input covariance, correlation, and raw-moment matrices to be analyzed by the sem function. The matrix is input in lower-triangular form on as many lines as is convenient, omitting the above-diagonal elements. The elements on the diagonal may also optionally be omitted, in which case they are taken to be 1.

Usage

readMoments(file="", text, diag=TRUE, 
    names=as.character(paste("X", 1:n, sep = "")))

Arguments

Value

Returns a lower-triangular matrix (i.e., with zeroes above the main diagonal) suitable for input to sem.

Author(s)

John Fox jfox@mcmaster.ca

See Also

sem

Examples

R.DHP <- readMoments(diag=FALSE, names=c("ROccAsp", "REdAsp", "FOccAsp", 
                "FEdAsp", "RParAsp", "RIQ", "RSES", "FSES", "FIQ", "FParAsp"),
                text="
    .6247     
    .3269  .3669       
    .4216  .3275  .6404
    .2137  .2742  .1124  .0839
    .4105  .4043  .2903  .2598  .1839
    .3240  .4047  .3054  .2786  .0489  .2220
    .2930  .2407  .4105  .3607  .0186  .1861  .2707
    .2995  .2863  .5191  .5007  .0782  .3355  .2302  .2950
    .0760  .0702  .2784  .1988  .1147  .1021  .0931 -.0438  .2087
")   
R.DHP

#the following will work only in an interactive sessions:
    ## Not run: 
R.DHP <- readMoments(diag=FALSE, names=c("ROccAsp", "REdAsp", "FOccAsp", 
                "FEdAsp", "RParAsp", "RIQ", "RSES", "FSES", "FIQ", "FParAsp"))
    .6247     
    .3269  .3669       
    .4216  .3275  .6404
    .2137  .2742  .1124  .0839
    .4105  .4043  .2903  .2598  .1839
    .3240  .4047  .3054  .2786  .0489  .2220
    .2930  .2407  .4105  .3607  .0186  .1861  .2707
    .2995  .2863  .5191  .5007  .0782  .3355  .2302  .2950
    .0760  .0702  .2784  .1988  .1147  .1021  .0931 -.0438  .2087
    
R.DHP
    
## End(Not run)

Residual Covariances for a Structural Equation Model

Description

These functions compute residual covariances, variance-standardized residual covariances, and normalized residual covariances for the observed variables in a structural-equation model fit by sem.

Usage

## S3 method for class 'sem'
residuals(object, ...)
## S3 method for class 'msem'
residuals(object, ...)

## S3 method for class 'sem'
standardizedResiduals(object, ...)
## S3 method for class 'msem'
standardizedResiduals(object, ...)

## S3 method for class 'objectiveML'
normalizedResiduals(object, ...)
## S3 method for class 'objectiveGLS'
normalizedResiduals(object, ...)
## S3 method for class 'msemObjectiveML'
normalizedResiduals(object, ...)

Arguments

Details

Residuals are defined as S - C, where S is the sample covariance matrix of the observed variables and C is the model-reproduced covariance matrix. The standardized residual covariance for a pair of variables divides the residual covariance by the product of the sample standard deviations of the two variables, (s_{ij} - c_{ij})/(s_{ii}s_{jj})^{1/2}. The normalized residual is given by

\frac{s_{ij}-c_{ij}} {[(c_{ii}c_{jj}-c_{ij}^2)/N^{*}]^{1/2}}

where N^{*} is the number of observations minus one if the model is fit to a covariance matrix, or the number of observations if it is fit to a raw moment matrix.

Value

Each function returns a matrix of residuals.

Author(s)

John Fox jfox@mcmaster.ca

References

Bollen, K. A. (1989) Structural Equations With Latent Variables. Wiley.

See Also

sem

Examples

# In the first example, readMoments() and specifyModel() read from the
# input stream. This example cannot be executed via example() but can be entered
# at the command prompt. The example is repeated using file input;
# this example can be executed via example(). 
    ## Not run: 
# Duncan, Haller, and Portes peer-influences model

R.DHP <- readMoments(diag=FALSE, names=c("ROccAsp", "REdAsp", "FOccAsp", 
                "FEdAsp", "RParAsp", "RIQ", "RSES", "FSES", "FIQ", "FParAsp"))
    .6247     
    .3269  .3669       
    .4216  .3275  .6404
    .2137  .2742  .1124  .0839
    .4105  .4043  .2903  .2598  .1839
    .3240  .4047  .3054  .2786  .0489  .2220
    .2930  .2407  .4105  .3607  .0186  .1861  .2707
    .2995  .2863  .5191  .5007  .0782  .3355  .2302  .2950
    .0760  .0702  .2784  .1988  .1147  .1021  .0931 -.0438  .2087
            
model.dhp <- specifyModel()
    RParAsp  -> RGenAsp, gam11,  NA
    RIQ      -> RGenAsp, gam12,  NA
    RSES     -> RGenAsp, gam13,  NA
    FSES     -> RGenAsp, gam14,  NA
    RSES     -> FGenAsp, gam23,  NA
    FSES     -> FGenAsp, gam24,  NA
    FIQ      -> FGenAsp, gam25,  NA
    FParAsp  -> FGenAsp, gam26,  NA
    FGenAsp  -> RGenAsp, beta12, NA
    RGenAsp  -> FGenAsp, beta21, NA
    RGenAsp  -> ROccAsp,  NA,     1
    RGenAsp  -> REdAsp,  lam21,  NA
    FGenAsp  -> FOccAsp,  NA,     1
    FGenAsp  -> FEdAsp,  lam42,  NA
    RGenAsp <-> RGenAsp, ps11,   NA
    FGenAsp <-> FGenAsp, ps22,   NA
    RGenAsp <-> FGenAsp, ps12,   NA
    ROccAsp <-> ROccAsp, theta1, NA
    REdAsp  <-> REdAsp,  theta2, NA
    FOccAsp <-> FOccAsp, theta3, NA
    FEdAsp  <-> FEdAsp,  theta4, NA

sem.dhp <- sem(model.dhp, R.DHP, 329,
    fixed.x=c('RParAsp', 'RIQ', 'RSES', 'FSES', 'FIQ', 'FParAsp'))
residuals(sem.dhp)
normalizedResiduals(sem.dhp) 
standardizedResiduals(sem.dhp) # same as residuals because model is fit to correlations
    
## End(Not run)
# The following example can be executed via example():

etc <- system.file(package="sem", "etc") # path to data and model files
   
(R.DHP <- readMoments(file=file.path(etc, "R-DHP.txt"),
				diag=FALSE, names=c("ROccAsp", "REdAsp", "FOccAsp", 
                "FEdAsp", "RParAsp", "RIQ", "RSES", "FSES", "FIQ", "FParAsp")))
(model.dhp <- specifyModel(file=file.path(etc, "model-DHP.txt")))
(sem.dhp <- sem(model.dhp, R.DHP, 329,
    fixed.x=c('RParAsp', 'RIQ', 'RSES', 'FSES', 'FIQ', 'FParAsp')))

residuals(sem.dhp)

normalizedResiduals(sem.dhp) 

standardizedResiduals(sem.dhp)  # same as residuals because model is fit to correlations

General Structural Equation Models

Description

sem fits general structural equation models (with both observed and unobserved variables). Observed variables are also called indicators or manifest variables; unobserved variables are also called factors or latent variables. Normally, the generic function (sem) is called directly with a semmod first argument produced by specifyModel, specifyEquations, or cfa, invoking the sem.semmod method, which in turn sets up a call to the sem.default method; thus, the user may wish to specify arguments accepted by the semmod and default methods. Similarly, for a multigroup model, sem would normally be called with a semmodList object produced by multigroupModel as its first argument, and would then generate a call to the code msemmod method.

Usage

## S3 method for class 'semmod'
sem(model, S, N, data, 
    raw=identical(na.action, na.pass), obs.variables=rownames(S), 
  	fixed.x=NULL, formula= ~ ., na.action=na.omit, 
    robust=!missing(data), debug=FALSE, 
    optimizer=optimizerSem, objective=objectiveML, ...)
    
## Default S3 method:
sem(model, S, N, raw=FALSE, data=NULL, start.fn=startvalues,
    pattern.number=NULL, valid.data.patterns=NULL,
    use.means=TRUE, param.names, 
  	var.names, fixed.x=NULL, robust=!is.null(data), semmod=NULL, debug=FALSE,
		analytic.gradient=!identical(objective, objectiveFIML), 
        warn=FALSE, maxiter=1000, par.size=c("ones", "startvalues"), 
		start.tol=1E-6, optimizer=optimizerSem, objective=objectiveML, cls, ...)
		
## S3 method for class 'semmodList'
sem(model, S, N, data, raw=FALSE, fixed.x=NULL, 
		robust=!missing(data), formula, group="Group", debug=FALSE, ...)
		
## S3 method for class 'msemmod'
sem(model, S, N, start.fn=startvalues,
        group="Group", groups=names(model), raw=FALSE, fixed.x, 
		param.names, var.names, debug=FALSE, analytic.gradient=TRUE, warn=FALSE,
		maxiter=5000, par.size = c("ones", "startvalues"), start.tol = 1e-06, 
		start=c("initial.fit", "startvalues"), initial.maxiter=1000,
		optimizer = optimizerMsem, objective = msemObjectiveML, ...)
    
startvalues(S, ram, debug=FALSE, tol=1E-6)
startvalues2(S, ram, debug=FALSE, tol=1E-6)

## S3 method for class 'sem'
coef(object, standardized=FALSE, ...)
## S3 method for class 'msem'
coef(object, ...)
## S3 method for class 'sem'
vcov(object, robust=FALSE, 
	analytic=inherits(object, "objectiveML") && object$t <= 500, ...)
## S3 method for class 'msem'
vcov(object, robust=FALSE, 
    analytic=inherits(object, "msemObjectiveML") && object$t <= 500, ...)
## S3 method for class 'sem'
df.residual(object, ...)
## S3 method for class 'msem'
df.residual(object, ...)

Arguments

Details

The model is set up using either RAM (“reticular action model” – don't ask!) notation – a simple format for specifying general structural equation models by coding the “arrows” in the path diagram for the model (see, e.g., McArdle and McDonald, 1984) – typically using the specifyModel function; in equation format using the specifyEquations function; or, for a simple confirmatory factor analysis model, via the cfa function. In any case, the model is represented internally in RAM format.

The variables in the v vector in the model (typically, the observed and unobserved variables, but not error variables) are numbered from 1 to m. the RAM matrix contains one row for each (free or constrained) parameter of the model, and may be specified either in symbolic format or in numeric format.

A symbolic ram matrix consists of three columns, as follows:

1. Arrow specification:

This is a simple formula, of the form "A -> B" or, equivalently, "B <- A" for a regression coefficient (i.e., a single-headed or directional arrow); "A <-> A" for a variance or "A <-> B" for a covariance (i.e., a double-headed or bidirectional arrow). Here, A and B are variable names in the model. If a name does not correspond to an observed variable, then it is assumed to be a latent variable. Spaces can appear freely in an arrow specification, and there can be any number of hyphens in the arrows, including zero: Thus, e.g., "A->B", "A --> B", and "A>B" are all legitimate and equivalent.

2. Parameter name:

The name of the regression coefficient, variance, or covariance specified by the arrow. Assigning the same name to two or more arrows results in an equality constraint. Specifying the parameter name as NA produces a fixed parameter.

3. Value:

start value for a free parameter or value of a fixed parameter. If given as NA, sem will compute the start value.

It is simplest to construct the RAM matrix with the specifyModel, specifyEquations, or cfa function, all of which return an object of class semmod, and also incorporate some model-specification convenience shortcuts. This process is illustrated in the examples below.

A numeric ram matrix consists of five columns, as follows:

1. Number of arrow heads:

1 (directed arrow) or 2 (covariance).

2. Arrow to:

index of the variable at the head of a directional arrow, or at one end of a bidirectional arrow. Observed variables should be assigned the numbers 1 to n, where n is the number of rows/columns in the covariance matrix S, with the indices corresponding to the variables' positions in S. Variable indices above n represent latent variables.

3. Arrow from:

the index of the variable at the tail of a directional arrow, or at the other end of a bidirectional arrow.

4. Parameter number:

free parameters are numbered from 1 to t, but do not necessarily appear in consecutive order. Fixed parameters are given the number 0. Equality contraints are specified by assigning two or more parameters the same number.

5. Value:

start value for a free parameter, or value of a fixed parameter. If given as NA, the program will compute a start value, by a slight modification of the method described by McDonald and Hartmann (1992). Note: In some circumstances, some start values are selected randomly; this might produce small differences in the parameter estimates when the program is rerun.

The numeric ram matrix is normally generated automatically, not specified directly by the user.

For specifyEquations, each input line is either a regression equation or the specification of a variance or covariance. Regression equations are of the form

y = par1*x1 + par2*x2 + ... + park*xk

where y and the xs are variables in the model (either observed or latent), and the pars are parameters. If a parameter is given as a numeric value (e.g., 1) then it is treated as fixed. Note that no “error” variable is included in the equation; “error variances” are specified via either the covs argument, via V(y) = par (see immediately below), or are added automatically to the model when, as by default, endog.variances=TRUE.

Variances are specified in the form V(var) = par and covariances in the form C(var1, var2) = par, where the vars are variables (observed or unobserved) in the model. The symbols V and C may be in either lower- or upper-case. If par is a numeric value (e.g., 1) then it is treated as fixed. In conformity with the RAM model, a variance or covariance for an endogenous variable in the model is an “error” variance or covariance.

To set a start value for a free parameter, enclose the numeric start value in parentheses after the parameter name, as parameter(value).

sem fits the model by calling the optimizer specified in the optimizer argument to minimize the objective function specified in the objective argument. If the optimization fails to converge, a warning message is printed.

The RAM formulation of the general structural equation model is given by the basic equation

v = Av + u

where v and u are vectors of random variables (observed or unobserved), and the parameter matrix A contains regression coefficients, symbolized by single-headed arrows in a path diagram. Another parameter matrix,

P = E(uu')

contains covariances among the elements of u (assuming that the elements of u have zero means). Usually v contains endogenous and exogenous observed and unobserved variables, but not error variables (see the examples below).

The startvalues function may be called directly, but is usually called by sem.default; startvalues2 is an older version of this function that may be used alternatively; see the startvalues argument to sem.

Value

sem returns an object of class c(objective, "sem"), where objective is the name of the objective function that was optimized (e.g., "objectiveML"), with the following elements:

For multigroup models, sem returns an object of class c("msemObjectiveML", "msem").

Warning

A common error is to fail to specify variance or covariance terms in the model, which are denoted by double-headed arrows, <->.

In general, every observed or latent variable in the model should be associated with a variance or error variance. This may be a free parameter to estimate or a fixed constant (as in the case of a latent exogenous variable for which you wish to fix the variance, e.g., to 1). Again in general, there will be an error variance associated with each endogenous variable in the model (i.e., each variable to which at least one single-headed arrow points — including observed indicators of latent variables), and a variance associated with each exogenous variable (i.e., each variable that appears only at the tail of single-headed arrows, never at the head).

To my knowledge, the only apparent exception to this rule is for observed variables that are declared to be fixed exogenous variables. In this case, the program generates the necessary (fixed-constant) variances and covariances automatically.

If there are missing variances, a warning message will be printed, and estimation will almost surely fail in some manner. Missing variances might well indicate that there are missing covariances too, but it is not possible to deduce this in a mechanical manner. The specifyModel funciton will by default supply error-variance parameters if these are missing.

Author(s)

John Fox jfox@mcmaster.ca, Zhenghua Nie, and Jarrett Byrnes

References

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

Bollen, K. A. (1989) Structural Equations With Latent Variables. Wiley.

Bollen, K. A. and Long, J. S. (eds.) Testing Structural Equation Models, Sage.

McArdle, J. J. and Epstein, D. (1987) Latent growth curves within developmental structural equation models. Child Development 58, 110–133.

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

McDonald, R. P. and Hartmann, W. M. (1992) A procedure for obtaining initial values of parameters in the RAM model. Multivariate Behavioral Research 27, 57–76.

Raftery, A. E. (1993) Bayesian model selection in structural equation models. In Bollen, K. A. and Long, J. S. (eds.) Testing Structural Equation Models, Sage.

Raftery, A. E. (1995) Bayesian model selection in social research (with discussion). Sociological Methodology 25, 111–196.

Satorra, A. (2000) Scaled and adjusted restricted tests in multi-sample analysis of moment structures. pp. 233–247 in Heijmans, R.D.H., Pollock, D.S.G. & Satorra, A. (eds.) Innovations in Multivariate Statistical Analysis. A Festschrift for Heinz Neudecker , Kluwer.

See Also

rawMoments, startvalues, objectiveML, objectiveGLS, optimizerNlm, optimizerOptim, optimizerNlminb, nlm, optim, nlminb, specifyModel, specifyEquations, cfa

Examples

# The following example illustrates the use the text argument to 
#   readMoments() and specifyEquations():

R.DHP <- readMoments(diag=FALSE, names=c("ROccAsp", "REdAsp", "FOccAsp", 
                "FEdAsp", "RParAsp", "RIQ", "RSES", "FSES", "FIQ", "FParAsp"),
                text="
    .6247     
    .3269  .3669       
    .4216  .3275  .6404
    .2137  .2742  .1124  .0839
    .4105  .4043  .2903  .2598  .1839
    .3240  .4047  .3054  .2786  .0489  .2220
    .2930  .2407  .4105  .3607  .0186  .1861  .2707
    .2995  .2863  .5191  .5007  .0782  .3355  .2302  .2950
    .0760  .0702  .2784  .1988  .1147  .1021  .0931 -.0438  .2087
")

model.dhp.1 <- specifyEquations(covs="RGenAsp, FGenAsp", text="
RGenAsp = gam11*RParAsp + gam12*RIQ + gam13*RSES + gam14*FSES + beta12*FGenAsp
FGenAsp = gam23*RSES + gam24*FSES + gam25*FIQ + gam26*FParAsp + beta21*RGenAsp
ROccAsp = 1*RGenAsp
REdAsp = lam21(1)*RGenAsp  # to illustrate setting start values
FOccAsp = 1*FGenAsp
FEdAsp = lam42(1)*FGenAsp
")

sem.dhp.1 <- sem(model.dhp.1, R.DHP, 329,
    fixed.x=c('RParAsp', 'RIQ', 'RSES', 'FSES', 'FIQ', 'FParAsp'))
summary(sem.dhp.1)

# Note: The following set of examples can't be run via example() because the default file
#  argument of specifyeEquations, specifyModel(), and readMoments() requires that the model 
#  specification and covariances, correlations, or raw moments be entered in an interactive
#  session at the command prompt. The examples can be copied and run in the R console,
#  however. See ?specifyModel and ?readMoments for further information.
#  These examples are repeated below using file input to specifyModel() and
#  readMoments(). The second version of the examples may be executed through example().

    ## Not run: 

# ------------- Duncan, Haller and Portes peer-influences model ----------------------
# A nonrecursive SEM with unobserved endogenous variables and fixed exogenous variables

R.DHP <- readMoments(diag=FALSE, names=c("ROccAsp", "REdAsp", "FOccAsp", 
                "FEdAsp", "RParAsp", "RIQ", "RSES", "FSES", "FIQ", "FParAsp"))
    .6247     
    .3269  .3669       
    .4216  .3275  .6404
    .2137  .2742  .1124  .0839
    .4105  .4043  .2903  .2598  .1839
    .3240  .4047  .3054  .2786  .0489  .2220
    .2930  .2407  .4105  .3607  .0186  .1861  .2707
    .2995  .2863  .5191  .5007  .0782  .3355  .2302  .2950
    .0760  .0702  .2784  .1988  .1147  .1021  .0931 -.0438  .2087
            
# Fit the model using a symbolic ram specification

model.dhp <- specifyModel()
    RParAsp  -> RGenAsp, gam11,  NA
    RIQ      -> RGenAsp, gam12,  NA
    RSES     -> RGenAsp, gam13,  NA
    FSES     -> RGenAsp, gam14,  NA
    RSES     -> FGenAsp, gam23,  NA
    FSES     -> FGenAsp, gam24,  NA
    FIQ      -> FGenAsp, gam25,  NA
    FParAsp  -> FGenAsp, gam26,  NA
    FGenAsp  -> RGenAsp, beta12, NA
    RGenAsp  -> FGenAsp, beta21, NA
    RGenAsp  -> ROccAsp,  NA,     1
    RGenAsp  -> REdAsp,  lam21,  NA
    FGenAsp  -> FOccAsp,  NA,     1
    FGenAsp  -> FEdAsp,  lam42,  NA
    RGenAsp <-> RGenAsp, ps11,   NA
    FGenAsp <-> FGenAsp, ps22,   NA
    RGenAsp <-> FGenAsp, ps12,   NA
    ROccAsp <-> ROccAsp, theta1, NA
    REdAsp  <-> REdAsp,  theta2, NA
    FOccAsp <-> FOccAsp, theta3, NA
    FEdAsp  <-> FEdAsp,  theta4, NA
    
# an equivalent specification, allowing specifyModel() to generate
#  variance parameters for endogenous variables (and suppressing the
#  unnecessary NAs):
 
model.dhp <- specifyModel()
RParAsp  -> RGenAsp, gam11
RIQ      -> RGenAsp, gam12
RSES     -> RGenAsp, gam13
FSES     -> RGenAsp, gam14
RSES     -> FGenAsp, gam23
FSES     -> FGenAsp, gam24
FIQ      -> FGenAsp, gam25
FParAsp  -> FGenAsp, gam26
FGenAsp  -> RGenAsp, beta12
RGenAsp  -> FGenAsp, beta21
RGenAsp  -> ROccAsp,  NA,     1
RGenAsp  -> REdAsp,  lam21
FGenAsp  -> FOccAsp,  NA,     1
FGenAsp  -> FEdAsp,  lam42
RGenAsp <-> FGenAsp, ps12

# Another equivalent specification, telling specifyModel to add paths for 
#   variances and covariance of RGenAsp and FGenAsp:
 
model.dhp <- specifyModel(covs="RGenAsp, FGenAsp")
RParAsp  -> RGenAsp, gam11
RIQ      -> RGenAsp, gam12
RSES     -> RGenAsp, gam13
FSES     -> RGenAsp, gam14
RSES     -> FGenAsp, gam23
FSES     -> FGenAsp, gam24
FIQ      -> FGenAsp, gam25
FParAsp  -> FGenAsp, gam26
FGenAsp  -> RGenAsp, beta12
RGenAsp  -> FGenAsp, beta21
RGenAsp  -> ROccAsp,  NA,     1
RGenAsp  -> REdAsp,  lam21
FGenAsp  -> FOccAsp,  NA,     1
FGenAsp  -> FEdAsp,  lam42

# Yet another equivalent specification using specifyEquations():

model.dhp.1 <- specifyEquations(covs="RGenAsp, FGenAsp")
RGenAsp = gam11*RParAsp + gam12*RIQ + gam13*RSES + gam14*FSES + beta12*FGenAsp
FGenAsp = gam23*RSES + gam24*FSES + gam25*FIQ + gam26*FParAsp + beta21*RGenAsp
ROccAsp = 1*RGenAsp
REdAsp = lam21(1)*RGenAsp  # to illustrate setting start values
FOccAsp = 1*FGenAsp
FEdAsp = lam42(1)*FGenAsp
 
sem.dhp.1 <- sem(model.dhp.1, R.DHP, 329,
    fixed.x=c('RParAsp', 'RIQ', 'RSES', 'FSES', 'FIQ', 'FParAsp'))
summary(sem.dhp.1)

# Fit the model using a numerical ram specification (not recommended!)

ram.dhp <- matrix(c(
#               heads   to      from    param  start
                1,       1,     11,      0,     1,
                1,       2,     11,      1,     NA, # lam21
                1,       3,     12,      0,     1,
                1,       4,     12,      2,     NA, # lam42
                1,      11,      5,      3,     NA, # gam11
                1,      11,      6,      4,     NA, # gam12
                1,      11,      7,      5,     NA, # gam13
                1,      11,      8,      6,     NA, # gam14
                1,      12,      7,      7,     NA, # gam23
                1,      12,      8,      8,     NA, # gam24
                1,      12,      9,      9,     NA, # gam25
                1,      12,     10,     10,     NA, # gam26
                1,      11,     12,     11,     NA, # beta12
                1,      12,     11,     12,     NA, # beta21
                2,       1,      1,     13,     NA, # theta1
                2,       2,      2,     14,     NA, # theta2
                2,       3,      3,     15,     NA, # theta3
                2,       4,      4,     16,     NA, # theta4
                2,      11,     11,     17,     NA, # psi11
                2,      12,     12,     18,     NA, # psi22
                2,      11,     12,     19,     NA  # psi12
                ), ncol=5, byrow=TRUE)

params.dhp <- c('lam21', 'lam42', 'gam11', 'gam12', 'gam13', 'gam14',
                 'gam23',  'gam24',  'gam25',  'gam26',
                 'beta12', 'beta21', 'theta1', 'theta2', 'theta3', 'theta4',
                 'psi11', 'psi22', 'psi12')
                 
vars.dhp <- c('ROccAsp', 'REdAsp', 'FOccAsp', 'FEdAsp', 'RParAsp', 'RIQ',
                'RSES', 'FSES', 'FIQ', 'FParAsp', 'RGenAsp', 'FGenAsp')
                
sem.dhp.2 <- sem(ram.dhp, R.DHP, 329, param.names=params.dhp, var.names=vars.dhp, 
	fixed.x=5:10)
summary(sem.dhp.2)


# -------------------- Wheaton et al. alienation data ----------------------
    

S.wh <- readMoments(names=c('Anomia67','Powerless67','Anomia71',
                                    'Powerless71','Education','SEI'))
   11.834                                    
    6.947    9.364                            
    6.819    5.091   12.532                    
    4.783    5.028    7.495    9.986            
   -3.839   -3.889   -3.841   -3.625   9.610     
  -21.899  -18.831  -21.748  -18.775  35.522  450.288

# This is the model in the SAS manual for PROC CALIS: A Recursive SEM with
# latent endogenous and exogenous variables.
# Curiously, both factor loadings for two of the latent variables are fixed.

model.wh.1 <- specifyModel()
    Alienation67   ->  Anomia67,      NA,     1
    Alienation67   ->  Powerless67,   NA,     0.833
    Alienation71   ->  Anomia71,      NA,     1
    Alienation71   ->  Powerless71,   NA,     0.833 
    SES            ->  Education,     NA,     1     
    SES            ->  SEI,           lamb,   NA
    SES            ->  Alienation67,  gam1,   NA
    Alienation67   ->  Alienation71,  beta,   NA
    SES            ->  Alienation71,  gam2,   NA
    Anomia67       <-> Anomia67,      the1,   NA
    Anomia71       <-> Anomia71,      the1,   NA
    Powerless67    <-> Powerless67,   the2,   NA
    Powerless71    <-> Powerless71,   the2,   NA
    Education      <-> Education,     the3,   NA
    SEI            <-> SEI,           the4,   NA
    Anomia67       <-> Anomia71,      the5,   NA
    Powerless67    <-> Powerless71,   the5,   NA
    Alienation67   <-> Alienation67,  psi1,   NA
    Alienation71   <-> Alienation71,  psi2,   NA
    SES            <-> SES,           phi,    NA
                           
sem.wh.1 <- sem(model.wh.1, S.wh, 932)
summary(sem.wh.1)

# The same model in equation format:

model.wh.1 <- specifyEquations()
Anomia67 = 1*Alienation67
Powerless67 = 0.833*Alienation67
Anomia71 = 1*Alienation71
Powerless71 = 0.833*Alienation71
Education = 1*SES
SEI = lamb*SES
Alienation67 = gam1*SES
Alienation71 = gam2*SES + beta*Alienation67
V(Anomia67) = the1
V(Anomia71) = the1
V(Powerless67) = the2
V(Powerless71) = the2
V(SES) = phi
C(Anomia67, Anomia71) = the5
C(Powerless67, Powerless71) = the5

# The same model, but treating one loading for each latent variable as free
# (and equal to each other).

model.wh.2 <- specifyModel()
    Alienation67   ->  Anomia67,      NA,        1
    Alienation67   ->  Powerless67,   lamby,    NA
    Alienation71   ->  Anomia71,      NA,        1
    Alienation71   ->  Powerless71,   lamby,    NA 
    SES            ->  Education,     NA,        1     
    SES            ->  SEI,           lambx,    NA
    SES            ->  Alienation67,  gam1,     NA
    Alienation67   ->  Alienation71,  beta,     NA
    SES            ->  Alienation71,  gam2,     NA
    Anomia67       <-> Anomia67,      the1,     NA
    Anomia71       <-> Anomia71,      the1,     NA
    Powerless67    <-> Powerless67,   the2,     NA
    Powerless71    <-> Powerless71,   the2,     NA
    Education      <-> Education,     the3,     NA
    SEI            <-> SEI,           the4,     NA
    Anomia67       <-> Anomia71,      the5,     NA
    Powerless67    <-> Powerless71,   the5,     NA
    Alienation67   <-> Alienation67,  psi1,     NA
    Alienation71   <-> Alienation71,  psi2,     NA
    SES            <-> SES,           phi,      NA 


sem.wh.2 <- sem(model.wh.2, S.wh, 932)
summary(sem.wh.2)

# And again, in equation format:

model.wh <- specifyEquations()
Anomia67 = 1*Alienation67
Powerless67 = lamby*Alienation67
Anomia71 = 1*Alienation71
Powerless71 = lamby*Alienation71
Education = 1*SES
SEI = lambx*SES
Alienation67 = gam1*SES
Alienation71 = gam2*SES + beta*Alienation67
V(Anomia67) = the1
V(Anomia71) = the1
V(Powerless67) = the2
V(Powerless71) = the2
V(SES) = phi
C(Anomia67, Anomia71) = the5
C(Powerless67, Powerless71) = the5


# Compare the two models by a likelihood-ratio test:

anova(sem.wh.1, sem.wh.2)


# ----------------------- Thurstone data ---------------------------------------
#  Second-order confirmatory factor analysis, from the SAS manual for PROC CALIS

R.thur <- readMoments(diag=FALSE, names=c('Sentences','Vocabulary',
        'Sent.Completion','First.Letters','4.Letter.Words','Suffixes',
        'Letter.Series','Pedigrees', 'Letter.Group'))
    .828                                              
    .776   .779                                        
    .439   .493    .46                                 
    .432   .464    .425   .674                           
    .447   .489    .443   .59    .541                    
    .447   .432    .401   .381    .402   .288              
    .541   .537    .534   .35    .367   .32   .555        
    .38   .358    .359   .424    .446   .325   .598   .452  
            
model.thur <- specifyModel()
    F1 -> Sentences,                      lam11
    F1 -> Vocabulary,                     lam21
    F1 -> Sent.Completion,                lam31
    F2 -> First.Letters,                  lam42
    F2 -> 4.Letter.Words,                 lam52
    F2 -> Suffixes,                       lam62
    F3 -> Letter.Series,                  lam73
    F3 -> Pedigrees,                      lam83
    F3 -> Letter.Group,                   lam93
    F4 -> F1,                             gam1
    F4 -> F2,                             gam2
    F4 -> F3,                             gam3
    F1 <-> F1,                            NA,     1
    F2 <-> F2,                            NA,     1
    F3 <-> F3,                            NA,     1
    F4 <-> F4,                            NA,     1

sem.thur <- sem(model.thur, R.thur, 213)
summary(sem.thur)

# The model in equation format:

model.thur <- specifyEquations()
Sentences = lam11*F1
Vocabulary = lam21*F1
Sent.Completion = lam31*F1
First.Letters = lam42*F2
4.Letter.Words = lam52*F2
Suffixes = lam62*F2
Letter.Series = lam73*F3
Pedigrees = lam83*F3
Letter.Group = lam93*F3
F1 = gam1*F4
F2 = gam2*F4
F3 = gam3*F4
V(F1) = 1
V(F2) = 1
V(F3) = 1
V(F4) = 1


#------------------------- Kerchoff/Kenney path analysis ---------------------
# An observed-variable recursive SEM from the LISREL manual

R.kerch <- readMoments(diag=FALSE, names=c('Intelligence','Siblings',
                        'FatherEd','FatherOcc','Grades','EducExp','OccupAsp'))
    -.100                                
     .277  -.152                          
     .250  -.108  .611                     
     .572  -.105  .294   .248               
     .489  -.213  .446   .410   .597         
     .335  -.153  .303   .331   .478   .651   
    
model.kerch <- specifyModel()
    Intelligence -> Grades,       gam51
    Siblings -> Grades,           gam52
    FatherEd -> Grades,           gam53
    FatherOcc -> Grades,          gam54
    Intelligence -> EducExp,      gam61
    Siblings -> EducExp,          gam62
    FatherEd -> EducExp,          gam63
    FatherOcc -> EducExp,         gam64
    Grades -> EducExp,            beta65
    Intelligence -> OccupAsp,     gam71
    Siblings -> OccupAsp,         gam72
    FatherEd -> OccupAsp,         gam73
    FatherOcc -> OccupAsp,        gam74
    Grades -> OccupAsp,           beta75
    EducExp -> OccupAsp,          beta76
                       
sem.kerch <- sem(model.kerch, R.kerch, 737, 
    fixed.x=c('Intelligence', 'Siblings', 'FatherEd', 'FatherOcc'))
summary(sem.kerch)

# The model in equation format:

model.kerch <- specifyEquations()
Grades = gam51*Intelligence + gam52*Siblings + gam53*FatherEd 
           + gam54*FatherOcc
EducExp = gam61*Intelligence + gam62*Siblings + gam63*FatherEd 
            + gam64*FatherOcc + beta65*Grades
OccupAsp = gam71*Intelligence + gam72*Siblings + gam73*FatherEd 
            + gam74*FatherOcc + beta75*Grades + beta76*EducExp


#------------------- McArdle/Epstein latent-growth-curve model -----------------
# This model, from McArdle and Epstein (1987, p.118), illustrates the use of a 
# raw moment matrix to fit a model with an intercept. (The example was suggested
# by Mike Stoolmiller.)

M.McArdle <- readMoments(
    names=c('WISC1', 'WISC2', 'WISC3', 'WISC4', 'UNIT'))
    365.661                                      
    503.175     719.905                           
    675.656     958.479    1303.392                
    890.680    1265.846    1712.475    2278.257     
     18.034      25.819      35.255      46.593     1.000
 
mod.McArdle <- specifyModel()
    C -> WISC1, NA, 6.07
    C -> WISC2, B2, NA
    C -> WISC3, B3, NA
    C -> WISC4, B4, NA
    UNIT -> C, Mc, NA
    C <-> C, Vc, NA,
    WISC1 <-> WISC1, Vd, NA
    WISC2 <-> WISC2, Vd, NA
    WISC3 <-> WISC3, Vd, NA
    WISC4 <-> WISC4, Vd, NA

sem.McArdle <- sem(mod.McArdle, M.McArdle, 204, fixed.x="UNIT", raw=TRUE)
summary(sem.McArdle)

# The model in equation format:

mod.McArdle <- specifyEquations()
WISC1 = 6.07*C
WISC2 = B2*C
WISC3 = B3*C
WISC4 = b4*C
C = Mc*UNIT
v(C) = Vc
v(WISC1) = Vd
v(WISC2) = Vd
v(WISC3) = Vd
v(WISC4) = Vd

    
#------------ Bollen industrialization and democracy example -----------------
# This model, from Bollen (1989, Ch. 8), illustrates the use in sem() of a
# case-by-variable data (see ?Bollen) set rather than a covariance or moment matrix

model.bollen <- specifyModel()
	Demo60 -> y1, NA, 1
	Demo60 -> y2, lam2, 
	Demo60 -> y3, lam3, 
	Demo60 -> y4, lam4, 
	Demo65 -> y5, NA, 1
	Demo65 -> y6, lam2, 
	Demo65 -> y7, lam3, 
	Demo65 -> y8, lam4, 
	Indust -> x1, NA, 1
	Indust -> x2, lam6, 
	Indust -> x3, lam7, 
	y1 <-> y5, theta15
	y2 <-> y4, theta24
	y2 <-> y6, theta26
	y3 <-> y7, theta37
	y4 <-> y8, theta48
	y6 <-> y8, theta68
	Indust -> Demo60, gamma11, 
	Indust -> Demo65, gamma21, 
	Demo60 -> Demo65, beta21, 
	Indust <-> Indust, phi
	
sem.bollen <- sem(model.bollen, data=Bollen)
summary(sem.bollen)
summary(sem.bollen, robust=TRUE) # robust SEs and tests
summary(sem.bollen, analytic.se=FALSE) # uses numeric rather than analytic Hessian

  # GLS rather than ML estimator:
sem.bollen.gls <- sem(model.bollen, data=Bollen, objective=objectiveGLS) 
summary(sem.bollen.gls)

# The model in equation format:

model.bollen <- specifyEquations()
y1 = 1*Demo60
y2 = lam2*Demo60
y3 = lam3*Demo60
y4 = lam4*Demo60
y5 = 1*Demo65
y6 = lam2*Demo65
y7 = lam3*Demo65
y8 = lam4*Demo65
x1 = 1*Indust
x2 = lam6*Indust
x3 = lam7*Indust
c(y1, y5) = theta15
c(y2, y4) = theta24
c(y2, y6) = theta26
c(y3, y7) = theta37
c(y4, y8) = theta48
c(y6, y8) = theta68
Demo60 = gamma11*Indust
Demo65 = gamma21*Indust + beta21*Demo60
v(Indust) = phi


# -------------- A simple CFA model for the Thurstone mental tests data --------------

R.thur <- readMoments(diag=FALSE, 
  names=c('Sentences','Vocabulary',
          'Sent.Completion','First.Letters','4.Letter.Words','Suffixes',
          'Letter.Series','Pedigrees', 'Letter.Group'))
.828                                              
.776   .779                                        
.439   .493    .46                                 
.432   .464    .425   .674                           
.447   .489    .443   .59    .541                    
.447   .432    .401   .381    .402   .288              
.541   .537    .534   .35    .367   .32   .555        
.38   .358    .359   .424    .446   .325   .598   .452

	#  (1) in CFA format:

mod.cfa.thur.c <- cfa(reference.indicators=FALSE)
FA: Sentences, Vocabulary, Sent.Completion
FB: First.Letters, 4.Letter.Words, Suffixes
FC: Letter.Series, Pedigrees, Letter.Group

cfa.thur.c <- sem(mod.cfa.thur.c, R.thur, 213)
summary(cfa.thur.c)

	#  (2) in equation format:

mod.cfa.thur.e <- specifyEquations(covs="F1, F2, F3")
Sentences = lam11*F1
Vocabulary = lam21*F1
Sent.Completion = lam31*F1
First.Letters = lam42*F2
4.Letter.Words = lam52*F2
Suffixes = lam62*F2
Letter.Series = lam73*F3
Pedigrees = lam83*F3
Letter.Group = lam93*F3
V(F1) = 1
V(F2) = 1
V(F3) = 1

cfa.thur.e <- sem(mod.cfa.thur.e, R.thur, 213)
summary(cfa.thur.e)

	#  (3) in path format:

mod.cfa.thur.p <- specifyModel(covs="F1, F2, F3")
F1 -> Sentences,                      lam11
F1 -> Vocabulary,                     lam21
F1 -> Sent.Completion,                lam31
F2 -> First.Letters,                  lam41
F2 -> 4.Letter.Words,                 lam52
F2 -> Suffixes,                       lam62
F3 -> Letter.Series,                  lam73
F3 -> Pedigrees,                      lam83
F3 -> Letter.Group,                   lam93
F1 <-> F1,                            NA,     1
F2 <-> F2,                            NA,     1
F3 <-> F3,                            NA,     1

cfa.thur.p <- sem(mod.cfa.thur.p, R.thur, 213)
summary(cfa.thur.p)

# -----  a CFA model fit by FIML to the mental-tests dataset with missing data -----

mod.cfa.tests <- cfa(raw=TRUE)
verbal: x1, x2, x3
math: y1, y2, y3

cfa.tests <- sem(mod.cfa.tests, data=Tests, na.action=na.pass, 
                objective=objectiveFIML, fixed.x="Intercept")
summary(cfa.tests)
summary(cfa.tests, saturated=TRUE) # takes time to fit saturated model for comparison


# ---  a multigroup CFA model fit to the Holzinger-Swineford mental-tests data  -----

mod.hs <- cfa()
spatial: visual, cubes, paper, flags
verbal: general, paragrap, sentence, wordc, wordm
memory: wordr, numberr, figurer, object, numberf, figurew
math: deduct, numeric, problemr, series, arithmet

mod.mg <- multigroupModel(mod.hs, groups=c("Female", "Male")) 

sem.mg <- sem(mod.mg, data=HS.data, group="Gender",
              formula = ~ visual + cubes + paper + flags +
              general + paragrap + sentence + wordc + wordm +
              wordr + numberr + figurer + object + numberf + figurew +
              deduct + numeric + problemr + series + arithmet
              )
summary(sem.mg)

	# with cross-group equality constraints:
	
mod.mg.eq <- multigroupModel(mod.hs, groups=c("Female", "Male"), allEqual=TRUE)

sem.mg.eq <- sem(mod.mg.eq, data=HS.data, group="Gender",
              formula = ~ visual + cubes + paper + flags +
                general + paragrap + sentence + wordc + wordm +
                wordr + numberr + figurer + object + numberf + figurew +
                deduct + numeric + problemr + series + arithmet
              )
summary(sem.mg.eq)

anova(sem.mg, sem.mg.eq) # test equality constraints
	
## End(Not run)

## ===============================================================================
	
# The following examples use file input and may be executed via example():

etc <- system.file(package="sem", "etc") # path to data and model files

#   to get all fit indices (not recommended, but for illustration):

opt <- options(fit.indices = c("GFI", "AGFI", "RMSEA", "NFI", "NNFI", 
         "CFI", "RNI", "IFI", "SRMR", "AIC", "AICc", "BIC", "CAIC"))

# ------------- Duncan, Haller and Portes peer-influences model ----------------------
# A nonrecursive SEM with unobserved endogenous variables and fixed exogenous variables

(R.DHP <- readMoments(file=file.path(etc, "R-DHP.txt"),
				diag=FALSE, names=c("ROccAsp", "REdAsp", "FOccAsp", 
                "FEdAsp", "RParAsp", "RIQ", "RSES", "FSES", "FIQ", "FParAsp")))
(model.dhp <- specifyModel(file=file.path(etc, "model-DHP.txt")))
sem.dhp.1 <- sem(model.dhp, R.DHP, 329,
    fixed.x=c('RParAsp', 'RIQ', 'RSES', 'FSES', 'FIQ', 'FParAsp'))
summary(sem.dhp.1)


# -------------------- Wheaton et al. alienation data ----------------------

(S.wh <- readMoments(file=file.path(etc, "S-Wheaton.txt"),
					names=c('Anomia67','Powerless67','Anomia71',
                            'Powerless71','Education','SEI')))

# This is the model in the SAS manual for PROC CALIS: A Recursive SEM with
# latent endogenous and exogenous variables.
# Curiously, both factor loadings for two of the latent variables are fixed.

(model.wh.1 <- specifyModel(file=file.path(etc, "model-Wheaton-1.txt")))                    
sem.wh.1 <- sem(model.wh.1, S.wh, 932)
summary(sem.wh.1)

# The same model, but treating one loading for each latent variable as free
# (and equal to each other).

(model.wh.2 <- specifyModel(file=file.path(etc, "model-Wheaton-2.txt")))
sem.wh.2 <- sem(model.wh.2, S.wh, 932)
summary(sem.wh.2)

# Compare the two models by a likelihood-ratio test:

anova(sem.wh.1, sem.wh.2)


# ----------------------- Thurstone data ---------------------------------------

#  Second-order confirmatory factor analysis, from the SAS manual for PROC CALIS

(R.thur <- readMoments(file=file.path(etc, "R-Thurstone.txt"),
		diag=FALSE, names=c('Sentences','Vocabulary',
        'Sent.Completion','First.Letters','4.Letter.Words','Suffixes',
        'Letter.Series','Pedigrees', 'Letter.Group')))
(model.thur <- specifyModel(file=file.path(etc, "model-Thurstone.txt")))
sem.thur <- sem(model.thur, R.thur, 213)
summary(sem.thur)


#------------------------- Kerchoff/Kenney path analysis ---------------------

# An observed-variable recursive SEM from the LISREL manual

(R.kerch <- readMoments(file=file.path(etc, "R-Kerchoff.txt"),
					   diag=FALSE, names=c('Intelligence','Siblings',
                        'FatherEd','FatherOcc','Grades','EducExp','OccupAsp')))
(model.kerch <- specifyModel(file=file.path(etc, "model-Kerchoff.txt")))
sem.kerch <- sem(model.kerch, R.kerch, 737, 
    fixed.x=c('Intelligence', 'Siblings', 'FatherEd', 'FatherOcc'))
summary(sem.kerch)


#------------------- McArdle/Epstein latent-growth-curve model -----------------

# This model, from McArdle and Epstein (1987, p.118), illustrates the use of a 
# raw moment matrix to fit a model with an intercept. (The example was suggested
# by Mike Stoolmiller.)

(M.McArdle <- readMoments(file=file.path(etc, "M-McArdle.txt"),
    names=c('WISC1', 'WISC2', 'WISC3', 'WISC4', 'UNIT')))
(mod.McArdle <- specifyModel(file=file.path(etc, "model-McArdle.txt")))
sem.McArdle <- sem(mod.McArdle, M.McArdle, 204, fixed.x="UNIT", raw=TRUE)
summary(sem.McArdle)


#------------ Bollen industrialization and democracy example -----------------

# This model, from Bollen (1989, Ch. 8), illustrates the use in sem() of a
# case-by-variable data set (see ?Bollen) rather than a covariance or moment matrix

(model.bollen <- specifyModel(file=file.path(etc, "model-Bollen.txt")))
sem.bollen <- sem(model.bollen, data=Bollen)
summary(sem.bollen)
summary(sem.bollen, robust=TRUE) # robust SEs and tests
summary(sem.bollen, analytic.se=FALSE) # uses numeric rather than analytic Hessian

  # GLS rather than ML estimator:
sem.bollen.gls <- sem(model.bollen, data=Bollen, objective=objectiveGLS) 
summary(sem.bollen.gls)

# -----  a CFA model fit by FIML to the mental-tests dataset with missing data -----

(mod.cfa.tests <- cfa(file=file.path(etc, "model-Tests.txt"), raw=TRUE))
cfa.tests <- sem(mod.cfa.tests, data=Tests, na.action=na.pass, 
                optimizer=optimizerNlm, objective=objectiveFIML, fixed.x="Intercept")
summary(cfa.tests)

#------------ Holzinger and Swineford muiltigroup CFA example ----------------

mod.hs <- cfa(file=file.path(etc, "model-HS.txt"))

mod.mg <- multigroupModel(mod.hs, groups=c("Female", "Male")) 

sem.mg <- sem(mod.mg, data=HS.data, group="Gender",
              formula = ~ visual + cubes + paper + flags +
              general + paragrap + sentence + wordc + wordm +
              wordr + numberr + figurer + object + numberf + figurew +
              deduct + numeric + problemr + series + arithmet
              )
summary(sem.mg)

	# with cross-group equality constraints:
	
mod.mg.eq <- multigroupModel(mod.hs, groups=c("Female", "Male"), allEqual=TRUE)

sem.mg.eq <- sem(mod.mg.eq, data=HS.data, group="Gender",
              formula = ~ visual + cubes + paper + flags +
                general + paragrap + sentence + wordc + wordm +
                wordr + numberr + figurer + object + numberf + figurew +
                deduct + numeric + problemr + series + arithmet
              )
summary(sem.mg.eq)

anova(sem.mg, sem.mg.eq) # test equality constraints

options(opt) # restore fit.indices option

Deprecated Functions in the sem Package

Description

These functions are provided for compatibility with older versions of the sem package only, and may be removed eventually. Although an effort has been made to insure backwards-compatibility, commands that worked in versions of the sem package prior to version 2.0-0 will not necessarily work in version 2.0-0 and beyond, or may not work in the same manner.

Usage

boot.sem(...)
mod.indices(...)
normalized.residuals(...)
path.diagram(...)
raw.moments(...)
read.moments(...)
specify.model(...)
standardized.coefficients(...)
standardized.residuals(...)
std.coef(...)

Arguments

Details

boot.sem is now a synonym for the bootSem function.

mod.indices is now a synonym for modIndices.

normalized.residuals is now a synonym for normalizedResiduals.

path.diagram is now a synonym for pathDiagram.

raw.moments is now a synonym for rawMoments.

read.moments is now a synonym for readMoments.

specify.model is now a synonym for specifyModel.

standardized.coefficients and std.coef are now synonyms for the standardizedCoefficients and stdCoef functions.

standardized.residuals is now a synonym for standardizedResiduals.

Specify a Structural Equation Model

Description

Create the RAM specification of a structural equation model.

Usage

specifyModel(file="", text, exog.variances=FALSE, endog.variances=TRUE, covs, 
	suffix="", quiet=FALSE)

specifyEquations(file="", text, ...)

cfa(file="", text, covs=paste(factors, collapse=","), 
    reference.indicators=TRUE, raw=FALSE, 
    subscript=c("name", "number"), ...)

multigroupModel(..., groups=names(models), allEqual=FALSE)

classifyVariables(model)

removeRedundantPaths(model, warn=TRUE)
## S3 method for class 'semmod'
combineModels(..., warn=TRUE)
## S3 method for class 'semmod'
update(object, file = "", text, ...)
## S3 method for class 'semmod'
edit(name, ...)

## S3 method for class 'semmod'
print(x, ...)
## S3 method for class 'semmodList'
print(x, ...)

Arguments

Details

The principal functions for model specification are specifyModel, to specify a model in RAM (path) format via single- and double-headed arrows; specifyEquations, to specify a model in equation format, which is then translated by the function into RAM format; and cfa, for compact specification of simple confirmatory factor analysis models.

specifyModel:

Each line of the RAM specification for specifyModel consists of three (unquoted) entries, separated by commas:

1. Arrow specification:

This is a simple formula, of the form A -> B or, equivalently, B <- A for a regression coefficient (i.e., a single-headed or directional arrow); A <-> A for a variance or A <-> B for a covariance (i.e., a double-headed or bidirectional arrow). Here, A and B are variable names in the model. If a name does not correspond to an observed variable, then it is assumed to be a latent variable. Spaces can appear freely in an arrow specification, and there can be any number of hyphens in the arrows, including zero: Thus, e.g., A->B, A --> B, and A>B are all legitimate and equivalent.

2. Parameter name:

The name of the regression coefficient, variance, or covariance specified by the arrow. Assigning the same name to two or more arrows results in an equality constraint. Specifying the parameter name as NA produces a fixed parameter.

3. Value:

start value for a free parameter or value of a fixed parameter. If given as NA (or simply omitted), sem will compute the start value.

Lines may end in a comment following #.

specifyEquations:

For specifyEquations, each input line is either a regression equation or the specification of a variance or covariance. Regression equations are of the form

y = par1*x1 + par2*x2 + ... + park*xk

where y and the xs are variables in the model (either observed or latent), and the pars are parameters. If a parameter is given as a numeric value (e.g., 1) then it is treated as fixed. Note that no “error” variable is included in the equation; “error variances” are specified via either the covs argument, via V(y) = par (see immediately below), or are added automatically to the model when, as by default, endog.variances=TRUE. A regression equation may be split over more than one input by breaking at a +, so that + is either the last non-blank character on a line or the first non-blank character on the subsequent line.

Variances are specified in the form V(var) = par and covariances in the form C(var1, var2) = par, where the vars are variables (observed or unobserved) in the model. The symbols V and C may be in either lower- or upper-case. If par is a numeric value (e.g., 1) then it is treated as fixed. In conformity with the RAM model, a variance or covariance for an endogenous variable in the model is an “error” variance or covariance.

Warning: If the covs argument to specifyEquations is used to specify variances and covariances, please be aware that covs="x1, x2" and covs=c("x1", "x2") are not equivalent: covs="x1, x2" specifies the variance of x1, the variance of x2, and their covariance, while covs=c("x1", "x2") specifies the variance of x1 and the variance of x2 but not their covariance.

To set a start value for a free parameter, enclose the numeric start value in parentheses after the parameter name, as parameter(value).

cfa:

For cfa, each input line includes the names of the variables, separated by commas, that load on the corresponding factor; the name of the factor is given optionally at the beginning of the line, followed by a colon. If necessary, the variables that load on a factor may be continued across two or more input lines; in this case, each such line but the last must end in a comma. A variable may load on more than one factor (as long as the resulting model is identified, of course), but each factor may appear in only one input line (or set of input lines, if the variable list is continued onto the next line).

Equality constraints for factor loadings can be set by using equal-signs (=) rather than commas to separate observed variable names. For example, fac1: x1=x2=x3, x4=x5 sets the loadings for x1, x2, and x3 equal to each other, and the loadings for x4 and x5 equal to each other.

Equality constraints among error variances can similarly be specified by using var: or variance: at the beginning of a line (actually, any character string beginning with var will do, and thus no factor name may begin with the characters var). For example, var: x1=x2=x3, x4=x5 sets the error variances for x1, x2, and x3 equal to each other, and the error variances for x4 and x5 equal to each other. There may be several lines beginning with var:.

If the argument reference.indicators=FALSE, the default, cfa will fix the variance of each factor to 1, and by default include covariances (i.e., correlations) among all pairs of factors. Alternatively, if reference.indicators=TRUE, then the factor variances are free parameters to be estimated from the data, and the first loading for each factor is set to 1 to identify the model. These two approaches produce equivalent models, with the same fit to the data, but alternative parametrizations. Specifying the argument covs=NULL implicitly fixes the factor intercorrelations to 0.

See sem and the examples for further details on model specification.

Other Functions:

classifyVariables classifies the variables in a model as endogenous or exogenous.

combineModels and removeRedundantPaths take semmod objects as arguments and do what their names imply.

The file input argument to the update method for semmod objects, which by default comes from standard input, is a set of update directives, one per line. There are five kinds of directives. In each case the directive begins with the directive name, followed by one or more fields separated by commas.

1. delete:

Remove a path from the model. Example: delete, RSES -> FGenAsp

2. add:

Add a path to the model. Example (the NA for the start value is optional): add, RSES -> FGenAsp, gam14, NA

3. replace:

Replace every occurrence of the first string with the second in the variables and parameters of the model. This directive may be used, for example, to change one variable to another or to rename a parameter. Example: replace, gam, gamma, substitutes the string "gamma" for "gam" wherever the latter appears, presumably in parameter names.

4. fix:

Fix a parameter that was formerly free. Example: fix, RGenAsp -> REdAsp, 1

5. free:

Free a parameter that was formerly fixed. Example (the NA for the start value is optional): free, RGenAsp -> ROccAsp, lam11, NA

The edit method for semmod objects opens the model in the R editor.

Value

specifyModel, specifyEquations, cfa, removeRedundantPaths, combineModels, update, and edit return an object of class semmod, suitable as input for sem.

multigroupModel returns an object of class semmodList, also suitable as input for sem.

classifyVariables returns a list with two character vectors: endogenous, containing the names of endogenous variables in the model; and exogenous, containing the names of exogenous variables.

Author(s)

John Fox jfox@mcmaster.ca and Jarrett Byrnes

See Also

sem

Examples

# example using the text argument:

model.dhp <- specifyModel(text="
    RParAsp  -> RGenAsp, gam11,  NA
    RIQ      -> RGenAsp, gam12,  NA
    RSES     -> RGenAsp, gam13,  NA
    FSES     -> RGenAsp, gam14,  NA
    RSES     -> FGenAsp, gam23,  NA
    FSES     -> FGenAsp, gam24,  NA
    FIQ      -> FGenAsp, gam25,  NA
    FParAsp  -> FGenAsp, gam26,  NA
    FGenAsp  -> RGenAsp, beta12, NA
    RGenAsp  -> FGenAsp, beta21, NA
    RGenAsp  -> ROccAsp,  NA,     1
    RGenAsp  -> REdAsp,  lam21,  NA
    FGenAsp  -> FOccAsp,  NA,     1
    FGenAsp  -> FEdAsp,  lam42,  NA
    RGenAsp <-> RGenAsp, ps11,   NA
    FGenAsp <-> FGenAsp, ps22,   NA
    RGenAsp <-> FGenAsp, ps12,   NA
    ROccAsp <-> ROccAsp, theta1, NA
    REdAsp  <-> REdAsp,  theta2, NA
    FOccAsp <-> FOccAsp, theta3, NA
    FEdAsp  <-> FEdAsp,  theta4, NA
")    
model.dhp

   # same model in equation form:
model.dhp.1 <- specifyEquations(covs="RGenAsp, FGenAsp", text="
RGenAsp = gam11*RParAsp + gam12*RIQ + gam13*RSES + gam14*FSES + beta12*FGenAsp
FGenAsp = gam23*RSES + gam24*FSES + gam25*FIQ + gam26*FParAsp + beta21*RGenAsp
ROccAsp = 1*RGenAsp
REdAsp = lam21(1)*RGenAsp  # to illustrate setting start values
FOccAsp = 1*FGenAsp
FEdAsp = lam42(1)*FGenAsp
")
model.dhp

# Note: The following examples can't be run via example() because the 
#  default file argument requires that the model specification be entered
#  at the command prompt. The examples can be copied and run in an interactive 
#  session in the R console, however.

    ## Not run: 
model.dhp <- specifyModel()
    RParAsp  -> RGenAsp, gam11,  NA
    RIQ      -> RGenAsp, gam12,  NA
    RSES     -> RGenAsp, gam13,  NA
    FSES     -> RGenAsp, gam14,  NA
    RSES     -> FGenAsp, gam23,  NA
    FSES     -> FGenAsp, gam24,  NA
    FIQ      -> FGenAsp, gam25,  NA
    FParAsp  -> FGenAsp, gam26,  NA
    FGenAsp  -> RGenAsp, beta12, NA
    RGenAsp  -> FGenAsp, beta21, NA
    RGenAsp  -> ROccAsp,  NA,     1
    RGenAsp  -> REdAsp,  lam21,  NA
    FGenAsp  -> FOccAsp,  NA,     1
    FGenAsp  -> FEdAsp,  lam42,  NA
    RGenAsp <-> RGenAsp, ps11,   NA
    FGenAsp <-> FGenAsp, ps22,   NA
    RGenAsp <-> FGenAsp, ps12,   NA
    ROccAsp <-> ROccAsp, theta1, NA
    REdAsp  <-> REdAsp,  theta2, NA
    FOccAsp <-> FOccAsp, theta3, NA
    FEdAsp  <-> FEdAsp,  theta4, NA
    
model.dhp
    
# an equivalent specification, allowing specifyModel() to generate
#  variance parameters for endogenous variables (and suppressing
#  the unnecessary trailing NAs):
 
model.dhp <- specifyModel()
RParAsp  -> RGenAsp, gam11
RIQ      -> RGenAsp, gam12
RSES     -> RGenAsp, gam13
FSES     -> RGenAsp, gam14
RSES     -> FGenAsp, gam23
FSES     -> FGenAsp, gam24
FIQ      -> FGenAsp, gam25
FParAsp  -> FGenAsp, gam26
FGenAsp  -> RGenAsp, beta12
RGenAsp  -> FGenAsp, beta21
RGenAsp  -> ROccAsp,  NA,     1
RGenAsp  -> REdAsp,  lam21
FGenAsp  -> FOccAsp,  NA,     1
FGenAsp  -> FEdAsp,  lam42
RGenAsp <-> FGenAsp, ps12

model.dhp

# Another equivalent specification, telling specifyModel to add paths for 
#   variances and covariance of RGenAsp and FGenAsp:
 
model.dhp <- specifyModel(covs="RGenAsp, FGenAsp")
RParAsp  -> RGenAsp, gam11
RIQ      -> RGenAsp, gam12
RSES     -> RGenAsp, gam13
FSES     -> RGenAsp, gam14
RSES     -> FGenAsp, gam23
FSES     -> FGenAsp, gam24
FIQ      -> FGenAsp, gam25
FParAsp  -> FGenAsp, gam26
FGenAsp  -> RGenAsp, beta12
RGenAsp  -> FGenAsp, beta21
RGenAsp  -> ROccAsp,  NA,     1
RGenAsp  -> REdAsp,  lam21
FGenAsp  -> FOccAsp,  NA,     1
FGenAsp  -> FEdAsp,  lam42

model.dhp

# The same model in equation format:

model.dhp.1 <- specifyEquations(covs="RGenAsp, FGenAsp")
RGenAsp = gam11*RParAsp + gam12*RIQ + gam13*RSES + gam14*FSES + beta12*FGenAsp
FGenAsp = gam23*RSES + gam24*FSES + gam25*FIQ + gam26*FParAsp + beta21*RGenAsp
ROccAsp = 1*RGenAsp
REdAsp = lam21(1)*RGenAsp  # to illustrate setting start values
FOccAsp = 1*FGenAsp
FEdAsp = lam42(1)*FGenAsp

model.dhp

classifyVariables(model.dhp)

# updating the model to impose equality constraints
#  and to rename the latent variables and gamma parameters

model.dhp.eq <- update(model.dhp)
delete, RSES -> FGenAsp
delete, FSES -> FGenAsp
delete, FIQ  -> FGenAsp
delete, FParAsp -> FGenAs
delete, RGenAsp  -> FGenAsp
add, RSES     -> FGenAsp, gam14,  NA
add, FSES     -> FGenAsp, gam13,  NA
add, FIQ      -> FGenAsp, gam12,  NA
add, FParAsp  -> FGenAsp, gam26,  NA
add, RGenAsp  -> FGenAsp, beta12, NA
replace, gam, gamma
replace, Gen, General

model.dhp.eq

# A three-factor CFA model for the Thurstone mental-tests data, 
#    specified three equivalent ways:

R.thur <- readMoments(diag=FALSE, 
    names=c('Sentences','Vocabulary',
            'Sent.Completion','First.Letters','4.Letter.Words','Suffixes',
            'Letter.Series','Pedigrees', 'Letter.Group'))
.828                                              
.776   .779                                        
.439   .493    .46                                 
.432   .464    .425   .674                           
.447   .489    .443   .59    .541                    
.447   .432    .401   .381    .402   .288              
.541   .537    .534   .35    .367   .32   .555        
.38   .358    .359   .424    .446   .325   .598   .452

	#  (1a) in CFA format:

mod.cfa.thur.c <- cfa(reference.indicators=FALSE)
FA: Sentences, Vocabulary, Sent.Completion
FB: First.Letters, 4.Letter.Words, Suffixes
FC: Letter.Series, Pedigrees, Letter.Group

cfa.thur.c <- sem(mod.cfa.thur.c, R.thur, 213)
summary(cfa.thur.c)

	#  (1b) in CFA format, using reference indicators:
	
mod.cfa.thur.r <- cfa()
FA: Sentences, Vocabulary, Sent.Completion
FB: First.Letters, 4.Letter.Words, Suffixes
FC: Letter.Series, Pedigrees, Letter.Group

cfa.thur.r <- sem(mod.cfa.thur.r, R.thur, 213)
summary(cfa.thur.r)

	#  (2) in equation format:

mod.cfa.thur.e <- specifyEquations(covs="F1, F2, F3")
Sentences = lam11*F1
Vocabulary = lam21*F1
Sent.Completion = lam31*F1
First.Letters = lam42*F2
4.Letter.Words = lam52*F2
Suffixes = lam62*F2
Letter.Series = lam73*F3
Pedigrees = lam83*F3
Letter.Group = lam93*F3
V(F1) = 1
V(F2) = 1
V(F3) = 1

cfa.thur.e <- sem(mod.cfa.thur.e, R.thur, 213)
summary(cfa.thur.e)

	#  (3) in path format:

mod.cfa.thur.p <- specifyModel(covs="F1, F2, F3")
F1 -> Sentences,                      lam11
F1 -> Vocabulary,                     lam21
F1 -> Sent.Completion,                lam31
F2 -> First.Letters,                  lam41
F2 -> 4.Letter.Words,                 lam52
F2 -> Suffixes,                       lam62
F3 -> Letter.Series,                  lam73
F3 -> Pedigrees,                      lam83
F3 -> Letter.Group,                   lam93
F1 <-> F1,                            NA,     1
F2 <-> F2,                            NA,     1
F3 <-> F3,                            NA,     1

cfa.thur.p <- sem(mod.cfa.thur.p, R.thur, 213)
summary(cfa.thur.p)

# The Thursstone CFA model with equality constraints on the
#  factor loadings and error variances

mod.cfa.thur.ceq <- cfa(reference.indicators=FALSE)
FA: Sentences = Vocabulary = Sent.Completion
FB: First.Letters = 4.Letter.Words = Suffixes
FC: Letter.Series = Pedigrees = Letter.Group
var: Sentences = Vocabulary = Sent.Completion
var: First.Letters = 4.Letter.Words = Suffixes
var: Letter.Series = Pedigrees = Letter.Group

cfa.thur.ceq <- sem(mod.cfa.thur.ceq, R.thur, 213)
summary(cfa.thur.ceq)
anova(cfa.thur.c, cfa.thur.ceq)
pathDiagram(cfa.thur.ceq, ignore.double=FALSE, ignore.self=TRUE,
    min.rank="FA, FB, FC", edge.labels="values")

# a multigroup CFA model fit to the Holzinger-Swineford
#   mental-tests data 

mod.hs <- cfa()
spatial: visual, cubes, paper, flags
verbal: general, paragrap, sentence, wordc, wordm
memory: wordr, numberr, figurer, object, numberf, figurew
math: deduct, numeric, problemr, series, arithmet

mod.mg <- multigroupModel(mod.hs, groups=c("Female", "Male")) 

sem.mg <- sem(mod.mg, data=HS.data, group="Gender",
              formula = ~ visual + cubes + paper + flags +
              general + paragrap + sentence + wordc + wordm +
              wordr + numberr + figurer + object + numberf + figurew +
              deduct + numeric + problemr + series + arithmet
              )
summary(sem.mg)

	# with cross-group equality constraints:
	
mod.mg.eq <- multigroupModel(mod.hs, groups=c("Female", "Male"), allEqual=TRUE)

sem.mg.eq <- sem(mod.mg.eq, data=HS.data, group="Gender",
              formula = ~ visual + cubes + paper + flags +
                general + paragrap + sentence + wordc + wordm +
                wordr + numberr + figurer + object + numberf + figurew +
                deduct + numeric + problemr + series + arithmet
              )
summary(sem.mg.eq)
    
## End(Not run)

Standardized Coefficients for Structural Equation Models

Description

These functions calculate standardized regression coefficients for structural equation models. The function stdCoef is simply an abbreviation for standardizedCoefficients.

Usage

standardizedCoefficients(object, ...)
## S3 method for class 'sem'
standardizedCoefficients(object, 
    digits = getOption("digits"), oneheaded = TRUE, twoheaded = TRUE, ...)
## S3 method for class 'msem'
standardizedCoefficients(object, ...)

stdCoef(...)

Arguments

Value

Returns a data frame with the coefficients, labelled both by parameter names and by arrows in the path diagram for the model. The msem (multigroup) method computes and prints the standardized coefficients for each group; it does not return a useful result.

Author(s)

John Fox jfox@mcmaster.ca and Adam Kramer

References

Bollen, K. A. (1989) Structural Equations With Latent Variables. Wiley.

See Also

sem

Examples

# In the first example, readMoments() and specifyModel() read from the
# input stream. This example cannot be executed via example() but can be entered
# at the command prompt. The example is repeated using file input;
# this example can be executed via example(). 
    ## Not run: 
# Duncan, Haller, and Portes peer-influences model

R.DHP <- readMoments(diag=FALSE, names=c("ROccAsp", "REdAsp", "FOccAsp", 
                "FEdAsp", "RParAsp", "RIQ", "RSES", "FSES", "FIQ", "FParAsp"))
    .6247     
    .3269  .3669       
    .4216  .3275  .6404
    .2137  .2742  .1124  .0839
    .4105  .4043  .2903  .2598  .1839
    .3240  .4047  .3054  .2786  .0489  .2220
    .2930  .2407  .4105  .3607  .0186  .1861  .2707
    .2995  .2863  .5191  .5007  .0782  .3355  .2302  .2950
    .0760  .0702  .2784  .1988  .1147  .1021  .0931 -.0438  .2087
            
model.dhp <- specifyModel()
    RParAsp  -> RGenAsp, gam11,  NA
    RIQ      -> RGenAsp, gam12,  NA
    RSES     -> RGenAsp, gam13,  NA
    FSES     -> RGenAsp, gam14,  NA
    RSES     -> FGenAsp, gam23,  NA
    FSES     -> FGenAsp, gam24,  NA
    FIQ      -> FGenAsp, gam25,  NA
    FParAsp  -> FGenAsp, gam26,  NA
    FGenAsp  -> RGenAsp, beta12, NA
    RGenAsp  -> FGenAsp, beta21, NA
    RGenAsp  -> ROccAsp,  NA,     1
    RGenAsp  -> REdAsp,  lam21,  NA
    FGenAsp  -> FOccAsp,  NA,     1
    FGenAsp  -> FEdAsp,  lam42,  NA
    RGenAsp <-> RGenAsp, ps11,   NA
    FGenAsp <-> FGenAsp, ps22,   NA
    RGenAsp <-> FGenAsp, ps12,   NA
    ROccAsp <-> ROccAsp, theta1, NA
    REdAsp  <-> REdAsp,  theta2, NA
    FOccAsp <-> FOccAsp, theta3, NA
    FEdAsp  <-> FEdAsp,  theta4, NA

sem.dhp <- sem(model.dhp, R.DHP, 329,
    fixed.x=c('RParAsp', 'RIQ', 'RSES', 'FSES', 'FIQ', 'FParAsp'))
standardizedCoefficients(sem.dhp) 
	
## End(Not run)
# The following example can be executed via example():

etc <- system.file(package="sem", "etc") # path to data and model files
   
(R.DHP <- readMoments(file=file.path(etc, "R-DHP.txt"),
				diag=FALSE, names=c("ROccAsp", "REdAsp", "FOccAsp", 
                "FEdAsp", "RParAsp", "RIQ", "RSES", "FSES", "FIQ", "FParAsp")))
(model.dhp <- specifyModel(file=file.path(etc, "model-DHP.txt")))
(sem.dhp <- sem(model.dhp, R.DHP, 329,
    fixed.x=c('RParAsp', 'RIQ', 'RSES', 'FSES', 'FIQ', 'FParAsp')))
standardizedCoefficients(sem.dhp) 

Two-Stage Least Squares

Description

Fits a regression equation, such as an equation in a structural-equation model, by two-stage least squares. This is equivalent to direct instrumental-variables estimation when the number of instruments is equal to the number of predictors.

Usage

## S3 method for class 'formula'
tsls(formula, instruments, data, subset, weights, 
	na.action, contrasts=NULL, ...)
## Default S3 method:
tsls(y, X, Z, w, names=NULL, ...)

## S3 method for class 'tsls'
print(x, ...)
## S3 method for class 'tsls'
summary(object, digits=getOption("digits"), ...)
## S3 method for class 'summary.tsls'
print(x, ...)
## S3 method for class 'tsls'
anova(object, model.2, s2, dfe, ...)

## S3 method for class 'tsls'
fitted(object, ...)
## S3 method for class 'tsls'
residuals(object, ...)
## S3 method for class 'tsls'
coef(object, ...)
## S3 method for class 'tsls'
vcov(object, ...)

Arguments

Value

tsls.formula returns an object of class tsls, with the following components:

Author(s)

John Fox jfox@mcmaster.ca

References

Fox, J. (1979) Simultaneous equation models and two-stage least-squares. In Schuessler, K. F. (ed.) Sociological Methodology 1979, Jossey-Bass.

Greene, W. H. (1993) Econometric Analysis, Second Edition, Macmillan.

See Also

sem

Examples

summary(tsls(Q ~ P + D, ~ D + F + A, data=Kmenta))     # demand equation

summary(tsls(Q ~ P + F + A, ~ D + F + A, data=Kmenta)) # supply equation

anova(tsls(Q ~ P + F + A, ~ D + F + A, data=Kmenta),
    tsls(Q ~ 1, ~ D + F + A, data=Kmenta))