| sem {sem} | R Documentation |
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
model |
RAM specification, which is a simple encoding of the path
diagram for the model. The model may be given either in symbolic
form (as a |
S |
covariance matrix among observed variables; may be input as a symmetric matrix,
or as a lower- or upper-triangular matrix. |
N |
number of observations on which the covariance matrix is based; for a multigroup model, a vector
of group |
data |
As a generally preferable alternative to specifying |
start.fn |
a function to compute startvalues for the free parameters of the model;
two functions are supplied, |
na.action |
a function to process missing data, if raw data are supplied in the |
raw |
|
pattern.number, valid.data.patterns |
these arguments pass information about valid (i.e., non-missing) data patterns and normally would not be specified directly by the user. |
use.means |
When raw data are supplied and intercepts are included in the model, use the
observed-variable means as start values for the intercepts; the default is |
obs.variables |
names of observed variables, by default taken from the row names of
the covariance or moment matrix |
fixed.x |
names (if the |
formula |
a one-sided formula, to be applied to |
robust |
In |
semmod |
a |
debug |
if |
... |
arguments to be passed down, including from |
param.names |
names of the |
var.names |
names of the |
analytic.gradient |
if |
warn |
if |
maxiter |
the maximum number of iterations for the optimization of the objective function, to be passed to the optimizer. |
par.size |
the anticipated size of the free parameters; if |
start.tol, tol |
if the magnitude of an automatic start value is less than |
optimizer |
a function to be used to minimize the objective function; the default for single-group models is
|
objective |
An objective function to be minimized, sometimes called a “fit” function
in the SEM literature. The default for single-group models is |
cls |
primary class to be assigned to the result; normally this is not specified directly, but raither is inferred from the objective function. |
ram |
numeric RAM matrix. |
object |
an object of class |
standardized |
if |
analytic |
return an analytic (as opposed to numeric) estimate of the coefficient covariance matrix;
at present only available for the |
group |
for a multigroup model, the quoted name of the group variable; if the |
groups |
a character vector giving the names of the groups; will be ignored if |
start |
if |
initial.maxiter |
if |
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,AandBare 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
NAproduces a fixed parameter.- 3. Value:
start value for a free parameter or value of a fixed parameter. If given as
NA,semwill 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, wherenis the number of rows/columns in the covariance matrixS, with the indices corresponding to the variables' positions inS. Variable indices abovenrepresent 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:
var.names |
vector of variable names. |
ram |
RAM matrix, including any rows generated for covariances among fixed exogenous variables; column 5 includes computed start values. |
S |
observed covariance matrix. |
J |
RAM selection matrix, |
n.fix |
number of fixed exogenous variables. |
n |
number of observed variables. |
N |
number of observations. |
m |
number of variables (observed plus unobserved). |
t |
number of free parameters. |
raw |
|
data |
the observed-variable data matrix, or |
semmod |
the |
optimizer |
the optimizer function. |
objective |
the objective function. |
coeff |
estimates of free parameters. |
vcov |
estimated asymptotic covariance matrix of parameter estimates, based on a numeric Hessian,
if supplied by the optimizer; otherwise |
par.posn |
indices of free parameters. |
convergence |
|
iterations |
number of iterations performed. |
criterion |
value of the objective function at the minimum. |
C |
model-reproduced covariance matrix. |
A |
RAM |
P |
RAM |
adj.obj |
robust adjusted value of the objective function; |
robust.vcov |
robust estimated coefficient covariance matrix; |
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