Description Usage Arguments Details Value Warning Author(s) References See Also Examples
sem
fits general structural equation models (with both observed and
unobserved variables) by the method of maximum likelihood, assuming
multinormal errors. Observed variables are also called indicators or
manifest variables; unobserved variables are also called factors
or latent variables. Normally, the generic function (sem
) would
be used.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 | sem(ram, ...)
## S3 method for class 'mod'
sem(ram, S, N, obs.variables=rownames(S), fixed.x=NULL, debug=FALSE, ...)
## Default S3 method:
sem(ram, S, N, param.names = paste("Param", 1:t, sep = ""),
var.names = paste("V", 1:m, sep = ""), fixed.x = NULL, raw=FALSE,
debug = FALSE, analytic.gradient = TRUE, warn = FALSE, maxiter = 500,
par.size=c('ones', 'startvalues'), refit=TRUE, start.tol=1E-6, ...)
startvalues(S, ram, debug = FALSE, tol=1E-6)
## S3 method for class 'sem'
print(x, ...)
## S3 method for class 'sem'
summary(object, digits=5, conf.level=0.9, ...)
## S3 method for class 'sem'
deviance(object, ...)
## S3 method for class 'sem'
df.residual(object, ...)
## S3 method for class 'sem'
anova(object, model.2, ...)
## S3 method for class 'sem'
coef(object, ...)
## S3 method for class 'sem'
vcov(object, ...)
|
ram |
RAM specification, which is a simple encoding of the path
diagram for the model. The |
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. |
obs.variables |
names of observed variables, by default taken from the row names of
the covariance matrix |
fixed.x |
names (if the |
raw |
|
debug |
if |
... |
arguments to be passed down, including from |
param.names |
names of the t free parameters, given in their numerical order;
default names are |
var.names |
names of the m entries of the v vector
(typically the observed and latent variables — see below), given in their
numerical order; default names are |
analytic.gradient |
if |
warn |
if |
maxiter |
the maximum number of iterations for the optimization performed by the
|
par.size |
the anticipated size of the free parameters; if |
refit |
if |
start.tol, tol |
if the magnitude of an automatic start value is less than |
object, x |
an object of class |
digits |
number of digits for printed output. |
conf.level |
level for confidence interval for the RMSEA index (default is .9). |
model.2 |
an object of class |
The model is set up using 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).
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:
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.
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.
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 specify.model
function,
which returns an object of class mod
. This process is illustrated in the examples
below.
A numeric ram
matrix consists of five columns, as follows:
1 (directed arrow) or 2 (covariance).
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.
the index of the variable at the tail of a directional arrow, or at the other end of a bidirectional arrow.
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.
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.
sem
fits the model by calling the nlm
optimizer
to minimize the negative log-likelihood for the model.
If nlm
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
.
The sem
methods for the generic functions deviance
and df.residual
functions
return, respectively, minus twice the difference in the log-likelihoods for the fitted model and
a saturated model, and the degrees of freedom associated with the deviance.
sem
returns an object of class sem
, with the following elements:
ram |
RAM matrix, including any rows generated for covariances among fixed exogenous variables; column 5 includes computed start values. |
coeff |
estimates of free parameters. |
criterion |
fitting criterion — minus twice the difference in the log-liklihood between the fitted model and a saturated model, divided by N - 1 (or N if the model is fit to a raw moment matrix). |
cov |
estimated asymptotic covariance matrix of parameter estimates. |
S |
observed covariance matrix. |
J |
RAM selection matrix, J, which picks out observed variables. |
C |
model-reproduced covariance matrix. |
A |
RAM A matrix. |
P |
RAM P 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. |
par.posn |
indices of free parameters. |
var.names |
vector of variable names. |
observed |
indices of observed variables. |
convergence |
convergence code returned by |
iterations |
number of iterations performed. |
raw |
|
chisqNull |
Unless the model is fit to a raw moment matrix, the chisquare value associated with a null model in which all of the observed variables are uncorrelated. |
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.
John Fox jfox@mcmaster.ca
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,
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# argument of specify.model() and read.moments() requires that the model
# specification and covariances, correlations, or raw moments be entered
# at the command prompt. The examples can be copied and run in the R console,
# however. See ?specify.model and ?read.moments for further information.
## Not run:
# ------------- Duncan, Haller and Portes peer-influences model ----------------------
# A nonrecursive SEM with unobserved endogenous variables and fixed exogenous variables
R.DHP <- read.moments(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 <- specify.model()
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.1 <- sem(model.dhp, R.DHP, 329,
fixed.x=c('RParAsp', 'RIQ', 'RSES', 'FSES', 'FIQ', 'FParAsp'))
summary(sem.dhp.1)
## Model Chisquare = 26.697 Df = 15 Pr(>Chisq) = 0.031302
## Chisquare (null model) = 872 Df = 45
## Goodness-of-fit index = 0.98439
## Adjusted goodness-of-fit index = 0.94275
## RMSEA index = 0.048759 90 % CI: (0.014517, 0.07831)
## Bentler-Bonnett NFI = 0.96938
## Tucker-Lewis NNFI = 0.95757
## Bentler CFI = 0.98586
## SRMR = 0.020204
## BIC = -60.244
##
## Normalized Residuals
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -0.7990 -0.1180 0.0000 -0.0120 0.0397 1.5700
##
## Parameter Estimates
## Estimate Std Error z value Pr(>|z|)
## gam11 0.161224 0.038487 4.1890 2.8019e-05 RGenAsp <--- RParAsp
## gam12 0.249653 0.044580 5.6001 2.1428e-08 RGenAsp <--- RIQ
## gam13 0.218404 0.043476 5.0235 5.0730e-07 RGenAsp <--- RSES
## gam14 0.071843 0.050335 1.4273 1.5350e-01 RGenAsp <--- FSES
## gam23 0.061894 0.051738 1.1963 2.3158e-01 FGenAsp <--- RSES
## gam24 0.228868 0.044495 5.1437 2.6938e-07 FGenAsp <--- FSES
## gam25 0.349039 0.044551 7.8346 4.6629e-15 FGenAsp <--- FIQ
## gam26 0.159535 0.040129 3.9755 7.0224e-05 FGenAsp <--- FParAsp
## beta12 0.184226 0.096207 1.9149 5.5506e-02 RGenAsp <--- FGenAsp
## beta21 0.235458 0.119742 1.9664 4.9255e-02 FGenAsp <--- RGenAsp
## lam21 1.062674 0.091967 11.5549 0.0000e+00 REdAsp <--- RGenAsp
## lam42 0.929727 0.071152 13.0668 0.0000e+00 FEdAsp <--- FGenAsp
## ps11 0.280987 0.046311 6.0674 1.2999e-09 RGenAsp <--> RGenAsp
## ps22 0.263836 0.044902 5.8759 4.2067e-09 FGenAsp <--> FGenAsp
## ps12 -0.022601 0.051649 -0.4376 6.6168e-01 FGenAsp <--> RGenAsp
## theta1 0.412145 0.052211 7.8939 2.8866e-15 ROccAsp <--> ROccAsp
## theta2 0.336148 0.053323 6.3040 2.9003e-10 REdAsp <--> REdAsp
## theta3 0.311194 0.046665 6.6687 2.5800e-11 FOccAsp <--> FOccAsp
## theta4 0.404604 0.046733 8.6578 0.0000e+00 FEdAsp <--> FEdAsp
##
## Iterations = 28
# Fit the model using a numerical ram specification
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, params.dhp, vars.dhp, fixed.x=5:10)
summary(sem.dhp.2)
## Model Chisquare = 26.697 Df = 15 Pr(>Chisq) = 0.031302
## Chisquare (null model) = 872 Df = 45
## Goodness-of-fit index = 0.98439
## Adjusted goodness-of-fit index = 0.94275
## RMSEA index = 0.048759 90 % CI: (0.014517, 0.07831)
## Bentler-Bonnett NFI = 0.96938
## Tucker-Lewis NNFI = 0.95757
## Bentler CFI = 0.98586
## SRMR = 0.020204
## BIC = -60.244
##
## Normalized Residuals
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -0.7990 -0.1180 0.0000 -0.0120 0.0397 1.5700
##
## Parameter Estimates
## Estimate Std Error z value Pr(>|z|)
## lam21 1.062674 0.091967 11.5549 0.0000e+00 REdAsp <--- RGenAsp
## lam42 0.929727 0.071152 13.0668 0.0000e+00 FEdAsp <--- FGenAsp
## gam11 0.161224 0.038487 4.1890 2.8019e-05 RGenAsp <--- RParAsp
## gam12 0.249653 0.044580 5.6001 2.1428e-08 RGenAsp <--- RIQ
## gam13 0.218404 0.043476 5.0235 5.0730e-07 RGenAsp <--- RSES
## gam14 0.071843 0.050335 1.4273 1.5350e-01 RGenAsp <--- FSES
## gam23 0.061894 0.051738 1.1963 2.3158e-01 FGenAsp <--- RSES
## gam24 0.228868 0.044495 5.1437 2.6938e-07 FGenAsp <--- FSES
## gam25 0.349039 0.044551 7.8346 4.6629e-15 FGenAsp <--- FIQ
## gam26 0.159535 0.040129 3.9755 7.0224e-05 FGenAsp <--- FParAsp
## beta12 0.184226 0.096207 1.9149 5.5506e-02 RGenAsp <--- FGenAsp
## beta21 0.235458 0.119742 1.9664 4.9255e-02 FGenAsp <--- RGenAsp
## theta1 0.412145 0.052211 7.8939 2.8866e-15 ROccAsp <--> ROccAsp
## theta2 0.336148 0.053323 6.3040 2.9003e-10 REdAsp <--> REdAsp
## theta3 0.311194 0.046665 6.6687 2.5800e-11 FOccAsp <--> FOccAsp
## theta4 0.404604 0.046733 8.6578 0.0000e+00 FEdAsp <--> FEdAsp
## psi11 0.280987 0.046311 6.0674 1.2999e-09 RGenAsp <--> RGenAsp
## psi22 0.263836 0.044902 5.8759 4.2067e-09 FGenAsp <--> FGenAsp
## psi12 -0.022601 0.051649 -0.4376 6.6168e-01 RGenAsp <--> FGenAsp
## Iterations = 28
# -------------------- Wheaton et al. alienation data ----------------------
S.wh <- read.moments(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 <- specify.model()
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)
## Model Chisquare = 13.485 Df = 9 Pr(>Chisq) = 0.14186
## Chisquare (null model) = 2131.4 Df = 15
## Goodness-of-fit index = 0.99527
## Adjusted goodness-of-fit index = 0.98896
## RMSEA index = 0.023136 90 % CI: (NA, 0.046997)
## Bentler-Bonnett NFI = 0.99367
## Tucker-Lewis NNFI = 0.99647
## Bentler CFI = 0.99788
## SRMR = 0.014998
## BIC = -48.051
##
## Normalized Residuals
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -1.26000 -0.13100 0.00014 -0.02870 0.11400 0.87400
##
## Parameter Estimates
## Estimate Std Error z value Pr(>|z|)
## lamb 5.36880 0.433982 12.3710 0.0000e+00 SEI <--- SES
## gam1 -0.62994 0.056128 -11.2233 0.0000e+00 Alienation67 <--- SES
## beta 0.59312 0.046820 12.6680 0.0000e+00 Alienation71 <--- Alienation67
## gam2 -0.24086 0.055202 -4.3632 1.2817e-05 Alienation71 <--- SES
## the1 3.60787 0.200589 17.9864 0.0000e+00 Anomia67 <--> Anomia67
## the2 3.59494 0.165234 21.7567 0.0000e+00 Powerless67 <--> Powerless67
## the3 2.99366 0.498972 5.9996 1.9774e-09 Education <--> Education
## the4 259.57583 18.321121 14.1681 0.0000e+00 SEI <--> SEI
## the5 0.90579 0.121710 7.4422 9.9032e-14 Anomia71 <--> Anomia67
## psi1 5.67050 0.422906 13.4084 0.0000e+00 Alienation67 <--> Alienation67
## psi2 4.51481 0.334993 13.4773 0.0000e+00 Alienation71 <--> Alienation71
## phi 6.61632 0.639506 10.3460 0.0000e+00 SES <--> SES
##
## Iterations = 78
# The same model, but treating one loading for each latent variable as free
# (and equal to each other).
model.wh.2 <- specify.model()
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)
## Model Chisquare = 12.673 Df = 8 Pr(>Chisq) = 0.12360
## Chisquare (null model) = 2131.4 Df = 15
## Goodness-of-fit index = 0.99553
## Adjusted goodness-of-fit index = 0.98828
## RMSEA index = 0.025049 90 % CI: (NA, 0.04985)
## Bentler-Bonnett NFI = 0.99405
## Tucker-Lewis NNFI = 0.99586
## Bentler CFI = 0.9978
## SRMR = 0.012981
## BIC = -42.026
##
## Normalized Residuals
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -0.997000 -0.140000 0.000295 -0.028800 0.100000 0.759000
##
## Parameter Estimates
## Estimate Std Error z value Pr(>|z|)
## lamby 0.86261 0.033383 25.8402 0.0000e+00 Powerless67 <--- Alienation67
## lambx 5.35302 0.432591 12.3743 0.0000e+00 SEI <--- SES
## gam1 -0.62129 0.056142 -11.0663 0.0000e+00 Alienation67 <--- SES
## beta 0.59428 0.047040 12.6335 0.0000e+00 Alienation71 <--- Alienation67
## gam2 -0.23580 0.054684 -4.3121 1.6173e-05 Alienation71 <--- SES
## the1 3.74499 0.249823 14.9906 0.0000e+00 Anomia67 <--> Anomia67
## the2 3.49378 0.200754 17.4033 0.0000e+00 Powerless67 <--> Powerless67
## the3 2.97409 0.499661 5.9522 2.6456e-09 Education <--> Education
## the4 260.13252 18.298141 14.2163 0.0000e+00 SEI <--> SEI
## the5 0.90377 0.121818 7.4190 1.1791e-13 Anomia71 <--> Anomia67
## psi1 5.47380 0.464073 11.7951 0.0000e+00 Alienation67 <--> Alienation67
## psi2 4.36410 0.362722 12.0315 0.0000e+00 Alienation71 <--> Alienation71
## phi 6.63576 0.640425 10.3615 0.0000e+00 SES <--> SES
##
## Iterations = 79
##
# Compare the two models by a likelihood-ratio test:
anova(sem.wh.1, sem.wh.2)
## LR Test for Difference Between Models
##
## Model Df Model Chisq Df LR Chisq Pr(>Chisq)
## Model 1 9 13.4851
## Model 2 8 12.6731 1 0.8119 0.3676
# ----------------------- Thurstone data ---------------------------------------
# Second-order confirmatory factor analysis, from the SAS manual for PROC CALIS
R.thur <- read.moments(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 <- specify.model()
F1 -> Sentences, lam11, NA
F1 -> Vocabulary, lam21, NA
F1 -> Sent.Completion, lam31, NA
F2 -> First.Letters, lam41, NA
F2 -> 4.Letter.Words, lam52, NA
F2 -> Suffixes, lam62, NA
F3 -> Letter.Series, lam73, NA
F3 -> Pedigrees, lam83, NA
F3 -> Letter.Group, lam93, NA
F4 -> F1, gam1, NA
F4 -> F2, gam2, NA
F4 -> F3, gam3, NA
Sentences <-> Sentences, th1, NA
Vocabulary <-> Vocabulary, th2, NA
Sent.Completion <-> Sent.Completion, th3, NA
First.Letters <-> First.Letters, th4, NA
4.Letter.Words <-> 4.Letter.Words, th5, NA
Suffixes <-> Suffixes, th6, NA
Letter.Series <-> Letter.Series, th7, NA
Pedigrees <-> Pedigrees, th8, NA
Letter.Group <-> Letter.Group, th9, NA
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)
## Model Chisquare = 38.196 Df = 24 Pr(>Chisq) = 0.033101
## Chisquare (null model) = 1101.9 Df = 36
## Goodness-of-fit index = 0.95957
## Adjusted goodness-of-fit index = 0.9242
## RMSEA index = 0.052822 90 % CI: (0.015262, 0.083067)
## Bentler-Bonnett NFI = 0.96534
## Tucker-Lewis NNFI = 0.98002
## Bentler CFI = 0.98668
## SRMR = 0.043595
## BIC = -90.475
##
## Normalized Residuals
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -9.72e-01 -4.16e-01 -4.20e-07 4.01e-02 9.39e-02 1.63e+00
##
## Parameter Estimates
## Estimate Std Error z value Pr(>|z|)
## lam11 0.51512 0.064964 7.9293 2.2204e-15 Sentences <--- F1
## lam21 0.52031 0.065162 7.9849 1.3323e-15 Vocabulary <--- F1
## lam31 0.48743 0.062422 7.8087 5.7732e-15 Sent.Completion <--- F1
## lam41 0.52112 0.063137 8.2538 2.2204e-16 First.Letters <--- F2
## lam52 0.49707 0.059673 8.3298 0.0000e+00 4.Letter.Words <--- F2
## lam62 0.43806 0.056479 7.7562 8.6597e-15 Suffixes <--- F2
## lam73 0.45244 0.071371 6.3392 2.3100e-10 Letter.Series <--- F3
## lam83 0.41729 0.061037 6.8367 8.1020e-12 Pedigrees <--- F3
## lam93 0.40763 0.064524 6.3175 2.6584e-10 Letter.Group <--- F3
## gam1 1.44381 0.264173 5.4654 4.6184e-08 F1 <--- F4
## gam2 1.25383 0.216597 5.7888 7.0907e-09 F2 <--- F4
## gam3 1.40655 0.279331 5.0354 4.7681e-07 F3 <--- F4
## th1 0.18150 0.028400 6.3907 1.6517e-10 Sentences <--> Sentences
## th2 0.16493 0.027797 5.9334 2.9678e-09 Vocabulary <--> Vocabulary
## th3 0.26713 0.033468 7.9816 1.5543e-15 Sent.Completion <--> Sent.Completion
## th4 0.30150 0.050686 5.9484 2.7073e-09 First.Letters <--> First.Letters
## th5 0.36450 0.052358 6.9617 3.3618e-12 4.Letter.Words <--> 4.Letter.Words
## th6 0.50642 0.059963 8.4455 0.0000e+00 Suffixes <--> Suffixes
## th7 0.39033 0.061599 6.3367 2.3474e-10 Letter.Series <--> Letter.Series
## th8 0.48137 0.065388 7.3618 1.8141e-13 Pedigrees <--> Pedigrees
## th9 0.50510 0.065227 7.7437 9.5479e-15 Letter.Group <--> Letter.Group
##
## Iterations = 54
##
#------------------------- Kerchoff/Kenney path analysis ---------------------
# An observed-variable recursive SEM from the LISREL manual
R.kerch <- read.moments(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 <- specify.model()
Intelligence -> Grades, gam51, NA
Siblings -> Grades, gam52, NA
FatherEd -> Grades, gam53, NA
FatherOcc -> Grades, gam54, NA
Intelligence -> EducExp, gam61, NA
Siblings -> EducExp, gam62, NA
FatherEd -> EducExp, gam63, NA
FatherOcc -> EducExp, gam64, NA
Grades -> EducExp, beta65, NA
Intelligence -> OccupAsp, gam71, NA
Siblings -> OccupAsp, gam72, NA
FatherEd -> OccupAsp, gam73, NA
FatherOcc -> OccupAsp, gam74, NA
Grades -> OccupAsp, beta75, NA
EducExp -> OccupAsp, beta76, NA
Grades <-> Grades, psi5, NA
EducExp <-> EducExp, psi6, NA
OccupAsp <-> OccupAsp, psi7, NA
sem.kerch <- sem(model.kerch, R.kerch, 737, fixed.x=c('Intelligence','Siblings',
'FatherEd','FatherOcc'))
summary(sem.kerch)
## Model Chisquare = 3.2685e-13 Df = 0 Pr(>Chisq) = NA
## Chisquare (null model) = 1664.3 Df = 21
## Goodness-of-fit index = 1
## BIC = 3.2685e-13
##
## Normalized Residuals
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -4.28e-15 0.00e+00 0.00e+00 7.62e-16 1.47e-15 5.17e-15
##
## Parameter Estimates
## Estimate Std Error z value Pr(>|z|)
## gam51 0.525902 0.031182 16.86530 0.0000e+00 Grades <--- Intelligence
## gam52 -0.029942 0.030149 -0.99314 3.2064e-01 Grades <--- Siblings
## gam53 0.118966 0.038259 3.10951 1.8740e-03 Grades <--- FatherEd
## gam54 0.040603 0.037785 1.07456 2.8257e-01 Grades <--- FatherOcc
## gam61 0.160270 0.032710 4.89979 9.5940e-07 EducExp <--- Intelligence
## gam62 -0.111779 0.026876 -4.15899 3.1966e-05 EducExp <--- Siblings
## gam63 0.172719 0.034306 5.03461 4.7882e-07 EducExp <--- FatherEd
## gam64 0.151852 0.033688 4.50758 6.5571e-06 EducExp <--- FatherOcc
## beta65 0.405150 0.032838 12.33799 0.0000e+00 EducExp <--- Grades
## gam71 -0.039405 0.034500 -1.14215 2.5339e-01 OccupAsp <--- Intelligence
## gam72 -0.018825 0.028222 -0.66700 5.0477e-01 OccupAsp <--- Siblings
## gam73 -0.041333 0.036216 -1.14126 2.5376e-01 OccupAsp <--- FatherEd
## gam74 0.099577 0.035446 2.80924 4.9658e-03 OccupAsp <--- FatherOcc
## beta75 0.157912 0.037443 4.21738 2.4716e-05 OccupAsp <--- Grades
## beta76 0.549593 0.038260 14.36486 0.0000e+00 OccupAsp <--- EducExp
## psi5 0.650995 0.033946 19.17743 0.0000e+00 Grades <--> Grades
## psi6 0.516652 0.026943 19.17590 0.0000e+00 EducExp <--> EducExp
## psi7 0.556617 0.029026 19.17644 0.0000e+00 OccupAsp <--> OccupAsp
##
## Iterations = 0
#------------------- 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 <- read.moments(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 <- specify.model()
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)
## Model fit to raw moment matrix.
##
## Model Chisquare = 83.791 Df = 8 Pr(>Chisq) = 8.4377e-15
## BIC = 41.246
##
## Normalized Residuals
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -0.15300 -0.01840 0.00132 -0.00576 0.02400 0.07760
##
## Parameter Estimates
## Estimate Std Error z value Pr(>|z|)
## B2 8.61354 0.135438 63.5976 0 WISC2 <--- C
## B3 11.64054 0.168854 68.9387 0 WISC3 <--- C
## B4 15.40323 0.213071 72.2916 0 WISC4 <--- C
## Mc 3.01763 0.060690 49.7219 0 C <--- UNIT
## Vc 0.44343 0.047704 9.2955 0 C <--> C
## Vd 11.78832 0.674060 17.4885 0 WISC1 <--> WISC1
##
## Iterations = 37
## End(Not run)
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