| sem {sem} | R Documentation |
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.
sem(ram, ...)
sem.mod(ram, S, N, obs.variables=rownames(S), fixed.x=NULL, debug=F, ...)
sem.default(ram, S, N, param.names=paste("Param", 1:t, sep = ""),
var.names=paste("V", 1:m, sep = ""), fixed.x=NULL,
analytic.gradient=T, heywood=F, warn=F, control=list())
startvalues(S, ram)
print.sem(x, ...)
summary.sem(object, digits=5, ...)
ram |
RAM specification, which is a simple encoding of the path
diagram for the model. The ram matrix may be given either in symbolic (character)
form, invoking sem.mod, which calls sem.default after setting up the model,
or (less conveniently) in numeric form, invoking sem.default directly
(see Details below). |
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 S. |
fixed.x |
names (if the ram matrix is given in symbolic form) or indices
(if it is in numeric form) of fixed exogenous variables. Specifying these obviates
the necessity of having to fix the variances and covariances among these
variables (and produces correct degrees of freedom for the model chisquare). |
debug |
if TRUE, some information is printed to help you debug the symbolic
model specification; for example, if a variable name is misspelled, sem will
assume that the variable is a (new) latent variable. The default is FALSE. |
... |
arguments to be passed down to sem.default (or ignored). |
param.names |
names of the t free parameters, given in their numerical order;
default names are Param1, ..., Paramt. Note: Should not be
specified when the ram matrix is given in symbolic form. |
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 Var1, ..., Varm.
Note: Should not be specified when the ram matrix is given in symbolic form. |
analytic.gradient |
if TRUE (the default), then analytic first derivatives are
used in the maximization of the likelihood; otherwise numeric derivatives are used. |
heywood |
if TRUE, variances are constrained to be non-negative in the
initial fitting process. The default is FALSE. |
warn |
if TRUE, warnings produced by the optimization functions will be printed.
This should generally not be necessary, since sem prints its own warnings, and saves
information about convergence. The default is FALSE. |
control |
a list with up to three elements:
optim.control, a list of control parameters to be passed to
the optim function via its control argument;
optim.method, the optimization method to be employed by optim;
if heywood is TRUE, then this defaults to
"L-BFGS-B", otherwise to "BFGS".
nlm.iterlim, the maximum number of iterations for the
nlm function, to be passed to it via its iterlim
argument. |
object, x |
an object of class sem returned by the sem function. |
digits |
number of digits for printed output. |
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 (character) format or in numeric format.
A symbolic ram matrix consists of three columns, as follows:
"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.NA produces a fixed parameter.NA, sem will compute the start value.
A numeric ram matrix consists of five columns, as follows:
S,
with the indices corresponding to the variables' positions in S.
Variable indices above n represent latent variables.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 optim
and nlm optimizers
sequentially to minimize the negative log-likelihood for the model.
If nlm fails to converge using an analytic gradient, then a numeric gradient is tried
(and vice-versa if the numeric gradient is tried first). Under these circumstances, 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.
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 — negative log-liklihood divided by N - 1. |
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.1 |
convergence code returned by optim (a code > 0
indicates a problem). |
message.1 |
message returned by optim in the first stage of
optimization. |
coef.1 |
parameter estimates returned by optim. |
convergence.2 |
convergence code returned by nlm in the
second stage of optimization (a code > 2 indicates a problem). |
coef.2 |
parameter estimates returned by nlm in
second stage of optimization. Present only if second stage failed. |
convergence.3 |
convergence code returned by nlm in the
third stage of optimization. Present only if second stage failed. |
John Fox jfox@mcmaster.ca
Bollen, K. A. (1989) Structural Equations With Latent Variables. Wiley.
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.
# ------------- Duncan, Haller and Portes peer-influences model ----------------------
# A nonrecursive SEM with unobserved endogenous variables and fixed exogenous variables
R.DHP <- matrix(c( # lower triangle of correlation matrix
1, 0, 0, 0, 0, 0, 0, 0, 0, 0,
.6247, 1, 0, 0, 0, 0, 0, 0, 0, 0,
.3269, .3669, 1, 0, 0, 0, 0, 0, 0, 0,
.4216, .3275, .6404, 1, 0, 0, 0, 0, 0, 0,
.2137, .2742, .1124, .0839, 1, 0, 0, 0, 0, 0,
.4105, .4043, .2903, .2598, .1839, 1, 0, 0, 0, 0,
.3240, .4047, .3054, .2786, .0489, .2220, 1, 0, 0, 0,
.2930, .2407, .4105, .3607, .0186, .1861, .2707, 1, 0, 0,
.2995, .2863, .5191, .5007, .0782, .3355, .2302, .2950, 1, 0,
.0760, .0702, .2784, .1988, .1147, .1021, .0931, -.0438, .2087, 1
), ncol=10, byrow=T)
# Fit the model using a symbolic ram specification
model.dhp <- matrix(c(
'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),
ncol=3, byrow=T)
obs.vars.dhp <- c('ROccAsp', 'REdAsp', 'FOccAsp', 'FEdAsp', 'RParAsp', 'RIQ',
'RSES', 'FSES', 'FIQ', 'FParAsp')
sem.dhp.1 <- sem(model.dhp, R.DHP, 329, obs.vars.dhp,
fixed.x=c('RParAsp', 'RIQ', 'RSES', 'FSES', 'FIQ', 'FParAsp'))
summary(sem.dhp.1)
## Model Chisquare = 26.697 Df = 15 Pr(>Chisq) = 0.031302
## Goodness-of-fit index = 0.98439
## Adjusted goodness-of-fit index = 0.94275
## BIC = -94.782
##
## Normalized Residuals
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -8.01e-01 -1.18e-01 5.02e-16 -1.20e-02 3.98e-02 1.57e+00
##
## Parameter Estimates
## Estimate Std Error z value Pr(>|z|)
## gam11 0.161222 0.038487 4.18896 1.4012e-05 RGenAsp <--- RParAsp
## gam12 0.249649 0.044581 5.59995 1.0720e-08 RGenAsp <--- RIQ
## gam13 0.218404 0.043476 5.02355 2.5362e-07 RGenAsp <--- RSES
## gam14 0.071841 0.050335 1.42724 7.6755e-02 RGenAsp <--- FSES
## gam23 0.061885 0.051738 1.19612 1.1583e-01 FGenAsp <--- RSES
## gam24 0.228867 0.044495 5.14368 1.3471e-07 FGenAsp <--- FSES
## gam25 0.349036 0.044551 7.83455 2.3315e-15 FGenAsp <--- FIQ
## gam26 0.159535 0.040129 3.97552 3.5113e-05 FGenAsp <--- FParAsp
## beta12 0.184227 0.096207 1.91490 2.7753e-02 RGenAsp <--- FGenAsp
## beta21 0.235487 0.119744 1.96659 2.4616e-02 FGenAsp <--- RGenAsp
## lam21 1.062682 0.091969 11.55481 0.0000e+00 REdAsp <--- RGenAsp
## lam42 0.929717 0.071151 13.06687 0.0000e+00 FEdAsp <--- FGenAsp
## ps11 0.280987 0.046311 6.06740 6.5001e-10 RGenAsp <--> RGenAsp
## ps22 0.263838 0.044902 5.87587 2.1032e-09 FGenAsp <--> FGenAsp
## ps12 -0.022611 0.051650 -0.43778 3.3077e-01 FGenAsp <--> RGenAsp
## theta1 0.412147 0.052211 7.89389 1.4433e-15 ROccAsp <--> ROccAsp
## theta2 0.336144 0.053323 6.30390 1.4512e-10 REdAsp <--> REdAsp
## theta3 0.311191 0.046665 6.66866 1.2907e-11 FOccAsp <--> FOccAsp
## theta4 0.404607 0.046733 8.65782 0.0000e+00 FEdAsp <--> FEdAsp
# 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=T)
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
## Goodness-of-fit index = 0.98439
## Adjusted goodness-of-fit index = 0.94275
## BIC = -94.782
##
## Normalized Residuals
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -8.01e-01 -1.18e-01 7.53e-16 -1.20e-02 3.98e-02 1.57e+00
##
## Parameter Estimates
## Estimate Std Error z value Pr(>|z|)
## lam21 1.062682 0.091969 11.55481 0.0000e+00 REdAsp <--- RGenAsp
## lam42 0.929717 0.071151 13.06687 0.0000e+00 FEdAsp <--- FGenAsp
## gam11 0.161222 0.038487 4.18896 1.4012e-05 RGenAsp <--- RParAsp
## gam12 0.249649 0.044581 5.59995 1.0720e-08 RGenAsp <--- RIQ
## gam13 0.218404 0.043476 5.02355 2.5362e-07 RGenAsp <--- RSES
## gam14 0.071841 0.050335 1.42724 7.6755e-02 RGenAsp <--- FSES
## gam23 0.061885 0.051738 1.19612 1.1583e-01 FGenAsp <--- RSES
## gam24 0.228867 0.044495 5.14368 1.3471e-07 FGenAsp <--- FSES
## gam25 0.349036 0.044551 7.83455 2.3315e-15 FGenAsp <--- FIQ
## gam26 0.159535 0.040129 3.97552 3.5113e-05 FGenAsp <--- FParAsp
## beta12 0.184227 0.096207 1.91490 2.7753e-02 RGenAsp <--- FGenAsp
## beta21 0.235487 0.119744 1.96658 2.4616e-02 FGenAsp <--- RGenAsp
## theta1 0.412147 0.052211 7.89389 1.4433e-15 ROccAsp <--> ROccAsp
## theta2 0.336144 0.053323 6.30390 1.4512e-10 REdAsp <--> REdAsp
## theta3 0.311191 0.046665 6.66866 1.2907e-11 FOccAsp <--> FOccAsp
## theta4 0.404607 0.046733 8.65782 0.0000e+00 FEdAsp <--> FEdAsp
## psi11 0.280987 0.046311 6.06739 6.5001e-10 RGenAsp <--> RGenAsp
## psi22 0.263838 0.044902 5.87587 2.1032e-09 FGenAsp <--> FGenAsp
## psi12 -0.022611 0.051650 -0.43778 3.3077e-01 RGenAsp <--> FGenAsp
# -------------------- Wheaton et al. alienation data ----------------------
S.wh <- matrix(c(
11.834, 0, 0, 0, 0, 0,
6.947, 9.364, 0, 0, 0, 0,
6.819, 5.091, 12.532, 0, 0, 0,
4.783, 5.028, 7.495, 9.986, 0, 0,
-3.839, -3.889, -3.841, -3.625, 9.610, 0,
-21.899, -18.831, -21.748, -18.775, 35.522, 450.288),
6, 6)
# 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 <- matrix(c(
'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),
ncol=3, byrow=T)
obs.vars.wh <- c('Anomia67','Powerless67','Anomia71','Powerless71','Education','SEI')
sem.wh.1 <- sem(model.wh.1, S.wh, 932, obs.vars.wh)
## Warning messages:
## 1: second optimization DID NOT converge in:
## sem.default(ram = ram, S = S, N = N, param.names = pars, var.names = vars,
## 2: final optimization DID converge in:
## sem.default(ram = ram, S = S, N = N, param.names = pars, var.names = vars,
# Note failure of second optimization, but success of third
summary(sem.wh.1)
## Model Chisquare = 13.485 Df = 9 Pr(>Chisq) = 0.14186
## Goodness-of-fit index = 0.99527
## Adjusted goodness-of-fit index = 0.98896
## BIC = -64.177
##
## Normalized Residuals
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -1.26e+00 -1.31e-01 9.69e-05 -2.87e-02 1.14e-01 8.75e-01
##
## Parameter Estimates
## Estimate Std Error z value Pr(>|z|)
## lamb 5.36885 0.433991 12.3709 0.0000e+00 SEI <--- SES
## gam1 -0.62994 0.056128 -11.2232 0.0000e+00 Alienation67 <--- SES
## beta 0.59312 0.046820 12.6680 0.0000e+00 Alienation71 <--- Alienation67
## gam2 -0.24086 0.055202 -4.3632 6.4085e-06 Alienation71 <--- SES
## the1 3.60790 0.200591 17.9863 0.0000e+00 Anomia67 <--> Anomia67
## the2 3.59496 0.165236 21.7566 0.0000e+00 Powerless67 <--> Powerless67
## the3 2.99366 0.498977 5.9996 9.8905e-10 Education <--> Education
## the4 259.57742 18.321464 14.1679 0.0000e+00 SEI <--> SEI
## the5 0.90580 0.121712 7.4422 4.9516e-14 Anomia71 <--> Anomia67
## psi1 5.67051 0.422907 13.4084 0.0000e+00 Alienation67 <--> Alienation67
## psi2 4.51485 0.334999 13.4772 0.0000e+00 Alienation71 <--> Alienation71
## phi 6.61627 0.639511 10.3458 0.0000e+00 SES <--> SES
# The same model, but treating one loading for each latent variable as free.
model.wh.2 <- matrix(c(
'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),
ncol=3, byrow=T)
sem.wh.2 <- sem(model.wh.2, S.wh, 932, obs.vars.wh)
## Warning messages:
## 1: second optimization DID NOT converge in:
## sem.default(ram = ram, S = S, N = N, param.names = pars, var.names = vars,
## 2: final optimization DID converge in:
## sem.default(ram = ram, S = S, N = N, param.names = pars, var.names = vars,
summary(sem.wh.2)
## Model Chisquare = 12.673 Df = 8 Pr(>Chisq) = 0.12360
## Goodness-of-fit index = 0.99553
## Adjusted goodness-of-fit index = 0.98828
## BIC = -56.36
##
## Normalized Residuals
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -0.997000 -0.139000 0.000142 -0.028500 0.100000 0.759000
##
## Parameter Estimates
## Estimate Std Error z value Pr(>|z|)
## lamby 0.86262 0.033383 25.8399 0.0000e+00 Powerless67 <--- Alienation67
## lambx 5.35329 0.432588 12.3750 0.0000e+00 SEI <--- SES
## gam1 -0.62132 0.056144 -11.0666 0.0000e+00 Alienation67 <--- SES
## beta 0.59424 0.047041 12.6324 0.0000e+00 Alienation71 <--- Alienation67
## gam2 -0.23584 0.054688 -4.3126 8.0685e-06 Alienation71 <--- SES
## the1 3.74502 0.249826 14.9905 0.0000e+00 Anomia67 <--> Anomia67
## the2 3.49376 0.200757 17.4030 0.0000e+00 Powerless67 <--> Powerless67
## the3 2.97442 0.499623 5.9533 1.3137e-09 Education <--> Education
## the4 260.12964 18.297803 14.2164 0.0000e+00 SEI <--> SEI
## the5 0.90378 0.121818 7.4191 5.8953e-14 Anomia71 <--> Anomia67
## psi1 5.47354 0.464054 11.7951 0.0000e+00 Alienation67 <--> Alienation67
## psi2 4.36409 0.362720 12.0315 0.0000e+00 Alienation71 <--> Alienation71
## phi 6.63548 0.640408 10.3613 0.0000e+00 SES <--> SES
# ----------------------- Thurstone data ---------------------------------------
# Second-order confirmatory factor analysis, from the SAS manual for PROC CALIS
R.thur <- matrix(c(
1., 0, 0, 0, 0, 0, 0, 0, 0,
.828, 1., 0, 0, 0, 0, 0, 0, 0,
.776, .779, 1., 0, 0, 0, 0, 0, 0,
.439, .493, .460, 1., 0, 0, 0, 0, 0,
.432, .464, .425, .674, 1., 0, 0, 0, 0,
.447, .489, .443, .590, .541, 1., 0, 0, 0,
.447, .432, .401, .381, .402, .288, 1., 0, 0,
.541, .537, .534, .350, .367, .320, .555, 1., 0,
.380, .358, .359, .424, .446, .325, .598, .452, 1.
), ncol=9, byrow=T)
model.thur <- matrix(c(
'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),
ncol=3, byrow=T)
obs.vars.thur <- c('Sentences','Vocabulary','Sent.Completion','First.Letters',
'4.Letter.Words','Suffixes','Letter.Series','Pedigrees',
'Letter.Group')
sem.thur <- sem(model.thur, R.thur, 213, obs.vars.thur)
summary(sem.thur)
## Model Chisquare = 38.196 Df = 24 Pr(>Chisq) = 0.033101
## Goodness-of-fit index = 0.95957
## Adjusted goodness-of-fit index = 0.9242
## BIC = -143.21
##
## Normalized Residuals
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -9.75e-01 -4.18e-01 -6.45e-05 4.01e-02 9.40e-02 1.63e+00
##
## Parameter Estimates
## Estimate Std Error z value Pr(>|z|)
## lam11 0.51513 0.064963 7.9296 1.1102e-15 Sentences <--- F1
## lam21 0.52032 0.065160 7.9852 6.6613e-16 Vocabulary <--- F1
## lam31 0.48744 0.062420 7.8090 2.8866e-15 Sent.Completion <--- F1
## lam41 0.52110 0.063140 8.2532 1.1102e-16 First.Letters <--- F2
## lam52 0.49705 0.059676 8.3292 0.0000e+00 4.Letter.Words <--- F2
## lam62 0.43805 0.056481 7.7557 4.4409e-15 Suffixes <--- F2
## lam73 0.45243 0.071371 6.3391 1.1553e-10 Letter.Series <--- F3
## lam83 0.41729 0.061036 6.8367 4.0515e-12 Pedigrees <--- F3
## lam93 0.40763 0.064524 6.3175 1.3295e-10 Letter.Group <--- F3
## gam1 1.44380 0.264162 5.4656 2.3071e-08 F1 <--- F4
## gam2 1.25391 0.216623 5.7884 3.5521e-09 F2 <--- F4
## gam3 1.40657 0.279336 5.0354 2.3841e-07 F3 <--- F4
## th1 0.18150 0.028401 6.3907 8.2580e-11 Sentences <--> Sentences
## th2 0.16493 0.027797 5.9334 1.4840e-09 Vocabulary <--> Vocabulary
## th3 0.26713 0.033469 7.9816 7.7716e-16 Sent.Completion <--> Sent.Completion
## th4 0.30150 0.050686 5.9485 1.3531e-09 First.Letters <--> First.Letters
## th5 0.36450 0.052358 6.9617 1.6803e-12 4.Letter.Words <--> 4.Letter.Words
## th6 0.50641 0.059962 8.4455 0.0000e+00 Suffixes <--> Suffixes
## th7 0.39033 0.061598 6.3367 1.1735e-10 Letter.Series <--> Letter.Series
## th8 0.48137 0.065388 7.3618 9.0705e-14 Pedigrees <--> Pedigrees
## th9 0.50510 0.065227 7.7437 4.7740e-15 Letter.Group <--> Letter.Group
#------------------------- Kerchoff/Kenney path analysis ---------------------
# An observed-variable recursive SEM from the LISREL manual
R.kerch <- matrix(c(
1, 0, 0, 0, 0, 0, 0,
-.100, 1, 0, 0, 0, 0, 0,
.277, -.152, 1, 0, 0, 0, 0,
.250, -.108, .611, 1, 0, 0, 0,
.572, -.105, .294, .248, 1, 0, 0,
.489, -.213, .446, .410, .597, 1, 0,
.335, -.153, .303, .331, .478, .651, 1),
ncol=7, byrow=T)
rownames(R.kerch) <- colnames(R.kerch) <- c('Intelligence','Siblings',
'FatherEd','FatherOcc','Grades','EducExp','OccupAsp')
model.kerch <- matrix(c(
'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),
ncol=3, byrow=T)
sem.kerch <- sem(model.kerch, R.kerch, 737, fixed.x=c('Intelligence','Siblings',
'FatherEd','FatherOcc'))
summary(sem.kerch)
## Model Chisquare = 6.537e-13 Df = 0 Pr(>Chisq) = NA
## Goodness-of-fit index = 1
## Adjusted goodness-of-fit index = NA
## BIC = NA
##
## Normalized Residuals
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -1.70e-14 -3.75e-16 7.45e-16 3.53e-15 4.47e-15 3.62e-14
##
## 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 1.6032e-01 Grades <--- Siblings
## gam53 0.118966 0.038259 3.10951 9.3699e-04 Grades <--- FatherEd
## gam54 0.040603 0.037785 1.07456 1.4129e-01 Grades <--- FatherOcc
## gam61 0.160270 0.032710 4.89979 4.7970e-07 EducExp <--- Intelligence
## gam62 -0.111779 0.026876 -4.15899 1.5983e-05 EducExp <--- Siblings
## gam63 0.172719 0.034306 5.03461 2.3941e-07 EducExp <--- FatherEd
## gam64 0.151852 0.033688 4.50758 3.2785e-06 EducExp <--- FatherOcc
## beta65 0.405150 0.032838 12.33799 0.0000e+00 EducExp <--- Grades
## gam71 -0.039405 0.034500 -1.14215 1.2670e-01 OccupAsp <--- Intelligence
## gam72 -0.018825 0.028222 -0.66700 2.5238e-01 OccupAsp <--- Siblings
## gam73 -0.041333 0.036216 -1.14126 1.2688e-01 OccupAsp <--- FatherEd
## gam74 0.099577 0.035446 2.80924 2.4829e-03 OccupAsp <--- FatherOcc
## beta75 0.157912 0.037443 4.21738 1.2358e-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