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.
# S3 method for 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, ...)
# S3 method for default
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 semmodList
sem(model, S, N, data, raw=FALSE, fixed.x=NULL,
robust=!missing(data), formula, group="Group", debug=FALSE, ...)
# S3 method for 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 sem
coef(object, standardized=FALSE, ...)
# S3 method for msem
coef(object, ...)
# S3 method for sem
vcov(object, robust=FALSE,
analytic=inherits(object, "objectiveML") && object$t
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:
vector of variable names.
RAM matrix, including any rows generated for covariances among fixed exogenous variables; column 5 includes computed start values.
observed covariance matrix.
RAM selection matrix, \(J\), which picks out observed variables.
number of fixed exogenous variables.
number of observed variables.
number of observations.
number of variables (observed plus unobserved).
number of free parameters.
TRUE if the model is fit to a raw moment matrix, FALSE otherwise.
the observed-variable data matrix, or NULL if data are not supplied.
the semmod specification object for the model, if one was supplied; otherwise NULL.
the optimizer function.
the objective function.
estimates of free parameters.
estimated asymptotic covariance matrix of parameter estimates, based on a numeric Hessian,
if supplied by the optimizer; otherwise NULL.
indices of free parameters.
TRUE or FALSE, depending upon whether the optimization apparently converged.
number of iterations performed.
value of the objective function at the minimum.
model-reproduced covariance matrix.
RAM \(A\) matrix.
RAM \(P\) matrix.
robust adjusted value of the objective function; NULL if
robust is FALSE.
robust estimated coefficient covariance matrix; NULL if
robust is FALSE.
For multigroup models, sem returns an object of class c("msemObjectiveML", "msem").
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 semmod object, as returned by the specifyModel,
specifyEquations, or cfa function,
or as a character matrix), invoking sem.semmod, which calls sem.default
after setting up the model,
or (less conveniently) in numeric form, invoking sem.default directly, which is
not recommended (see Details below). The model argument may also be a
multigroup-model specification, as produced by multigroupModel.
covariance matrix among observed variables; may be input as a symmetric matrix,
or as a lower- or upper-triangular matrix. S may also be a raw (i.e., “uncorrected”)
moment matrix --- that is, a sum-of-squares-and-products matrix divided by N. This
form of input is useful for fitting models with intercepts, in which case the moment matrix
should include the mean square and cross-products for a unit variable all of whose entries are 1;
of course, the raw mean square for the unit variable is 1. Raw-moment matrices may be computed
by rawMoments. If the ram argument is given in symbolic form, then
the observed-variable covariance or raw-moment matrix may contain variables that do not appear in the model,
in which case a warning is printed. S may also be a list of covariance or moment matrices for each
group in a multigroup model.
As an alternative to specifying S the user may
supply a data frame containing the data for the model (see the argument data).
number of observations on which the covariance matrix is based; for a multigroup model, a vector of group \(N\)s.
As a generally preferable alternative to specifying S and
N, the user may supply a data frame containing the data to which the model is to be fit.
In a multigroup model, the data argument may be a list of data frames or a single data frame;
in the later event, the factor given as the group argument is used to split the data into groups.
a function to compute startvalues for the free parameters of the model;
two functions are supplied, startvalues and a older version, startvalues2,
the first of which is the default.
a function to process missing data, if raw data are supplied in the data argument.
The default is na.omit, which returns only complete cases; specify na.action=na.pass to
get FIML estimates in the presence of missing data from the objectiveFIML
and objectiveFIML2 objective functions.
TRUE if S is a raw moment matrix or if a raw moment matrix --- as
opposed to a covariance matrix --- is to be
computed from data; the default is FALSE unless the na.action argument
is set to na.pass.
these arguments pass information about valid (i.e., non-missing) data patterns and normally would not be specified directly by the user.
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 TRUE.
names of observed variables, by default taken from the row names of
the covariance or moment matrix S, which may be given directly or generated according to the
data and formula arguments.
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).
a one-sided formula, to be applied to data to generate the variables for
which covariances or raw moments are computed. The default formula is ~., i.e., all of the
variables in the data, including an implied intercept; if a covariance matrix is to be computed,
the constant is suppressed. In a multigroup model, alternatively a list one one-sided formulas as be
given, to be applied individually to the groups.
In sem: if TRUE, then quantities are calculated that can be used to compute robust
estimates of coefficient standard errors and robust tests when the model is fit by multinormal maximum likelihood;
the default is TRUE when the data argument is TRUE, and this option is only
available when the data argument is given. In vcov: if TRUE, return a robust coefficient covariance matrix
(if object contains the requisite information).
a semmod object containing the description of the model; optional, and normally supplied
not directly by the user but via the semmod method for sem.
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. Information about
the optimization will also be printed, but details will vary with the optimizer employed.
The default is FALSE.
arguments to be passed down, including from sem.default to the
optimizer.
names of the \(t\) free parameters, given in their numerical order;
default names are Param1, ..., Paramt. Note: Should not be
specified when the model is given in symbolic form.
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 model is given in symbolic form.
if TRUE (the default, except for the objectiveFIML objective function,
where, at present, an analytic gradient slows down the computation), then analytic first derivatives are
used in the maximization of the likelihood if the optimzer employed will accept them;
otherwise numeric derivatives are used, again if the optimizer will compute them.
if TRUE, warnings produced by the optimization function will be printed.
This should generally not be necessary, since sem prints its own warning, and saves
information about convergence. The default is FALSE.
the maximum number of iterations for the optimization of the objective function, to be passed to the optimizer.
the anticipated size of the free parameters; if "ones",
a vector of ones is used; if "startvalues", taken from the start values.
You can try changing this argument if you encounter convergence problems.
The default is "startvalues" if the largest input variance is at
least 100 times the smallest, and "ones" otherwise. Whether this argument is actually
used depends upon the optimizer employed.
if the magnitude of an automatic start value is less than start.tol, then
it is set to start.tol; defaults to 1E-6.
a function to be used to minimize the objective function; the default for single-group models is
optimizerSem.
Alternatives are nlm, which employs the standard R optimizer nlm;
optimizerOptim, which employs optim;
and optimizerNlminb, which uses nlminb --- or
the user can supply an optimizer. For multigroup model, the default is optimizerMsem, and
msemOptimizerNlm, based on nlm, is provided as an alternative.
An objective function to be minimized, sometimes called a “fit” function
in the SEM literature. The default for single-group models is objectiveML,
which produces maximum-likelihood estimates assuming multinormality. An alternative is
objectiveGLS, which produced generalized least squares estimates --- or the user
can supply an objective function to be minimized. For multigroup models, the default is
available is msemObjectiveML for ML estimates and an alternative is msemObjectiveGLS
for GLS estiamtes.
primary class to be assigned to the result; normally this is not specified directly, but raither is inferred from the objective function.
numeric RAM matrix.
an object of class "sem" or "msem", returned by sem.
if TRUE, return standardized coefficients.
return an analytic (as opposed to numeric) estimate of the coefficient covariance matrix;
at present only available for the objectiveML objective function. The default is FALSE
for this objective function if there are no more than 100 parameters and FALSE otherwise.
for a multigroup model, the quoted name of the group variable; if the data argument is given,
snd is a single data frame, then this should be a factor in the data set or a variable coercible to a factor,
to be used to split the data into groups; otherwise, the name is arbitrary.
a character vector giving the names of the groups; will be ignored if group is a factor
in data.
if "initial.fit" (the default), start values
for a multi-group model are computed by first fitting the intra-group models separately by group;
if "startvalues", then start values are computed as for a single-group model.
In some cases, the intra-group models may not be identified even if the multi-group model is, and
then start="initial.fit" should not be used.
if start="initial.fit" for a multi-group model, then
initial.maxiter gives the maximum number of iterations for each initial intra-group fit.
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.
John Fox jfox@mcmaster.ca, Zhenghua Nie, and Jarrett Byrnes
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:
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 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 (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.
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.
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.