R: General Structural Equation Models

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 `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`.
`S` | 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`).
`N` | number of observations on which the covariance matrix is based; for a multigroup model, a vector of group Ns.
`data` | 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.
`start.fn` | 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.
`na.action` | 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.
`raw` | `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`.
`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 `TRUE`.
`obs.variables` | 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.
`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).
`formula` | 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.
`robust` | 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).
`semmod` | 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`.
`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. 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.
`param.names` | names of the t free parameters, given in their numerical order; default names are `Param1`, ..., `Param`t. Note: Should not be specified when the model 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`, ..., `Var`m. Note: Should not be specified when the model is given in symbolic form.
`analytic.gradient` | 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.
`warn` | 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`.
`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 `"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.
`start.tol, tol` | if the magnitude of an automatic start value is less than `start.tol`, then it is set to `start.tol`; defaults to 1E-6.
`optimizer` | 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.
`objective` | 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.
`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 `"sem"` or `"msem"`, returned by `sem`.
`standardized` | if `TRUE`, return standardized coefficients.
`analytic` | 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.
`group` | 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.
`groups` | a character vector giving the names of the groups; will be ignored if `group` is a factor in `data`.
`start` | 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.
`initial.maxiter` | if `start="initial.fit"` for a multi-group model, then `initial.maxiter` gives the maximum number of iterations for each initial intra-group fit.

### Details

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

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

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

1. Arrow specification:

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

2. Parameter name:

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

3. Value:

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

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

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

1. Number of arrow heads:

  1 (directed arrow) or 2 (covariance).

2. Arrow to:

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

3. Arrow from:

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

4. Parameter number:

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

5. Value:

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

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

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

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

where `y` and the `x`s are variables in the model (either observed or latent), and the `par`s 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 `var`s 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, J, which picks out observed variables.
`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` | `TRUE` if the model is fit to a raw moment matrix, `FALSE` otherwise.
`data` | the observed-variable data matrix, or `NULL` if data are not supplied.
`semmod` | the `semmod` specification object for the model, if one was supplied; otherwise `NULL`.
`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 `NULL`.
`par.posn` | indices of free parameters.
`convergence` | `TRUE` or `FALSE`, depending upon whether the optimization apparently converged.
`iterations` | number of iterations performed.
`criterion` | value of the objective function at the minimum.
`C` | model-reproduced covariance matrix.
`A` | RAM A matrix.
`P` | RAM P matrix.
`adj.obj` | robust adjusted value of the objective function; `NULL` if `robust` is `FALSE`.
`robust.vcov` | robust estimated coefficient covariance matrix; `NULL` if `robust` is `FALSE`.

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

[Package sem version 3.1-13 Index]