This function creates a new MxLISRELObjective object.
mxLISRELObjective(LX=NA, LY=NA, BE=NA, GA=NA, PH=NA, PS=NA, TD=NA, TE=NA, TH=NA,
TX = NA, TY = NA, KA = NA, AL = NA,
dimnames = NA, thresholds = NA, vector = FALSE, threshnames = dimnames)Returns a new MxLISRELObjective object. MxLISRELObjective objects should be included with models with referenced MxAlgebra, MxData and MxMatrix objects.
An optional character string indicating the name of the 'LX' matrix.
An optional character string indicating the name of the 'LY' matrix.
An optional character string indicating the name of the 'BE' matrix.
An optional character string indicating the name of the 'GA' matrix.
An optional character string indicating the name of the 'PH' matrix.
An optional character string indicating the name of the 'PS' matrix.
An optional character string indicating the name of the 'TD' matrix.
An optional character string indicating the name of the 'TE' matrix.
An optional character string indicating the name of the 'TH' matrix.
An optional character string indicating the name of the 'TX' matrix.
An optional character string indicating the name of the 'TY' matrix.
An optional character string indicating the name of the 'KA' matrix.
An optional character string indicating the name of the 'AL' matrix.
An optional character vector that is currently ignored
An optional character string indicating the name of the thresholds matrix.
A logical value indicating whether the objective function result is the likelihood vector.
An optional character vector to be assigned to the column names of the thresholds matrix.
Objective functions are functions for which free parameter values are chosen such that the value of the objective function is minimized. The mxLISRELObjective provides maximum likelihood estimates of free parameters in a model of the covariance of a given MxData object. This model is defined by LInear Structural RELations (LISREL; Jöreskog & Sörbom, 1982, 1996). Arguments 'LX' through 'AL' must refer to MxMatrix objects with the associated properties of their respective matrices in the LISREL modeling approach.
The full LISREL specification has 13 matrices and is sometimes called the extended LISREL model. It is defined by the following equations.
$$\eta = \alpha + B \eta + \Gamma \xi + \zeta$$ $$y = \tau_y + \Lambda_y \eta + \epsilon$$ $$x = \tau_x + \Lambda_x \xi + \delta$$
The table below is provided as a quick reference to the numerous matrices in LISREL models. Note that NX is the number of manifest exogenous (independent) variables, the number of Xs. NY is the number of manifest endogenous (dependent) variables, the number of Ys. NK is the number of latent exogenous variables, the number of Ksis or Xis. NE is the number of latent endogenous variables, the number of etas.
| Matrix | Word | Abbreviation | Dimensions | Expression | Description |
| \(\Lambda_x\) | Lambda x | LX | NX x NK | Exogenous Factor Loading Matrix | |
| \(\Lambda_y\) | Lambda y | LY | NY x NE | Endogenous Factor Loading Matrix | |
| \(B\) | Beta | BE | NE x NE | Regressions of Latent Endogenous Variables Predicting Endogenous Variables | |
| \(\Gamma\) | Gamma | GA | NE x NK | Regressions of Latent Exogenous Variables Predicting Endogenous Variables | |
| \(\Phi\) | Phi | PH | NK x NK | cov(\(\xi\)) | Covariance Matrix of Latent Exogenous Variables |
| \(\Psi\) | Psi | PS | NE x NE | cov(\(\zeta\)) | Residual Covariance Matrix of Latent Endogenous Variables |
| \(\Theta_{\delta}\) | Theta delta | TD | NX x NX | cov(\(\delta\)) | Residual Covariance Matrix of Manifest Exogenous Variables |
| \(\Theta_{\epsilon}\) | Theta epsilon | TE | NY x NY | cov(\(\epsilon\)) | Residual Covariance Matrix of Manifest Endogenous Variables |
| \(\Theta_{\delta \epsilon}\) | Theta delta epsilson | TH | NX x NY | cov(\(\delta, \epsilon\)) | Residual Covariance Matrix of Manifest Exogenous with Endogenous Variables |
| \(\tau_x\) | tau x | TX | NX x 1 | Residual Means of Manifest Exogenous Variables | |
| \(\tau_y\) | tau y | TY | NY x 1 | Residual Means of Manifest Endogenous Variables | |
| \(\kappa\) | kappa | KA | NK x 1 | mean(\(\xi\)) | Means of Latent Exogenous Variables |
| \(\alpha\) | alpha | AL | NE x 1 | Residual Means of Latent Endogenous Variables |
From the extended LISREL model, several submodels can be defined. Subtypes of the LISREL model are defined by setting some of the arguments of the LISREL objective to NA. Note that because the default values of each LISREL matrix is NA, setting a matrix to NA can be accomplished by simply not giving it any other value.
The first submodel is the LISREL model without means.
$$\eta = B \eta + \Gamma \xi + \zeta$$ $$y = \Lambda_y \eta + \epsilon$$ $$x = \Lambda_x \xi + \delta$$
The LISREL model without means requires 9 matrices: LX, LY, BE, GA, PH, PS, TD, TE, and TH. Hence this LISREL model has TX, TY, KA, and AL as NA. This can be accomplished be leaving these matrices at their default values.
The TX, TY, KA, and AL matrices must be specified if either the mxData type is “cov” or “cor” and a means vector is provided, or if the mxData type is “raw”. Otherwise the TX, TY, KA, and AL matrices are ignored and the model without means is estimated.
A second submodel involves only endogenous variables.
$$\eta = B \eta + \zeta$$ $$y = \Lambda_y \eta + \epsilon$$
The endogenous-only LISREL model requires 4 matrices: LY, BE, PS, and TE. The LX, GA, PH, TD, and TH must be NA in this case. However, means can also be specified, allowing TY and AL if the data are raw or if observed means are provided.
Another submodel involves only exogenous variables.
$$x = \Lambda_x \xi + \delta$$
The exogenous-model model requires 3 matrices: LX, PH, and TD. The LY, BE, GA, PS, TE, and TH matrices must be NA. However, means can also be specified, allowing TX and KA if the data are raw or if observed means are provided.
The model that is run depends on the matrices that are not NA. If all 9 matrices are not NA, then the full model is run. If only the 4 endogenous matrices are not NA, then the endogenous-only model is run. If only the 3 exogenous matrices are not NA, then the exogenous-only model is run. If some endogenous and exogenous matrices are not NA, but not all of them, then appropriate errors are thrown. Means are included in the model whenever their matrices are provided.
The MxMatrix objects included as arguments may be of any type, but should have the properties described above. The mxLISRELObjective will not return an error for incorrect specification, but incorrect specification will likely lead to estimation problems or errors in the mxRun function.
Like the mxRAMObjective, the mxLISRELObjective evaluates with respect to an MxData object. The MxData object need not be referenced in the mxLISRELObjective function, but must be included in the MxModel object. mxLISRELObjective requires that the 'type' argument in the associated MxData object be equal to 'cov', 'cor', or 'raw'.
To evaluate, place MxLISRELObjective objects, the mxData object for which the expected covariance approximates, referenced MxAlgebra and MxMatrix objects, and optional MxBounds and MxConstraint objects in an MxModel object. This model may then be evaluated using the mxRun function. The results of the optimization can be found in the 'output' slot of the resulting model, and may be obtained using the mxEval function.
Jöreskog, K. G. & Sörbom, D. (1996). LISREL 8: User's Reference Guide. Lincolnwood, IL: Scientific Software International.
Jöreskog, K. G. & Sörbom, D. (1982). Recent developments in structural equation modeling. Journal of Marketing Research, 19, 404-416.
The OpenMx User's guide can be found at https://openmx.ssri.psu.edu/documentation/.
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##### Factor Model
mLX <- mxMatrix("Full", values=c(.5, .6, .8, rep(0, 6), .4, .7, .5),
name="LX", nrow=6, ncol=2,
free=c(TRUE,TRUE,TRUE,rep(FALSE, 6),TRUE,TRUE,TRUE))
mTD <- mxMatrix("Diag", values=c(rep(.2, 6)), name="TD", nrow=6, ncol=6,
free=TRUE)
mPH <- mxMatrix("Symm", values=c(1, .3, 1), name="PH", nrow=2, ncol=2,
free=c(FALSE, TRUE, FALSE))
# Create a LISREL objective with LX, TD, and PH matrix names
objective <- mxLISRELObjective(LX="LX", TD="TD", PH="PH")
testModel <- mxModel(model="testModel4", mLX, mTD, mPH, objective)
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