Estimate factor scores and standard errors
This function creates the factor scores and their standard errors under different methods for an MxModel object that has either a RAM or LISREL expectation.
An MxModel object with either an MxExpectationLISREL or MxExpectationRAM
The type of factor scores to compute
This is a helper function to compute or estimate factor scores along with their standard errors. The two maximum likelihood methods create a new model for each data row. They then estimate the factor scores as free parameters in a model with a single data row. For 'ML', the conditional likelihood of the data given the factor scores is optimized:
. For 'WeightedML', the joint likelihood of the data and the factor scores is optimized:
L(D, F) = L(D|F) L(F)
. The WeightedML scores are akin to the empirical Bayes random effects estimates from mixed effects modeling. They display the same kind of shrinkage as random effects estimates, and for the same reason: they account for the latent variable distribution in their estimation. In many cases, especially for ordinal data or missing data, the weighted ML scores are to be preferred over alternatives (Estabrook & Neale, 2013).
For type='Regression', with LISREL expectation, factor scores are computed based on a simple formula. This formula is equivalent to the formula for the Kalman updated scores in a state space model with zero dynamics (Priestly & Subba Rao, 1975). Thus, to compute the regression factor scores, the appropriate state space model is set-up and the mxKalmanScores function is used to produce the factor scores and their standard errors. With RAM expectation, factor scores are predicted from the non-missing manifest variables for each row of the raw data, using a general linear prediction formula analytically equivalent to that used with LISREL expectation. The standard errors for regression-predicted RAM factor scores are the square roots of the indeterminate variances of the latent variables, given the data row's missing-data pattern and the values of any relevant definition variables.
An array with dimensions (Number of Rows of Data, Number of Latent Variables, 2). The third dimension has the scores in the first slot and the standard errors in the second slot. The rows are in the order of the unsorted data. Multigroup models are an exception, in that the returned value is instead a list of such arrays, containing one per group.
Estabrook, R. & Neale, M. C. (2013). A Comparison of Factor Score Estimation Methods in the Presence of Missing Data: Reliability and an Application to Nicotine Dependence. Multivariate Behavioral Research, 48, 1-27.
Priestley, M. & Subba Rao, T. (1975). The estimation of factor scores and Kalman filtering for discrete parameter stationary processes. International Journal of Control, 21, 971-975.
The OpenMx User's guide can be found at http://openmx.psyc.virginia.edu/documentation.
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# Create and estimate a factor model require(OpenMx) data(demoOneFactor) manifests <- names(demoOneFactor) latents <- c("G") factorModel <- mxModel("OneFactor", type="LISREL", manifestVars=list(exo=manifests), latentVars=list(exo=latents), mxPath(from=latents, to=manifests), mxPath(from=manifests, arrows=2), mxPath(from=latents, arrows=2, free=FALSE, values=1.0), mxPath(from='one', to=manifests), mxData(observed=cov(demoOneFactor), type="cov", numObs=500, means = colMeans(demoOneFactor))) summary(factorRun <- mxRun(factorModel)) # Swap in raw data in place of summary data factorRun <- mxModel(factorRun, mxData(observed=demoOneFactor[1:50,], type="raw")) # Estimate factor scores for the model r1 <- mxFactorScores(factorRun, 'Regression')
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