lrv2ord {semTools}

Calculate Population Moments for Ordinal Data Treated as Numeric

Description

This function calculates ordinal-scale moments implied by LRV-scale moments

Usage

lrv2ord(Sigma, Mu, thresholds, cWts)

Arguments

Sigma

Population covariance matrix, with variable names saved in the dimnames attribute.

Mu

Optional numeric vector of population means. If missing, all means will be set to zero.

thresholds

Either a single numeric vector of population thresholds used to discretize each normally distributed variable, or a named list of each discretized variable's vector of thresholds. The discretized variables may be a subset of all variables in Sigma if the remaining variables are intended to be observed rather than latent normally distributed variables.

cWts

Optional (default when missing is to use 0 for the lowest category, followed by successive integers for each higher category). Either a single numeric vector of category weights (if they are identical across all variables) or a named list of each discretized variable's vector of category weights.

Details

Binary and ordinal data are frequently accommodated in SEM by incorporating a threshold model that links each observed categorical response variable to a corresponding latent response variable that is typically assumed to be normally distributed (Kamata & Bauer, 2008; Wirth & Edwards, 2007).

Value

A list including the LRV-scale population moments (means, covariance matrix, correlation matrix, and thresholds), the category weights, a data.frame of implied univariate moments (means, SDs, skewness, and excess kurtosis (i.e., in excess of 3, which is the kurtosis of the normal distribution) for discretized data treated as numeric, and the implied covariance and correlation matrix of discretized data treated as numeric.

Author(s)

Terrence D. Jorgensen (University of Amsterdam; [e-mail])

Andrew Johnson (Curtin University; [e-mail])

References

Kamata, A., & Bauer, D. J. (2008). A note on the relation between factor analytic and item response theory models. Structural Equation Modeling, 15(1), 136–153. doi:10.1080/10705510701758406

Wirth, R. J., & Edwards, M. C. (2007). Item factor analysis: Current approaches and future directions. Psychological Methods, 12(1), 58–79. doi:10.1037/1082-989X.12.1.58

Examples

## SCENARIO 1: DIRECTLY SPECIFY POPULATION PARAMETERS

## specify population model in LISREL matrices
Nu <- rep(0, 4)
Alpha <- c(1, -0.5)
Lambda <- matrix(c(1, 1, 0, 0, 0, 0, 1, 1), nrow = 4, ncol = 2,
                 dimnames = list(paste0("y", 1:4), paste0("eta", 1:2)))
Psi <- diag(c(1, .75))
Theta <- diag(4)
Beta <- matrix(c(0, .5, 0, 0), nrow = 2, ncol = 2)

## calculate model-implied population means and covariance matrix
## of latent response variables (LRVs)
IB <- solve(diag(2) - Beta) # to save time and space
Mu_LRV <- Nu + Lambda %*% IB %*% Alpha
Sigma_LRV <- Lambda %*% IB %*% Psi %*% t(IB) %*% t(Lambda) + Theta

## Specify (unstandardized) thresholds to discretize normally distributed data
## generated from Mu_LRV and Sigma_LRV, based on marginal probabilities
PiList <- list(y1 = c(.25, .5, .25),
               y2 = c(.17, .33, .33, .17),
               y3 = c(.1, .2, .4, .2, .1),
               ## make final variable highly asymmetric
               y4 = c(.33, .25, .17, .12, .08, .05))
sapply(PiList, sum) # all sum to 100%
CumProbs <- sapply(PiList, cumsum)
## unstandardized thresholds
TauList <- mapply(qnorm, p = lapply(CumProbs, function(x) x[-length(x)]),
                  m = Mu_LRV, sd = sqrt(diag(Sigma_LRV)))
for (i in 1:4) names(TauList[[i]]) <- paste0(names(TauList)[i], "|t",
                                             1:length(TauList[[i]]))

## assign numeric weights to each category (optional, see default)
NumCodes <- list(y1 = c(-0.5, 0, 0.5), y2 = 0:3, y3 = 1:5, y4 = 1:6)


## Calculate Population Moments for Numerically Coded Ordinal Variables
lrv2ord(Sigma = Sigma_LRV, Mu = Mu_LRV, thresholds = TauList, cWts = NumCodes)


## SCENARIO 2: USE ESTIMATED PARAMETERS AS POPULATION

data(datCat) # already stored as c("ordered","factor")
fit <- cfa(' f =~ 1*u1 + 1*u2 + 1*u3 + 1*u4 ', data = datCat)
lrv2ord(Sigma = fit, thresholds = fit) # use same fit for both
## or use estimated thresholds with specified parameters, but note that
## lrv2ord() will only extract standardized thresholds
dimnames(Sigma_LRV) <- list(paste0("u", 1:4), paste0("u", 1:4))
lrv2ord(Sigma = cov2cor(Sigma_LRV), thresholds = fit)