Estimate an Auxiliary Model Using Two-Step Maximum Likelihood

Estimates an auxiliary model based on a previously fitted latent class model, using a two-step maximum likelihood procedure.

Usage

two_step(x, model = NULL, data, ...)

Arguments

x: A fitted latent class model of class MxModel.
model: An auxiliary model specified as an OpenMx RAM model or a character string that can be converted to a RAM model using as_ram(). May be omitted when data is a single numeric, integer, or factor variable.
data: A data.frame containing the auxiliary variables, or a single numeric, integer, or factor variable. Observations must be in the same order as in the original latent class model.
...: Additional arguments passed to OpenMx::mxRun().

Value

An MxModel containing the original latent class model with fixed measurement parameters and freely estimated auxiliary model parameters.

Details

In the first step, the latent class model is estimated using the measurement indicators. In the second step, its parameter estimates (including the mixing proportions) are fixed, and the auxiliary model is added to each latent class. Only the parameters of the auxiliary model are freely estimated.

Unlike the one-step approach, the auxiliary variables cannot influence the measurement model parameters. However, auxiliary variables can influence posterior class membership probabilities because they contribute to the joint likelihood. Classification uncertainty is accounted for by integrating over latent class membership rather than assigning observations to their most likely class.

Note that conventional standard errors obtained from this procedure treat the first-step parameter estimates as known and therefore do not account for uncertainty in the estimation of the latent class model. Corrected standard errors can be obtained using an appropriate two-step sandwich estimator or a bootstrap that repeats both steps.

The auxiliary model is incorporated into each class-specific RAM model. By default, its parameters are freely estimated across latent classes, allowing class-specific means, variances, and regression coefficients.

The measurement indicators and auxiliary variables are assumed conditionally independent given latent class membership, unless additional relationships are explicitly specified.

References

Bakk, Z., & Kuha, J. (2018). Two-step estimation of models between latent classes and external variables. Psychometrika, 83, 871–892. doi:10.1007/s11336-017-9592-7

Examples

if (FALSE) { # \dontrun{
library(OpenMx)

# Estimate a latent profile model
dat <- iris[, 1:4]
names(dat) <- c("SL", "SW", "PL", "PW")

set.seed(123)
fit <- mx_profiles(
  data = dat[, c("PL", "PW")],
  classes = 2
)

# Continuous distal outcome
res <- two_step(
  x = fit,
  data = dat$SL
)
table_results(res)

# Equivalent specification using RAM syntax
res <- two_step(
  x = fit,
  model = "SL ~ 1",
  data = dat["SL"]
)

# Ordinal distal outcome
ordinal <- ordered(
  cut(dat$SL, breaks = 3)
)

res_ord <- two_step(
  x = fit,
  data = ordinal
)

# Custom auxiliary model with two continuous variables
res_custom <- two_step(
  x = fit,
  model = "
    SL ~ 1
    SW ~ 1
    SL ~~ SW
  ",
  data = dat[, c("SL", "SW")]
)
} # }