EM re-estimation of the shared random-effects covariance (Gaussian-copula extension)

Extends longitudinalSubVar()'s EM algorithm with one additional "inner" E-step (impute_latent_ordinal()), run before the existing "outer" E-step (posterior mean/second-moment of the random effects) at every iteration – an ECM (Expectation-Conditional-Maximization) scheme. The outer E-step and M-step formulas are otherwise completely unchanged from longitudinalSubVar(): only the stacked outcome vector yi fed into them differs, since ordinal biomarkers' observed category codes are replaced by the inner E-step's imputed latent scores before every outer-step update. As in longitudinalSubVar(), sigma2 is never updated (for ordinal biomarkers it stays fixed at 1, the identification constraint for a probit latent variable); only D is re-estimated.

Usage

longitudinalSubVarCopula(
  thetaLong,
  l,
  biomarker_type,
  thresholds,
  tol.em = 1e-04,
  max.iter = 500,
  verbose = FALSE
)