Low-level fitting functions for the censored HLM model.

Low-level fitting functions for the censored HLM model.

chlm_fit(y, delta, X, Z, beta0, gamma0, maxit = 100, epsilon = 1e-08,
  splitE = FALSE, nIRLS = 5)

chlm_control(epsilon = 1e-05, maxit = 100, nIRLS = 5,
  splitE = TRUE)

Arguments

y	Vector of observations of length n.
delta	Optional logical vector of length n indicating the censorring status (TRUE: uncensored, FALSE: uncensored).
X	Mean covariate matrix of size n x p.
Z	Variance covariate matrix of size n x q.
beta0	Optional initial mean parameter vector of length p.
gamma0	Optional initial variance parameter vector of length q.
maxit	Maximum number of iteration of the fitting algorithm (see Details).
epsilon	Tolerance threshold for termination of the algorithm (see Details).
splitE	If TRUE, perform the E-step after each conditional M-step (see Details).
nIRLS	Number of IRLS steps to take before switching to Fisher scoring. Can be 0 or greater than maxit to do only one or the other.

Value

A list with the following elements:

beta

The MLE of the mean parameter vector.

gamma

The MLE of the variance parameter vector.

loglik

The value of the loglikelihood at the fitted parameter values.

iter

The number of steps taken by the algorithm.

error

The value of the loglikelihood relative error at the end of the algorithm.

Details

The heteroscedastic linear model (HLM) is defined as $$ y_i \mid \boldsymbol{x}_i, \boldsymbol{z}_i \stackrel{\mathrm{ind}}{\sim} \mathcal N\big(\boldsymbol{x}_i'\boldsymbol{\beta}, \exp(\boldsymbol{z}_i'\boldsymbol{\gamma})\big), $$ where for each subject \(i\), \(y_i\) is the response, and \(\boldsymbol{x}_i \in \mathbb{R}^p\) and \(\boldsymbol{z}_i \in \mathbb{R}^q\) are mean and variance covariate vectors, respectively.

The fitting algorithm is an Expectation-Conditional-Maximization (ECM) algorithm extending the alternating weighted-LM/GLM updates of beta and gamma, proposed by Smyth (1989) for the uncensored setting. The ECM algorithm terminates when either maxit iterations have been reached, or when

|ll_curr - ll_prev| / (0.1 + |ll_curr|) < epsilon,

where ll_curr and ll_prev are the loglikelihood values at the current and previous iterations.

TODO:

• Input checking.

• print, summary, vcov methods.

• residual method. Perhaps use expected lifetime for the censored observations?

• Separate into chlm and hlm classes?

References

Smyth, G.K. "Generalized Linear Models with Varying Dispersion." Journal of the Royal Statistical Society Series B 51:1 (1989): 47-60. https://www.jstor.org/stable/2345840.