Statistical Modelling

In this paper, we propose a methodology which a) evaluates the effect of covariates on doubly interval-censored paired responses, b) is based on minimal parametric assumptions concerning the distributional parts of the model and c) evaluates the association between the two responses of the pair. Our methodology tackles three research questions arising from the Signal Tandmobiel® project, a prospective Flemish (Belgian) longitudinal dental study. The research questions are 1) What is the effect of baseline covariates on the time-to-caries of the permanent right first molars? 2) Is the effect of the covariates the same for the upper and lower teeth? 3) What is the association between the times-to-caries on the upper and lower teeth? Time-to-caries is defined as the difference of two interval-censored observations, caries time and emergence time, and hence it is a doubly interval-censored response. We suggest using an accelerated failure time model with a bivariate smooth error distribution being a mixture of bivariate normal components defined on a fine fixed grid. To deal with the problem of doubly interval censoring, we use Bayesian methodology and Markov chain Monte Carlo sampling.

Fitting multilevel models to discrete outcome data is problematic because the discrete distribution of the response variable implies an analytically intractable log-likelihood function. Among a number of approximate methods proposed, second-order penalised quasi-likelihood (PQL) is commonly used and is one of the most accurate. Unfortunately, even the second-order PQL approximation has been shown to produce estimates biased toward zero in certain circumstances. This bias can be marked especially when the data are sparse. One option to reduce this bias is to use Monte-Carlo simulation. A bootstrap bias correction method proposed by Kuk has been implemented in MLwiN. However, a similar technique based on the Robbins-Monro (RM) algorithm is potentially more efficient. An alternative is to use simulated maximum likelihood (SML), either alone or to refine estimates identified by other methods. In this article, we first compare bias correction using the RM algorithm, Kuk’s method and SML. We find that SML performs as efficiently as the other two methods and also yields standard errors of the bias-corrected parameter estimates and an estimate of the log-likelihood at the maximum, with which nested models can be compared. Secondly, using simulated and real data examples, we compare SML, second-order Laplace approximation (as implemented in HLM), Markov Chain Monte-Carlo (MCMC) (in MLwiN) and numerical integration using adaptive quadrature methods (in Stata’s GLLAMM and in SAS’s proc NLMIXED). We find that when the data are sparse, the second-order Laplace approximation produces markedly lower parameter estimates, whereas the MCMC method produces estimates that are noticeably higher than those from the SML and quadrature methods. Although proc NLMIXED is much faster than GLLAMM, it is not designed to fit models of more than two levels. SML produces parameter estimates and log-likelihoods very similar to those from quadrature methods. Further our SML approach extends to handle other link functions, discrete data distributions, non-normal random effects and higher-level models.

A convenient reparametrization of the marginal covariance matrix arising in longitudinal studies is discussed. The new parameters have transparent statistical interpretations, are unconstrained and may be modelled parsimoniously in terms of polynomials of time. We exploit this framework to model the dependence of the covariance structure on baseline covariates, time and their interaction. The rationale is based on the assumption that a homogeneous covariance structure with respect to the covariate space is a testable model choice. Accordingly, we provide methods for testing this assumption by incorporating covariates along with time into the model for the covariance structure. We also present new computational algorithms which can handle unbalanced longitudinal data, thereby extending existing methods. The new model is used to analyse Kenward’s (1987) cattle data, and the findings are compared with published analyses of the same data set.

The aim of this article is to discuss the problem of testing variance components in elliptical linear mixed models. The elliptical class includes all symmetrical continuous distributions, such as normal, Student-t, Pearson VII, power exponential, logistic I, logistic II, and so on. An approach for elliptical linear mixed models is proposed. The estimation procedure for the fixed parameters, variance components and random effects is also presented and a score-type test for one-sided alternatives to assess the variance components is investigated. Finally, two illustrative examples are given in which normal linear mixed models are compared with elliptical linear mixed models with heavy-tailed error distributions.