Maria Francesca Marino

Mixed hidden Markov quantile regression models for longitudinal data with possibly incomplete sequences

Quantile regression provides a detailed and robust picture of the distribution of a response variable, conditional on a set of observed covariates. Recently, it has be been extended to the analysis of longitudinal continuous outcomes using either time-constant or time-varying random parameters. However, in real-life data, we frequently observe both temporal shocks in the overall trend and individual-specific heterogeneity in model parameters. A benchmark dataset on HIV progression gives a clear example. Here, the evolution of the CD4 log counts exhibits both sudden temporal changes in the overall trend and heterogeneity in the effect of the time since seroconversion on the response dynamics. To accommodate such situations, we propose a quantile regression model, where time-varying and time-constant random coefficients are jointly considered. Since observed data may be incomplete due to early drop-out, we also extend the proposed model in a pattern mixture perspective. We assess the performance of the proposals via a large-scale simulation study and the analysis of the CD4 count data.

Gaussian quadrature approximations in mixed hidden Markov models for longitudinal data

Mixed hidden Markov models represent an interesting tool for the analysis of longitudinal data. They allow to account for both time-constant and time-varying sources of unobserved heterogeneity, which are frequent in this kind of studies. Individual-specific latent features, which may be either constant or varying over time, are included in the linear predictor and lead to a general form of dependence between individual measurements. When a parametric (continuous) distribution is associated to time-constant random parameters, the estimation process requires the calculation of (multiple) integrals. These, generally, have not a closed form and should be numerically approximated. The aim is to compare the standard, the adaptive and the pseudo-adaptive Gaussian quadrature approximations by means of a large scale simulation study, where continuous and discrete responses with (conditional) density in the exponential family are considered. Simulation results show that the approximation error is often substantially reduced when the adaptive quadrature rules are considered in place of the standard one. Such an improvement comes at the cost of a higher computational complexity when the fully adaptive scheme is applied. It is shown that, when a sufficient number of repeated measurements per unit is available, the pseudo-adaptive quadrature represents a convenient compromise between quality of results and computational complexity.

Latent drop-out based transitions in linear quantile hidden Markov models for longitudinal responses with attrition

Longitudinal data are characterized by the dependence between observations from the same individual. In a regression perspective, such a dependence can be usefully ascribed to unobserved features (covariates) specific to each individual. On these grounds, random parameter models with time-constant or time-varying structure are now well established in the generalized linear model context. In the quantile regression framework, specifications based on random parameters have only recently known a flowering interest. We start from the recent proposal by Farcomeni (Stat Comput 22:141–152, 2012) on longitudinal quantile hidden Markov models, and extend it to handle potentially informative missing data mechanisms. In particular, we focus on monotone missingness which may lead to selection bias and, therefore, to unreliable inferences on model parameters. We detail the proposed approach by re-analyzing a well known dataset on the dynamics of CD4 cell counts in HIV seroconverters and by means of a simulation study reported in the supplementary material.

A bidimensional finite mixture model for longitudinal data subject to dropout: A bidimensional finite mixture model

In longitudinal studies, subjects may be lost to follow up and, thus, present incomplete response sequences. When the mechanism underlying the dropout is nonignorable, we need to account for dependence between the longitudinal and the dropout process. We propose to model such a dependence through discrete latent effects, which are outcome-specific and account for heterogeneity in the univariate profiles. Dependence between profiles is introduced by using a probability matrix to describe the corresponding joint distribution. In this way, we separately model dependence within each outcome and dependence between outcomes. The major feature of this proposal, when compared with standard finite mixture models, is that it allows the nonignorable dropout model to properly nest its ignorable counterpart. We also discuss the use of an index of (local) sensitivity to nonignorability to investigate the effects that assumptions about the dropout process may have on model parameter estimates. The proposal is illustrated via the analysis of data from a longitudinal study on the dynamics of cognitive functioning in the elderly.

Dealing with reciprocity in dynamic stochastic block models

A stochastic block model for dynamic network data is introduced, where directed relations among a set of nodes are observed at different time occasions and the blocks are represented by a sequence of latent variables following a Markov chain. Dyads are explicitly modeled conditional on the states occupied by both nodes involved in the relation. With respect to the approaches already available in the literature, the main focus is on reciprocity. In this regard, three different parameterizations are proposed in which: (i) reciprocity is allowed to depend on the blocks of the nodes in the dyad; (ii) reciprocity is assumed to be constant across blocks; and (iii) reciprocity is ruled out. The assumption of conditional independence between dyads (referred to different pairs of nodes and time occasions) given the latent blocks is always retained. Given the complexity of the model, inference on its parameters is based on a variational approach, where a lower bound of the log-likelihood function is maximized instead of the intractable full model log-likelihood. An approximate likelihood ratio test statistic is proposed which compares the value at convergence of this lower bound under different model specifications. This allows us to formally test for both the hypothesis of no reciprocity and that of constant reciprocity with respect to the latent blocks. The proposed approach is illustrated via a simulation study based on different scenarios. The application to two benchmark datasets in the social network literature is also proposed to illustrate the effectiveness of the proposal in studying reciprocity and identifying groups of nodes having a similar social behavior.