Modelling Longitudinal Binary Outcomes for Analysing Covariate Dependence

Abstract

Correlated data which are very common in longitudinal studies should be analysed with models and methods that take the correlation into account. Most of the longitudinal models are based on marginal approaches, assuming an induced correlation between successive individuals, often lacking the proper specification of the dependence of binary outcomes. As a result, such models may fail to provide efficient estimation of parameters. Conditional models which is another commonly used approach for the above situation use a transition probability model to capture the dependence of outcome variables. While the selection of a model depends on the question under study, there is no clear directives in the existing literature about when to choose which model. Keeping in mind the limitations of the existing popular methods for analysing longitudinal data, the objectives of this study were set. The objectives of this study are (i) to examine how well the dependence of repeated response are addressed in selected methods including GEE and ALR, (ii) to propose joint models based on a marginal conditional approach enabling to incorporate the true dependence relationship, using likelihood methods, (iii) to propose a joint model based on a marginal conditional approach under a quasilikelihood setup appropriate for situation where the distribution of the outcome variables is unknown, (iv) to develop and demonstrate the inferential theories associated with all the proposed models, (v) make comparisons of the proposed models with the existing models and (vi) to illustrate the proposed models with applications to real life data. The proposed models demonstrated under objective (ii) link the marginal and sequence of conditional models to provide the joint model needed for predicting the covariate effects on dependent variable at different time points. In case of more than three repeated measurements, the regressive model approach was proposed that can be extended for any order of dependence without complicating the theory and keeping the number of parameters of the model for repeated measures minimum. This model has the flexibility such that one can easily add interaction terms among previous outcomes and predictors in the proposed framework if and when required. A number of simulation studies resulted that the proposed methods perform better than GEE and ALR in terms of bias and 95% coverage probability. The marginal conditional model developed under a quasi-likelihood setup captures the correlations among repeated observations in a built-in nature and unlike GEE or ALR, does not need to have a correlation parameter in the model. This model can be extended for any number of repeated measures without complicating the theory and keeping the number of parameters to a minimum. The simulation studies showed that, when the data are correlated or the distribution of the outcome variables are not identical at different time points, the estimates of this method has less bias than GEE or ALR. The marginal conditional feature of the proposed models make the models very useful for analysing big data, one can use the existing software for model fitting and risk prediction of a sequence of events. The application using Health and Retirement Study data illustrate the performance of the proposed models and prove the usefulness of such models for longitudinal data.