Estimating treatment e cacy over time : a logistic regression model for binary longitudinal outcomes 3

This paper presents a case study in longitudinal data analysis where the goal is to estimate the e cacy of a new drug for treatment of a severe chronic constipation. Data consist of long sequences of binary 11 outcomes (relief=no relief) on each of a large number of patients randomized to treatment (low and high dose) or placebo. Data characteristics indicate: (1) the treatment e ects vary non-linearly with 13 time; (2) there is substantial heterogeneity across subjects in their responses to treatment; and (3) there is a high proportion of subjects who never experience any relief (the non-responders). 15 To overcome these challenges, we develop a hierarchical model for binary longitudinal data with a mixture distribution on the probability of response to account for the high frequency of non-responders. 17 While the model is speci ed conditionally on subject-speci c latent variables, we also draw inferences on key population-average parameters for the assessment of the treatments’ e cacy in a population. 19 In addition we employ a model-checking method to compare the goodness-oft for our model against simpler modelling approaches for aggregated counts, such as the zero-in ated Poisson and zero-in ated 21 negative binomial models. We estimate subject-speci c and population-average rate ratios of relief for the treatment with respect 23 to the placebo as functions of time (RRt), and compare them with the rate ratios estimated from the models for aggregated counts. We nd that: (1) the treatment is e ective with respect to the placebo 25 with higher e cacy at the beginning of the study; (2) the estimated rate ratios from the models for aggregated counts appear to be similar to the average across time of the population-average rate ratios 27 estimated under our model; and (3) model-checking suggests that the hierarchical and zero-in ated negative binomial model t the data best. 29 If we are mainly interested to establish the overall e cacy (or safety) of a new drug, it is appropriate to aggregate the longitudinal data over time and analyse the count data by use of standard statistical 31 methods. However, the models for aggregated counts cannot capture time trend of treatment such as the initial treatment bene t or the development of tolerance during the early stage of the treatment which 33