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The stepped wedge design is a type of unidirectional crossover design where cluster units switch from control to intervention condition at different pre-specified time points. While a convention in study planning is to assume the cluster-period sizes are identical, stepped wedge cluster randomized trials (SW-CRTs) involving repeated cross-sectional designs frequently have unequal cluster-period sizes, which can impact the efficiency of the treatment effect estimator. In this article, we provide a comprehensive investigation of the efficiency impact of unequal cluster sizes for generalized estimating equation analyses of SW-CRTs, with a focus on binary outcomes as in the Washington State Expedited Partner Therapy trial. Several major distinctions between our work and existing work include: (i) we consider multilevel correlation structures in marginal models with binary outcomes; (ii) we study the implications of both the between-cluster and within-cluster imbalances in sizes; and (iii) we provide a comparison between the independence working correlation versus the true working correlation and detail the consequences of ignoring correlation estimation in SW-CRTs with unequal cluster sizes. We conclude that the working independence assumption can lead to substantial efficiency loss and a large sample size regardless of cluster-period size variability in SW-CRTs, and recommend accounting for correlations in the analysis. To improve study planning, we additionally provide a computationally efficient search algorithm to estimate the sample size in SW-CRTs accounting for unequal cluster-period sizes, and conclude by illustrating the proposed approach in the context of the Washington State study.
In medical studies, composite indices and/or scores are routinely used for predicting medical conditions of patients. These indices are usually developed from observed data of certain disease risk factors, and it has been demonstrated in the literature that single index models can provide a powerful tool for this purpose. In practice, the observed data of disease risk factors are often longitudinal in the sense that they are collected at multiple time points for individual patients, and there are often multiple aspects of a patient's medical condition that are of our concern. However, most existing single‐index models are developed for cases with independent data and a single response variable, which are inappropriate for the problem just described in which within‐subject observations are usually correlated and there are multiple mutually correlated response variables involved. This paper aims to fill this methodological gap by developing a single index model for analyzing longitudinal data with multiple responses. Both theoretical and numerical justifications show that the proposed new method provides an effective solution to the related research problem. It is also demonstrated using a dataset from the English Longitudinal Study of Aging.
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