Non-linear mixed models in the analysis of mediated longitudinal data with binary outcomes

Background: Structural equation models (SEMs) provide a general framework for analyzing mediated longitudinal data. However when interest is in the total effect (i.e. direct plus indirect) of a predictor on the binary outcome, alternative statistical techniques such as non-linear mixed models (NLMM) may be preferable, particularly if specific causal pathways are not hypothesized or specialized SEM software is not readily available. The purpose of this paper is to evaluate the performance of the NLMM in a setting where the SEM is presumed optimal. Methods: We performed a simulation study to assess the performance of NLMMs relative to SEMs with respect to bias, coverage probability, and power in the analysis of mediated binary longitudinal outcomes. Both logistic and probit models were evaluated. Models were also applied to data from a longitudinal study assessing the impact of alcohol consumption on HIV disease progression. Results: For the logistic model, the NLMM adequately estimated the total effect of a repeated predictor on the repeated binary outcome and were similar to the SEM across a variety of scenarios evaluating sample size, effect size, and distributions of direct vs. indirect effects. For the probit model, the NLMM adequately estimated the total effect of the repeated predictor, however, the probit SEM overestimated effects. Conclusions: Both logistic and probit NLMMs performed well relative to corresponding SEMs with respect to bias, coverage probability and power. In addition, in the probit setting, the NLMM may produce better estimates of the total effect than the probit SEM, which appeared to overestimate effects. Background SEMs are a general modeling framework often used in the social sciences to analyze complex relationships between variables, such as mediated relationships between variables. A mediator is a variable in the causal pathway between a predictor and the outcome of interest. SEMs are becoming more common in the clinical research setting and can be used to model hypothesized causal pathways between variables of interest. Extensions of SEMs have been developed to allow for more general types of dependent variables, including binary outcomes [1]. Common statistical techniques for nonmediated longitudinal binary data include non-linear mixed models (NLMM) [2] and generalized estimating equations (GEE) [3]. When interest is primarily in the total effect of a predictor on an outcome, even if mediation may be present, these commonly used techniques may be preferred over SEMs as they specify straightforward predictor-outcome variable relationships and do not require specialized software, as the SEM often does. It is therefore of interest to determine, in a setting conducive to using SEMs, whether a method such as NLMMs adequately models the total effect of a predictor on binary outcomes without directly modeling mediation. We focus on NLMM rather than GEE in this paper as it is more similar to the non-linear SEMs for longitudinal data available in SEM software-both are conditional rather than marginal models. Comparisons have been made between SEM and other statistical models in different contexts [4-13]. Mixed effect models have been evaluated against SEMs with continuous data [14,15], and found to adequately model mediated predictor-outcome relationships. MacKinnon et al. [16] examined the calculation of mediated effects in cross-sectional binary data with non-SEM techniques using two different methods (difference of coefficients