Inferential implications of normalizing binomial proportions in a structural equation model: A simulation study...

Most commercial software for implementation of structural equation models (SEM) cannot explicitly accommodate outcome variables of binomial nature. As a result, SEM modeling strategies of binomial outcomes are often based on normal approximations of empirical proportions. Inferential implications of these approximations are particularly relevant to health-related outcomes. The objective of this study was to assess the inferential implications of specifying a binomial variable as an empirical proportion (%) in predictor and outcome roles in a SEM. We addressed this objective first by a simulation study, and second by a proof-of-concept data application on beef feedlot morbidity to bovine respiratory disease (BRD). We simulated data on body weight at feedlot arrival (AW), morbidity count for BRD (Mb), and average daily gain (ADG). Alternative SEMs were fitted to the simulated data. Model 1 specified a directed acyclic causal diagram with morbidity fitted as a binomial outcome (Mb) and as a proportion (Mb_p) predictor. Model 2 specified a similar causal diagram with morbidity fitted as a proportion for both outcome and predictor roles within the network. Structural parameters for Model 1 were accurately estimated based on the nominal coverage probability of 95 % confidence intervals. In turn, there was poor coverage for most morbidity-related parameters under Model 2. Both SEM models showed adequate empirical power (>80 %) to detect parameters not equal to zero. Model 1 and Model 2 produced predictions that were reasonable from a management standpoint, as determined by calculating the root mean squared error (RMSE) through cross-validation. However, interpretability of parameter estimates in Model 2 was impaired due to the model misspecification relative to the data generation. The data application fitted SEM extensions, Model 1 * and Model 2 * , to a dataset from a group of feedlots in the Midwestern US. Models 1 * and 2 * included explanatory covariates, specifically percent shrink (PS), backgrounding type (BG), and season (SEA). Lastly, we tested if AW exerted both direct and BRD-mediated indirect effects on ADG using Model 2 * . In Model 1 * , mediation was not testable due to the incomplete path from morbidity as a binomial outcome through Mb_p as a predictor to ADG. Model 2 * supported a minor morbidity-mediated mechanism between AW and ADG, though parameter estimates were not directly interpretable. Our results indicate normal approximation to a binomial disease outcome in a SEM may be a viable option for inference on mediation hypotheses and for predictive purposes, despite limitations in interpretability due to inherent model misspecification.