Flexible derivative time-varying model in matched case-crossover studies for a small number of geographical locations among the participants

In matched case-crossover studies, any stratum effect is removed by conditioning on the fixed number of case-control sets in the stratum, and hence, the conditional logistic regression model is not able to detect any effects associated with matching covariates. However, some matching covariates such as time and location often modify the effect of covariates, making the estimations obtained by conditional logistic regression incorrect. Therefore, in this paper, we propose a flexible derivative time-varying coefficient model to evaluate effect modification by time and location, in order to make correct statistical inference, when the number of locations is small. Our proposed model is developed under the Bayesian hierarchical model framework and allows us to simultaneously detect relationships between the predictor and binary outcome and between the predictor and time. Inference is proposed based on the derivative function of the estimated function to determine whether there is an effect modification due to time and/or location, for a small number of locations among the participants. We demonstrate the accuracy of the estimation using a simulation study and an epidemiological example of a 1-4 bidirectional case-crossover study of childhood aseptic meningitis with drinking water turbidity.