# Probability and Statistics Seminar

### Nonparametric estimation of the odds-ratios assuming Generalized Additive Models: The multivariate case including interactions

Calculating odds-ratios (OR) and corresponding confidence intervals (CI) for exposures that have been measured using a continuous scale presents important limitations in the traditional practice of analytic epidemiology. Approximations based on linear models require making arbitrary assumptions about the shape of the relation curve, or about its breakpoints. Categorical analyses generally have low statistical efficiency and cut-off points for the categories are in most cases arbitrary and/or opportunistic. Recent publications in epidemiology have shown an interest in the application of a more general flexible regression technique such as the Generalized Additive Models (GAM) (Hastie and Tibshirani, 1990). This modern regression has the advantage of not assuming a parametric relation between exposure and effect, and eliminates the need for the investigator to impose functional assumptions in this respect. The only assumption required is that the effect of the continuous covariate follows an arbitrary continuous smooth function. In this talk we propose the use of GAM to derive the corresponding nonparametric estimates (and CIs) by means of the Local Scoring Algorithm. This procedure permits great flexibility and adequate statistical efficiency. Definition of the nonparametric OR curve is also extended to the case of having second-order interactions between two continuous covariates .In this context, inferential issues are solved by means of bootstrap resampling techniques. Finally, we illustrate these new methods through several real data sets.