Nonparametric surface estimation for quantitative bioassay, survival data, and location of extrema

Estimating the location of extrema of a multivariate nonlinear regression surface may be a primary motivation in fitting such a surface, particularly in fixed design regression. A goal may be to establish optimal conditions to maximize/minimize yield. The most common approach is to optimize a second order parametric regression surface (with product interactions), usually assuming normality of the response. This is the so-called response surface method that has become popular in various fields. However, not only can the normality assumption be problematic, but using a parametric structure can lead to bias in the extremum location estimate due to lack of model flexibility. Interactions between predictors are often too complex to be effectively modeled with a parametric form. These problems can be alleviated by using nonparametric regression to estimate the response surface. The study of inference for nonparametric extremum estimators so far has been limited to the univariate case.; Here we propose nonparametric estimates and establish asymptotic consistency and normality results for the location of extrema in a multivariate regression function. These results are used to establish multivariate confidence regions for extremum locations. Two biological application examples are presented, which include comparisons of parametric response surface analysis with the proposed nonparametric approach. One application is a folate bioassay study with various sources of folate as predictors. A second application refers to the age and calendar time at AIDS diagnosis that is associated with maximum AIDS incidence for two racial/ethnic groups in California.; The potential problems exhibited by multivariate regression models containing a parametric form associated with lack of flexibility, such as bias and difficulty with complex interactions, also extend to survival data. To explore this, we propose a nonparametric quantile surface estimate for censored lifetime data. The method is applied to an AIDS survival data set with two predictors, and the results are compared to those obtained from the usual proportional hazards regression model using a novel graphical diagnostic. The nonparametric surface approach produces seemingly better results for these data, particularly in expressing between-predictor interaction.