| Density ratio models have attracted much attention recently, and many statistical methods have been developed under these models. Density ratio models have a natural connection with generalized linear model, which has been widely used in biostatistics and other areas of applied statistics. In this study, we focus on semiparametric inferences under density ratio models.;First, by assuming a two-sample density ratio model of the diseased and non-diseased populations, we propose a semiparametric empirical likelihood confidence interval for the area under a receiver operating characteristic curve (AUC). We show that the limiting distribution of the semiparametric empirical log-likelihood ratio statistic for AUC has a scaled chi-square distribution. The proposed semiparametric empirical likelihood approach is shown, via a simulation study, to be more robust than a fully parametric approach and is more accurate than a fully nonparametric approach. Some results on a simulation study and an analysis of two real examples are presented.;Second, we propose a nonparametric method to compare the means of several populations. This nonparametric method does not make any assumption on the population distributions. The test statistic is easy to compute and has an asymptotic chi-square distribution. This nonparametric method is an alternative to the classical one-way ANOVA when the normal and equal variance assumptions are violated. We present some results on a simulation study and an analysis of two real examples.;Finally, we propose a semiparametric method to compare the means of several populations under a multiple-sample density ratio model. This method is built upon semiparametric estimation of the differences between the population means. We show that the semiparametric test statistic converges to a chi-square distribution. A simulation study and an analysis of three real data sets are provided. The simulation results show that our semiparametric method is comparable to the parametric ANOVA when data are normal, and is significantly better than the parametric ANOVA and Kruskal-Wallis test when data are not normal. |
