Inference for the relative treatment effect with the density ratio model

Consider the problem of estimating and testing the relative treatment effect between two populations based on a random sample from each distribution. Under the well–established normal theory, inference is based on analysis of variance methods. However, there are many examples of skewed data which show that normal theory is not applicable. Then the problem of inference regarding the treatment effect can be attacked by standard nonparametric methods. In this paper, we propose a semiparametric model, the so–called density ratio model, which specifies that the log–likelihood ratio of two densities is linear in some parameters. For testing hypotheses regarding the relative treatment effect, a robust test is obtained by employing the density ratio model for a suitable Box–Cox transformation of the data. The transformation, along with the density ratio model, are estimated by maximum empirical likelihood. The new test procedure is studied theoretically and it is applied to real and simulated data. It is further compared with some nonparametric competitors, and it is found to have relatively high power across a wide variety of distributions, including those outside the density ratio family.

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