Threshold estimation based on a $P$-value framework in Dose--Response and Regression Settings
Mouli Banerjee, University of Michigan
Monday, April 4, 2011
4:00 p.m., room 223, Weber Bldg
We use $p$--values as a discrepancy criterion for identifying the threshold value at which a regression function takes off from its baseline value -- a problem that is motivated by applications in omics experiments, pharmacological dose-response studies and environmental statistics. In this paper, we study the problem in two different sampling settings: one where multiple responses can be obtained at a number of different covariate-levels and the other being the standard regression setting(limited number of response at each covariate). Our procedure involves testing the hypothesis that the regression function is at its baseline at each covariate value and then computing the (potentially approximate) $p$--value of the test. An estimate of the threshold is provided by fitting a stump, i.e., a piecewise constant function with a single jump discontinuity, to a function of these observed $p$--values (or their surrogates), as they behave in markedly different ways on different sides of the threshold. The estimate is shown to be consistent and its finite sample properties are studied through an extensive simulation study. Our approach is computationally simple and can also be used to estimate the baseline value of the regression function. The procedure can handle heteroscedastic errors and is applicable to certain time--series models. We also illustrate the procedure on real data sets.