Testing for treatment effect in the presence of regression toward the mean

We are often faced with the statistical problem of evaluating the effect of a treatment in the extreme of a population. This requires taking measurements on truncated random variables and, hence, it becomes necessary to take proper account of the effect of regression toward the mean. The usual statistical procedures are inappropriate for testing treatment effect in the presence of regression toward the mean. Likelihood ratio and score tests based on truncated distributions should provide valid statistical inferences in these situations. We conducted simulation studies to investigate the properties of these methods and found that the likelihood ratio test performs well even when the sample size is moderate, whereas the score test does not seem to control the nominal significance level. We compared the likelihood ratio test to a regression-based t-test, assuming the mean of the baseline distribution to be known, and found the likelihood ratio test more powerful. In the case where the baseline mean is unknown, we also investigated Wald's test and compared it with the likelihood ratio test and score test with respect to validity and power using simulation. Wald's test and the score test do not control the nominal significance level unless the sample size is extremely large. Overall, the likelihood ratio test has the best performance among all the methods studied. The proposed likelihood ratio test is illustrated using an example of a cholesterol study.