Results of clinical studies are often obscured by the fact that some of the subjects improve with treatment while others do not. As a consequence, for example, we may obtain a contaminated distribution in a treatment group, having one component similar to the entire distribution for a control group and the other shifted by the treatment effect. Maximum-likelihood estimation and the likelihood ratio test for investigating the proportion of responders together with the treatment effect are based on asymptotic theory and use iterative maximization techniques. We investigate the sampling distribution of the estimates and the test and demonstrate their use in statistical inference. Our results suggest that the chi-square distribution with 1.4 degrees of freedom provides a useful test criterion even when the sample sizes are as small as 10. Only in obvious testing situations is thet-test adequate.