In many clinical experiments, some subjects are unaffected by the treatment. This so-called non-response phenomenon has attracted the attention of many researchers in recent years. This dissertation focuses on detecting the association between the dose level and the observed values in the case where there is a mixture in treatment groups. That is, there is a linear relation between dose and response in a fraction of the observations and the shift in mean increases as the dose level increases. We investigate the Likelihood Ratio Test (LRT) in the context of normal mixture models. We do this based on critical values for the LRT obtained through simulation. We customize the general Expected-Maximization (EM) Algorithm to our situation in order to obtain Maximum Likelihood Estimates (MLE) of the parameter values under the alternative. We note that as we expected MLE of the parameters are close to the true parameter values and the mean square error decreases as the sample size increases. For the power study we also conduct LRT, Spearman's correlation test, and Simple linear regression test on each simulated sample. The power of three tests is compared and the McNemar's test is conducted to test the difference between tests. Overall, the power of the LRT is greater than Spearman's test and Simple linear regression test, although the three tests are not powerful with small shifted proportion and small shifted mean. We conclude that the LRT is very powerful in those cases where the mixing proportion is greater than 0.5 and there is a linear dose response relationship with slope greater than or equal to 0.3 standard deviation units. At the same time, the Simple linear regression test works almost as well as the LRT in those cases where the power is greater than 0.5.