Response-adaptive designs are important in clinical trials for their ability to skew the allocation towards the better performing treatment. Most of the designs are derived under specific assumptions (e.g., normality, existence of first few moments, etc.) on the response model. However, responses may be nonnormal and consequently the existence of moments is not guaranteed. Moreover, in most of the cases, a treatment yielding a higher expected response is regarded as better, ignoring the possible effect of variability. In this article, we consider location and variability aspects simultaneously and redefine the criterion of the best treatment and develop a reasonable covariate adjusted allocation procedure for two treatment trials. A detailed study is also conducted to explore the performance.