This talk includes two testing problems of regression functions with responses missing at random. One problem is minimum distance model checking. The proposed lack-of-fit tests are based on a class of minimum integrated square distances between a kernel type estimator of a regression function and the parametric regression function being fitted. These tests are shown to be consistent against a large class of fixed alternatives. The corresponding test statistics are shown to have asymptotic normal distributions under the null hypothesis. The other problem considers testing the superiority between two regression curves against a one-sided alternative. A class of statistics using imputation and covariate-matching are shown to be asymptotically normal under the null hypothesis and a class of local alternatives. The corresponding tests are consistent against the general alternative. The covariate-matched statistics constructed by complete cases are also discussed. Some simulation results are presented.