tests for one-way analysis of covariance with heteroscedasticity

The one-way analysis of covariance model is used to evaluate the equality between multiple treatments in the presence of a covariate‎. ‎presenting a test that controls the size is a major consideration for this model especially when the variances are unequal‎, ‎since the actual size of the test depends on the values of the variances‎. ‎the traditional f test in the analysis of covariance problem can produce unreliable outcomes‎, ‎and there is no simple manipulation test that satisfactorily controls the size‎. ‎the calculations for the generalized f test and parametric bootstrap test are complicated and must be performed using monte carlo techniques‎. ‎in this study‎, ‎we first suggest a generalized test for the analysis of covariance problem and then construct three approximate tests that do not have the problems of earlier tests‎. ‎the performance of the generalized test and the three simple tests is shown using extensive simulations‎. ‎when the variances are very unequal‎, ‎the actual sizes of these tests can be close to the nominal size and rarely exceed the nominal level‎. ‎three data sets illustrate the application of the methods.