Exploring ANCOVA models with bimodal error structures

Analysis of covariance is a frequently employed statistical technique in experimental and quasi-experimental research. A key assumption in this analysis is that the error terms follow a normal distribution. This paper investigates parameter estimation and hypothesis testing within covariance analysis models when the error term distribution deviates from normality and instead follows an alpha skew-normal distribution. We consider a one-way deterministic analysis of covariance, a one-way deterministic analysis of covariance with two covariates, and a stochastic analysis of covariance. The unknown model parameters are estimated using the maximum likelihood method. Based on these estimators, new test statistics are proposed to assess both the treatment effect and the significance of the slope parameter. A Monte Carlo simulation study is conducted to compare the efficiency of the proposed estimators with traditional least squares estimators. The simulation results demonstrate that the maximum likelihood estimators exhibit greater efficiency compared to the least squares estimators. Furthermore, the test statistics derived from maximum likelihood estimators are found to be more powerful than those based on least squares. In the application section, two real-world datasets are analyzed to illustrate the proposed method.