A new Bartlett-based homogeneity test for linear regression models

Abstract

The assumption of homogeneous residuals is crucial in linear regression analysis. In fact, all statistical inferences around regression coefficients are built on the basis of this fundamental assumption. As a result, having a linear model with heterogeneous residuals would destruct all those beneficial reliable inferences, and this specifically means that all extracted predicted values, confidence regions and any other conclusions are then false, misleading and far from reality. Also, heterogeneity underestimates true significance levels, which could lead to considering the importance of an explanatory variable whereas truth is not. These are a few consequences of handling a linear regression model with heterogeneous residuals, and hence, researchers have focused on two things: first, defining a consistent statistical tool to catch deviations from homogeneity, and second, proposing approaches to deal with that violation efficiently. We are not interested here in the last goal, and rather, we are focusing on the first one. Indeed, there were enormous and remarkable efforts seeking to fulfil the first goal in different ways, from plotting residuals against fitted values to applying statistical tests, such as the Breusch-Pagan test, Koenker test, and other related tests. But in effect, analysing plots is highly dependent ‎on ‎self-experience and how one would draw conclusions and thoughts about the plot and, in fact, it is not an easy task in many real-world studies. On the other hand, each of the Breusch-Pagan and Koenker tests has its own deficiencies and misleading conclusions. Other related tests, which we will summarise shortly, are not easily implemented or programmed. So, we aim in the present paper‎ to present a simple statistical method for testing the homogeneity assumption of linear regression residuals, by just employing the well-known Bartlett's test‎, on the basis of defining two suitable disjoint subsets driven from the original dataset. We evaluate the proposed approach by a series of simulation studies and analysing a previous historic case study, and it will be shown that the proposed method‎ controls the error rate in a nominal level and has high performance in sense of both homogeneity and heterogeneity detection‎s.