Assumptions of multiple regression: Correcting two misconceptions - Williams MN; Grajales CAG; Kurkiewicz D (PARE)
In 2002, an article entitled “Four assumptions of multiple regression that researchers should always test” by Osborne and Waters was published in PARE. This article has gone on to be viewed over 264,000 times (as at June 2013), and it is one of the first results displayed in a Google search for “regression assumptions”. While Osborne and Waters’ efforts in raising awareness of the need to check assumptions when using regression are laudable, we note that the original article contained at least two fairly important misconceptions about the assumptions of multiple regression: Firstly, that multiple regression requires the assumption of normally distributed variables; and secondly, that measurement errors necessarily cause underestimation of simple regression coefficients. In this article, we clarify that multiple regression models estimated using ordinary least squares require the assumption of normally distributed errors in order for trustworthy inferences, at least in small samples, but not the assumption of normally distributed response or predictor variables. Secondly, we point out that regression coefficients in simple regression models will be biased (toward zero) estimates of the relationships between variables of interest when measurement error is uncorrelated across those variables, but that when correlated measurement error is present, regression coefficients may be either upwardly or downwardly biased. We conclude with a brief corrected summary of the assumptions of multiple regression when using ordinary least squares.
Keywords: 1303 Specialist Studies in Education