The Validity of Polynomial Regression in the Random Regression Model

Abstract

Sockloff (1976), in reviewing the appropriateness of fixed and random models in regression analysis, has concluded the analysis of nonlinearity via polynomial and product regression should be limited to experimental studies (p. 288), and that, [r]egarding the analysis of nonlinearity in observational data under the Random Model, the Random Normal Model cannot be used, and,... the appropriate counterpart inferential model does not currently exist (p. 288). These remarkable and misleading conclusions result from the belief that some assumption must be made about the joint distribution of the dependent and independent variables in a random regression model, e.g., multivariate normality. Although multivariate normality assures linearity of regression, it is not necessary to assume multivariate normality in order to obtain valid significance tests and estimates of regression coefficients in linear regression. Indeed the standard F tests and confidence intervals appropriate to fixed regression models remain valid under conditions that permit the distribution of the x variables to remain arbitrary. As a consequence of this, the ordinary methods of estimation and statistical testing appropriate to the fixed polynomial regression model remain valid under fairly unrestrictive conditions. Thus, in fact, there is no need to distinguish, for this purpose, between the random and fixed model, nor to restrict oneself to continuous x variables as is necessary with the assumption of multivariate normality. This is not to say that the random and fixed models are identical; they are not. The standard errors of the regression coefficients differ in the two cases and the regression coefficients are normally distributed only for fixed x variables. The distribution of regression coefficients in the random model is dependent on the distribution of the independent