Other Distributions for a Continuous Response Aside the Normal Distribution in a Linear Regression Model.

Abstract

In many instances, when one encounters a continuous response in model building, the normal distribution is often the preferred choice for the distribution of the response given the predictors. In particular, to some statisticians, the normal distribution is seen as the only distribution for a continuous response. Even when the assumption of normality is not met, various transformations are applied on the data so that it appears to be more nearly normal. This is sometimes not pleasant since, the model may no longer apply directly to the original scale of measurement, which is in most cases of interest. Likewise, in doing so, one tries to force the model framework and distributional assumption that may not be best for the data at hand. Aside transformations, other distributions exist and can equally (or even better) be used for a continuous response in a linear regression model. The theory in GLM extends the linear regression theory such that, a much broader family of distributions can be used for the error terms other than the normal distribution. In this paper, other continuous distributions are used to illustrate how they outperform the normal distribution in some instances. It is also shown that, occasionally (for a continuous response), the normal distribution does not seem to be a choice unless transformations are applied. As a tool for assessing which of the distributions provides the best fit, both AIC and BIC are used.

To fit a GLM in SAS, the GENMOD procedure is used. In R, this can be accomplished by using the glm function. With these tools, only a handful of distributions can be used for the error terms. However, with the GAMLSS package in R, a number of distributions can be utilized. In using GAMLSS, the distribution of the response variable does not necessarily have to belong to the exponential family.

Keywords: GLM, Exponential Family, AIC, BIC, GAMLSS.