181-2007: Using SAS® PROC NLMIXED for Robust Regression

We show how the NLMIXED procedure can be used to fit linear models (simple linear regression, multiple linear regression, analysis of variance and analysis of covariance) when data are not normally distributed or contaminated with potential outliers. This is accomplished using PROC NLMIXED’s capability of fitting user defined distributions through the general log likelihood option in the Model statement. This methodology can offer advantages over the Mestimation option in Proc ROBUSTREG which cannot analyze repeated measures and Proc NPAR1WAY which only provides p-values and cannot adjust for covariates or handle repeated measures. We provide some the basic theory and directions for specifying the likelihoods needed and then demonstrate the usefulness of the method on several examples. INTRODUCTION Linear models are often fitted using the REG, GLM, GENMOD, and MIXED procedures. With the exception of the GENMOD procedure the remaining procedures always assume that the data, Y, can be modeled as Y= mean function + error, where the errors have a normal distribution. When the data consist of independent observations the procedures will use either ordinary least squares (OLS) or weighted least squares (WLS), if weights are specified in the procedure call. The normality assumption and presence of outliers can be assessed by examining the residuals from the model. All of these procedures supply raw residuals, calculated as the difference between the observed data value and the estimated mean function, residual=Y-estimated mean function. The raw residuals can then be examined for non-normality and for outliers by graphical means and with Proc UNIVARIATE. It is not uncommon to find that the residuals display one of the following: heavier tails than expected for a normal distribution; skewness; or a small number of larger than expected residuals, which may reflect outliers. In this paper we focus on the presence of outliers and heavy tailed residual distributions. There is no single approved definition of an outlier; we will use the intuitive but imprecise definition that an outlier is an observation that does not fit in with the bulk of the data. In a regression model outlying data points are often identified as observations with residuals that are much larger (in absolute value) than the remaining residuals. Note that residuals are dependent upon the model. It is possible for apparent outliers to be produced by an incorrect model and can be corrected by replacing an incorrect model with the correct model. Another possibility is that the data have been contaminated in some way. For example, in a clinical trial we may need to measure a subject’s blood pressure. There can be measurement error due to equipment malfunction or operator error, or the blood pressure may simply be written down incorrectly on the Case Report Form. It is also possible that one or two subjects decide to go on an espresso and cappuccino binge artificially raising their blood pressure. Heavy tailed distributions are another difficulty that often arises in real world data. The term “heavy tail” is often used in comparison to the relatively “light tail” of a normal distribution. When plotted, the probability density function (pdf) of a heavy tailed distribution will exhibit thicker tails than a normal distribution, that is, the heavy tailed distribution allows for a higher probability of relatively extreme events. Almost all observations generated from a normal distribution will be within plus or minus three standard deviations from the mean (99.8% of the time to be precise). Heavy tailed distribution allow for a larger number of observations far from the mean. Heavy tailed distributions are not uncommon. They may be either an intrinsic characteristic of the data or they may be a byproduct of the analysis method. For an example of the first kind, many types of financial data empirically demonstrate heavy tails (Bouchaud and Potter, 2000). An example where the analyst creates a heavy tailed distribution can be found in clinical trials. Many clinicians like to look at percent changes, 100% X (post baselinebaseline)/baseline. If some baseline values are close to zero, extremely large percent changes (either positive or negative) can result. This is problematic enough, but if the measured variable can only take on positive values, the percent change variable can also be highly skewed. SAS Global Forum 2007 Statistics and Data Analysis