Analysis of Residuals

Plotting Residuals

The typical technique involves the production of a scatterplot permitting a visual examination in addition to more formal statistical analysis. Various possibilities exist that may require attention. Each separate test or assessment requires some effort. The possibility exists that the explanation for the residuals requires a combination of multiple explanations and consideration of several different possible sources.

The first option suggests that the residual pattern is random and within expected tolerance limits for the overall pattern and individual predictions. Under these conditions, the multiple regression equation provides a set of predictions that work without any of the problems identifiable on the basis of residuals. The errors that exist are random, based on the normal assumptions related to sampling error.

The first consideration is the existence of outliers. An outlier involves a single value whose error is so great that there is a distortion to the estimation of the process. Consider a set of values where the error value for one case is 100. Suppose that the next largest positive value for the error is 10. Remember, that the sum of the errors should be zero. The result is that the level of error for this single case becomes so large that the rest of the values are in a sense distorted because almost all of the error becomes associated with this single case. The most frequent remedy is removal of the identified case. The problem with a random sample is that under sampling, the extreme value may simply reflect a randomly drawn outlier.

In theory, 50% of the errors should be positive and 50% should be negative. Often this percentage is distorted by an outlier, but if the percentage is greater or lesser by a significant amount, some adjustment may be required. The assumption is that error should operate as a normal distribution around a mean (zero) and a departure from that indicates some element that may require adjustment or consideration. An extreme outlier in the analysis of residuals indicates the source of a significant potential departure from the normal curve considerations. The definition of a random extreme outlier would be that the mean of the sample adjusts only slightly while the variance (variability) observes a tremendous drop. Under those conditions, the outlier often is considered simply the result of a random chance factor associated with sampling.

...