Investigations of model validity using residuals

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Ponder, Wendell
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Procedures for assessing model adequacy have been investigated. Since any detection of model misspecification usually begins with an examination of a set of sample residuals resulting from a fitted model, residual predictors have been examined. A necessary and sufficient condition for a residual predictor to have zero expectation and a specified covariance matrix has been obtained. Within this class of residual predictors, the one that minimizes the expected sum of squared prediction errors was found;Towards the detection of model misspecification, it was shown that a set of predicted residuals can be transformed to a set of independent Beta variables. Since the expected values of the Beta variables have a known ordering under the null hypothesis, each ordering of a set of Beta values can be ranked from most likely to least likely based upon the probability of each possible ordering. Thus, if incorrect model specification causes an extreme ordering to appear, then in principle it can be detected by calculating the extremeness of the ordering;The particular type of misspecification caused by the choice of an inappropriate degree in polynomial regression has been investigated, and a new procedure, based upon the Durbin-Watson d-statistic, has been proposed for determining the appropriate polynomial degree. It was shown that the d-statistic can be transformed to another statistic F(,d), whose distribution, for the case of polynomial regression, differs from a central F-distribution only by quantities on the order of 1/n('1). In the context of selecting the proper polynomial degree, the power of the F(,d)-test was compared to that of the forward selection F-test through examination of the probability limits of the two test Statistics and Probability; This study showed that the F(,d)-test appears to be the more sensitive to underspecification. The probability limit of the Durbin-Watson d-statistic was also examined to derive a relationship between the amount of autocorrelation among the true residuals which would be needed to produce the same probability limit as that produced by an omitted variable. Finally, an example was given where the F(,d)-test did detect the real need for a quadratic term which the usual forward selection F-test failed to detect.

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Fri Jan 01 00:00:00 UTC 1982