The LOGSELECT Procedure
Predicted Probabilities and Regression Diagnostics
For binary response data, you can produce observationwise predicted probabilities, confidence limits, and regression diagnostics developed by Pregibon (1981) by specifying the OUTPUT statement. For multinomial response data, you can likewise produce observationwise predicted probabilities, confidence limits, and raw residuals.
For a binary response model, given a vector of covariates for the ith observation in your data table and the model-predicted parameter estimates , you can write the linear predictor . The mean of the ith observation , or the model-predicted event probability , is , where the link function g is chosen by specifying the LINK= option. The variance of the binary distribution is , and is the estimated covariance of . Denote the frequency of the ith observation as and the weight as .
For ordinal response models, the predicted cumulative probabilities are computed in the same fashion by using the appropriate model-predicted intercept parameters and letting consist of the slope parameters: and for .
For nominal response models, the predicted probabilities are computed by using the appropriate model-predicted intercept parameters and letting consist of the slope parameters: and for .
Confidence Intervals
Approximate confidence intervals for predicted probabilities can be computed as follows. The variance of the linear predictor is estimated by
For multinomial models, the variance also depends on the response function. Let be a (J–1) column vector whose jth entry is equal to 1 and all other entries are equal to 0. Redefine , , and . Then
The asymptotic confidence interval for is
where is the th percentile point of a standard normal distribution.
The predicted probability and the confidence limits for are obtained by back-transforming the corresponding measures for the linear predictor. So the confidence limits are
Hat-Matrix Diagonals
The diagonal elements of the hat matrix are useful in detecting extreme points in the design space, where they tend to have larger values. For the generalized linear model, the variance of the ith individual observation is
For the ith observation, let
where is the derivative of the link function evaluated at . The weight matrix is a diagonal matrix, with denoting the ith diagonal element, which is used in computing the expected information matrix. Define the leverage, or hat-matrix diagonal, , as the ith diagonal element of the matrix
If the estimated probability is extreme (less than 0.1 and greater than 0.9, approximately), then the hat-matrix diagonal might be greatly reduced in value. Consequently, when an observation has a very large or very small estimated probability, its leverage is not a good indicator of the observation’s distance from the design space (Hosmer and Lemeshow 2000, p. 171).
Residuals
Residuals are useful in identifying observations that are not explained well by the model. For binary and binomial response data, the raw residual is
where is the number of event responses out of trials for the ith observation. For single-trial syntax, and if the ordered response is 1 and otherwise. For multinomial response data, the raw residual is
where = 1 if the ith observation has response level j and = 0 otherwise, and are the model-predicted probabilities of response level j for observation i.
The Pearson residual is the square root of the ith observation’s contribution to Pearson’s chi-square:
The deviance residual is the square root of the contribution of the ith observation to the deviance, with the sign of the raw residual,
where
The working residual is
The Pearson residuals, standardized to have unit asymptotic variance, are
The deviance residuals, standardized to have unit asymptotic variance, are
The likelihood residuals, which estimate components of a likelihood ratio test of deleting an individual observation, are a weighted combination of the standardized Pearson and deviance residuals:
Other Regression Diagnostics
The CBAR statistic is a confidence interval displacement diagnostic that provides a scalar measure of the influence of an individual observation on . This diagnostic is based on the same idea as the Cook distance in linear regression theory (Cook and Weisberg 1982), but it uses the one-step estimate:
The DIFDEV and DIFCHISQ statistics are diagnostics for detecting ill-fitted observations—observations that contribute heavily to the disagreement between the data and the predicted values of the fitted model. DIFDEV is the change in the deviance that results from deleting an individual observation, and DIFCHISQ is the change in the Pearson chi-square statistic that results from the same deletion. By using the one-step estimate, DIFDEV and DIFCHISQ for the ith observation are computed as follows: