Introduction to Categorical Data Analysis

148 BUILDING AND APPLYING LOGISTIC REGRESSION MODELS
With categorical predictors, we can use residuals to compare observed and fitted
counts. This should be done with the grouped form of the data. Let yi denote the
number of “successes” for ni trials at setting i of the explanatory variables. Let ˆπi
denote the estimated probability of success for the model fit. Then, the estimated
binomial mean ni ˆπi is the fitted number of successes.
For a GLM with binomial random component, the Pearson residual (3.9)
comparing yi to its fit is
Pearson residual = ei =
yi − ni ˆπi
� [ni ˆπi(1 −ˆπi)]
Each Pearson residual divides the difference between an observed count and its fitted
value by the estimated binomial standard deviation of the observed count. When ni
is large, ei has an approximate normal distribution. When the model holds, {ei} has
an approximate expected value of zero but a smaller variance than a standard normal
variate.
The standardized residual divides (yi − ni ˆπi) by its SE,
standardized residual = yi − ni ˆπi
SE
=
yi − ni ˆπi
� [ni ˆπi(1 −ˆπi)(1 − hi)]
The term hi in this formula is the observation’s leverage, its element from the diagonal
of the so-called hat matrix. (Roughly speaking, the hat matrix is a matrix
that, when applied to the sample logits, yields the predicted logit values for the
model.) The greater an observation’s leverage, the greater its potential influence on
the model fit.
The standardized residual equals ei/ √ (1 − hi), so it is larger in absolute value
than the Pearson residual ei. Itis approximately standard normal when the model
holds. We prefer it. An absolute value larger than roughly 2 or 3 provides evidence of
lack of fit. This serves the same purpose as the standardized residual (2.9) defined in
Section 2.4.5 for detecting patterns of dependence in two-way contingency tables. It
is a special case of the standardized residual presented in Section 3.4.5 for describing
lack of fit in GLMs.
When fitted values are very small, we have noted that X 2 and G 2 do not have
approximate null chi-squared distributions. Similarly, residuals have limited meaning
in that case. For ungrouped binary data and often when explanatory variables are
continuous, each ni = 1. Then, yi can equal only 0 or 1, and a residual can assume only
two values and is usually uninformative. Plots of residuals also then have limited use,
consisting merely of two parallel lines of dots. The deviance itself is then completely
uninformative about model fit. When data can be grouped into sets of observations
having common predictor values, it is better to compute residuals for the grouped
data than for individual subjects.