Introduction to Categorical Data Analysis

5.2 MODEL CHECKING 145
increases. Models with multiple predictors would consider interaction terms. If more
complex models do not fit better, this provides some assurance that a chosen model
is adequate.
We illustrate for the model in Section 4.1.3 that used x = width alone to predict
the probability that a female crab has a satellite,
logit[π(x)]=α + βx
One check compares this model with the more complex model that contains a quadratic
term,
logit[ˆπ(x)]=α + β1x + β2x 2
For that model, ˆβ2 = 0.040 has SE = 0.046. There is not much evidence to support
adding that term. The likelihood-ratio statistic for testing H0: β2 = 0 equals 0.83
(df = 1, P -value = 0.36).
The model in Section 4.4.1 used width and color predictors, with three dummy
variables for color. Section 4.4.2 noted that an improved fit did not result from adding
three cross-product terms for the interaction between width and color in their effects.
5.2.2 Goodness of Fit and the Deviance
A more general way to detect lack of fit searches for any way the model fails.
A goodness-of-fit test compares the model fit with the data. This approach regards
the data as representing the fit of the most complex model possible – the saturated
model, which has a separate parameter for each observation.
Denote the working model by M. In testing the fit of M, we test whether all parameters
that are in the saturated model but not in M equal zero. In GLM terminology,
the likelihood-ratio statistic for this test is the deviance of the model (Section 3.4.3).
In certain cases, this test statistic has a large-sample chi-squared null distribution.
When the predictors are solely categorical, the data are summarized by counts in
a contingency table. For the ni subjects at setting i of the predictors, multiplying the
estimated probabilities of the two outcomes by ni yields estimated expected frequencies
for y = 0 and y = 1. These are the fitted values for that setting. The deviance
statistic then has the G 2 form introduced in equation (2.7), namely
G 2 (M) = 2 � observed [log(observed/fitted)]
for all the cells in that table. The corresponding Pearson statistic is
X 2 (M) = � (observed − fitted) 2 /fitted
For a fixed number of settings, when the fitted counts are all at least about 5,
X 2 (M) and G 2 (M) have approximate chi-squared null distributions. The degrees of