Introduction to Categorical Data Analysis

118 LOGISTIC REGRESSION
4.4.2 Model Comparison to Check Whether a Term is Needed
Are certain terms needed in a model? To test this, we can compare the maximized
log-likelihood values for that model and the simpler model without those terms.
To test whether color contributes to model (4.11), we test H0: β1 = β2 = β3 = 0.
This hypothesis states that, controlling for width, the probability of a satellite is
independent of color. The likelihood-ratio test compares the maximized log-likelihood
L1 for the full model (4.11) to the maximized log-likelihood L0 for the simpler
model in which those parameters equal 0. Table 4.6 shows that the test statistic is
−2(L0 − L1) = 7.0. Under H0, this test statistic has an approximate chi-squared
distribution with df = 3, the difference between the numbers of parameters in the
two models. The P -value of 0.07 provides slight evidence of a color effect. Since the
analysis in the previous subsection noted that estimated probabilities are quite different
for dark-colored crabs, it seems safest to leave the color predictor in the model.
4.4.3 Quantitative Treatment of Ordinal Predictor
Color has a natural ordering of categories, from lightest to darkest. Model (4.11)
ignores this ordering, treating color as nominal scale. A simpler model treats color in
a quantitative manner. It supposes a linear effect, on the logit scale, for a set of scores
assigned to its categories.
To illustrate, we use scores c ={1, 2, 3, 4} for the color categories and fit the model
The prediction equation is
logit[P(Y = 1)] =α + β1c + β2x (4.12)
logit[ ˆ
P(Y = 1)] =−10.071 − 0.509c + 0.458x
The color and width estimates have SE values of 0.224 and 0.104, showing strong
evidence of an effect for each. At a given width, for every one-category increase in
color darkness, the estimated odds of a satellite multiply by exp(−0.509) = 0.60. For
example, the estimated odds of a satellite for dark colored crabs are 60% of those for
medium-dark crabs.
A likelihood-ratio test compares the fit of this model to the more complex
model (4.11) that has a separate parameter for each color. The test statistic equals
−2(L0 − L1) = 1.7, based on df = 2. This statistic tests that the simpler model
(4.12) holds, given that model (4.11) is adequate. It tests that the color parameters
in equation (4.11), when plotted against the color scores, follow a linear
trend. The simplification seems permissible (P = 0.44).
The estimates of the color parameters in the model (4.11) that treats color as
nominal scale are (1.33, 1.40, 1.11, 0). The 0 value for the dark category reflects the
lack of an indicator variable for that category. Though these values do not depart
significantly from a linear trend, the first three are similar compared to the last one.