Introduction to Categorical Data Analysis

78 GENERALIZED LINEAR MODELS
Figure 3.4. Number of satellites by width of female crab.
Most software has more sophisticated ways of smoothing the data, revealing the
trend without grouping the width values. Smoothing methods based on generalized
additive models do this by providing even more general structural form than GLMs.
They find possibly complex functions of the explanatory variables that serve as the
best predictors of a certain type. Figure 3.5 also shows a curve based on smoothing the
data using this method. The sample means and the smoothed curve both show a strong
increasing trend. (The means tend to fall above the curve, since the response counts
in a category tend to be skewed to the right. The smoothed curve is less susceptible
to outlying observations.) The trend seems approximately linear, and we next discuss
models for which the mean or the log of the mean is linear in width.
Let μ denote the expected number of satellites for a female crab, and let x denote
her width. From GLM software, the ML fit of the Poisson loglinear model (3.5) is
log ˆμ =ˆα + ˆβx =−3.305 + 0.164x
The effect ˆβ = 0.164 of width has SE = 0.020. Since ˆβ >0, width has a positive
estimated effect on the number of satellites.
The model fit yields an estimated mean number of satellites ˆμ, afitted value, at
any width. For instance, from equation (3.6), the fitted value at the mean width of
x = 26.3 is
ˆμ = exp( ˆα + ˆβx) = exp[−3.305 + 0.164(26.3)] =e 1.01 = 2.7