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OTE/SPH
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JWBK119-12
August 31, 2006
Logistic Regression Approach 185
2
larger than the critical value of χ with 1 degree of freedom at the 5% significance
level. Hence the null hypothesis is rejected, implying that the explanatory variable
has significant effects on the response.
The odds ratio given in Table 12.8 as evaluated in MINITAB essentially gives the
rate of increase in the odds given a unit change in the explanatory variable. For the
horseshoe crab example, the rate of increase in the odds of finding satellites given a
unit increase in the width of the female horseshoe crab is 2.15. As shown in equation
β
(12.14), this rate of increase in odds is given by e . The confidence interval for the
odds ratio can thus be evaluated from
ˆ β±z α/2 ASE( ˆ β)
e .
With the parameter estimates, the probabilities of the presence of satellites given the
carapace width of the female crab can also be evaluated as:

exp ˆα + ˆ βx Width
ˆ π Presence (x Width ) =
1 + exp ˆα + ˆ βx Width
These are known as the event probabilities. These event probabilities are plotted against
the explanatory variables in Figure 12.1.
After fitting the logistic regression model, it is important to examine the suitability
of the model in light of the prediction generated. The different techniques available
for such tests are commonly referred to as goodness-of-fit (GOF) tests as they compare
the quality of fit between the predicted responses and the actual responses. The most
common GOF tests in categorical data analysis are those based on likelihood ratios
and Pearson residuals. Here, GOF tests based on likelihood ratios are discussed.
The motivation for likelihood ratio based GOF tests basically arises from the max-
imum likelihood theory underlying the parameter estimation process. A typical

1.0
Probability of presence of satellite 0.6
0.8

0.4

0.2
Event Probabilities
Observations
0.0
20 22 24 26 28 30
x-width

Figure 12.1 Plot of event probability against explanatory variable, x Width , for the horseshoe
crab example.