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[0.09, 0.31]

[0.11, 0.32]

2.5.2.1 Example: Mortality of moths exposed to cypermethrin

As an additional comparison of procedures for constructing confidence intervals, we compute 95 percent confidence intervals for 7, the parameter of the logistic regression model that denotes the effect of sex on mortality of moths exposed to cypermethrin (see Section 2.5.1.3). Recall that the MLE for the logit-scale effect of sex is 7 = 1.10, and an estimate of its uncertainty is SE(7) = 0.356. Therefore, the 95 percent confidence interval for 7 based on inverting the Wald test is [0.40,1.80] (=1.10 ± 1.96 * 0.356).

To compute a 95 percent confidence interval for 7 based on inverting the likelihood ratio test, we must find the values of 70 that satisfy the following inequality

—2{log L(a, 7 70|y) — log L(a, 7 7|y)} < 3.84. (2.5.6)

Using a numerical root-finding procedure, we find that [0.42,1.82] is the 95 percent confidence interval for 7. In Figure 2.7 we plot the left-hand side of Eq. (2.5.6) against fixed values of y0 to show that we have calculated this interval correctly.

2.5.3 A Bayesian Approach to Model Selection

In this section we develop a Bayesian approach to the problem of selecting between two nested models, which was described in Section 2.5.1. Recall that H0 and Hi denote two complementary hypotheses that can be specified in terms of a (possibly vector-valued) parameter 0:

Hi : 0 e ©0, where ©0 is a subset of the parameter space © and ©0 is the complement of ©0. The problem is to select either the null model represented by H0 or the alternative (more complex) model represented by Hi .

A Bayesian analysis of the problem requires a prior for each model's parameters. Let n(0|H0) and n(0|H1) denote prior density functions for the parameters of null

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