Description
1. (Adapted from Exercise 1.10 of Agresti (2015)) GLMs normally use a hierarchical structure by which
the presence of a higher-order term implies also including the lower-order terms. Explain why this is sensible,
by showing that
(a) a model that includes an x
2
explanatory variable but not x makes a strong assumption about where the
maximum or minimum of E[Y ] occurs.
(b) a model that includes x1x2 but not x2 makes a strong assumption about the effect of x2 when x1 = 0.
2. (Adapted from Exercise of Agresti (2015)) Show that the gamma distribution is a member of the
exponential dispersion family and identify the natural parameter. The p.d.f. for the gamma distribution can
be written as
f(y; k, µ) = (k/µ)
k
Γ(k)
e
−ky/µy
k−1
, y ≥ 0,
for which E[Y ] = µ, V ar(Y ) = µ
2/k.
3. (Adapted from Exercise 7.32 of Agresti (2015)) For the horseshoe crab data, the negative binomial
modeling shown in the R output below treats colour as nominal-scale and then in a quantitative manner, with
the category numbers as scores. Interpret the result of the likelihood-ratio test comparing the two models.
For the simpler model, interpret the colour effect and interpret results of the likelihood-ratio test of the null
hypothesis of no colour effect.
> fit.nb.color <- glm.nb(y ~ factor(color)) # Using Crabs.dat file
> summary(fit.nb.color)
Estimate Std. Error z value Pr(>|z|)
(Intercept) 1.4069 0.3526 3.990 6.61e-05f
factor(color)2 -0.2146 0.3750 -0.572 0.567
factor(color)3 -0.6061 0.4036 -1.502 0.133
factor(color)4 -0.6913 0.4508 -1.533 0.125
—
> fit.nb.color2 <- glm.nb(y ~ color) # using color scores (1,2,3,4)
> summary(fit.nb.color2)
Estimate Std. Error z value Pr(>|z|)
(Intercept) 1.7045 0.3095 5.507 3.66e-08
color -0.2689 0.1225 -2.194 0.0282
—
> anova(fit.nb.color2, fit.nb.color)
Likelihood ratio test of Negative Binomial Models
Response: y
Model theta Res.df 2 x log-lik. Test df LR stat. Pr(Chi)
1
1 0.7986 171 -762.6794
2 0.8019 169 -762.2960 1 vs. 2 2 0.3834 0.8256
—
> 1 – pchisq(767.409-762.679, df=172-171) # LR test vs. null model
[1] 0.0296
4. In the first nine decades of the twentieth century in baseball’s National league, the percentage of times the
starting pitcher pitched a complete game were: 72.7 (1900-1909), 63.4, 50.0, 44.3, 41.6, 32.8, 27.2, 22.5, 13.3
(1980-1989).
(a) Treating the number of games as the same in each decade, the linear probability model has ML fit
πˆ = 0.7578 − 0.0694x, where x = decade (x = 1, 2, . . . , 9). Interpret −0.0694.
(b) Substituting x = 13, predict the percentage of complete games for 2020-2029. Interpret.
(c) The logistic regression ML fit is ˆπ = exp(1.148 − 0.315x)/[1 + exp(1.148 − 0.315x)]. Obtain ˆπ for x = 13.
Which link function do you prefer?
5. For a study using the logistic regression model to determine characteristics associated with remission in cancer
patient, the following table shows the most important explanatory variable, a labeling index (LI). This index
measures proliferative activity of cells after a patient receives an injection of tritiated thymidine, representing
the percentage of cells that are “labelled” The response Y measured whether the patient achieved remission
(1 = yes). Software reports for a logistic regression model using LI to predict the probability of remission.
Table 1 contains the output.
Criterion Intercept Only Intercept and Covariate
−2 log L 34.372 26.073
Parameter Estimate S.E. Chi-Square pr > ChiSq
Intercept -3.7771 1.3786 7.5064 0.0061
LI 0.1449 0.0593 5.9594 0.0146
Odds Ratio Estimates
Effect Point Estimate 95% CI
LI 1.156 (1.029, 1.298)
Table 1: Computer Output for Cancer data
(a) Show how software obtained ˆπ = 0.068 when LI = 8.
(b) Show that ˆπ = 0.5 when LI = 26.06694.
(c) Show that the rate of change in ˆπ is 0.009 when LI = 8 and 0.036 when LI = 26.06694.
(d) The lower quartile and upper quartile for LI are 14 and 28. Show that ˆπ increases by 0.42, from 0.15 to
0.57, between those values.
(e) For a unit change in LI, show that the estimated odds of remission multiply by 1.156.
(f) Explain how to obtain the confidence interval reported for the odds ratio. Interpret.
(g) Conduct a likelihood ratio test for the effect (β = 0), showing how to construct the test statistic using
the −2 log L values reported.
6. (Adapted from Exercise 5.16 of Agresti (2015)) A study has ni
independent binary observations
{yi1, . . . , yini
} at level X = xi
, i = 1, . . . , N, with P
i
ni = n. Consider the model logit(πi) = β0 +β1xi
, where
πi = P(Yij = 1).
(a) Show that the kernel of the likelihood function is the same is treating the data as n Bernoulli observations
or N binomial observations.
(b) For the saturated model, explain why the likelihood function is different for these two data forms. Hence,
the deviance reported by software depends on the form of data entry.
2
(c) Explain why the difference between deviances for two unsaturated models does not depend on the form
of data entry.
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