MATH 523: Generalized Linear Models Midterm Exam VERSION B 2020
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Midterm Exam VERSION B
MATH 523: Generalized Linear Models
2020
Problem 1. Consider the Geometric family of distributions with parameter π e (0, 1) and probability mass function
f (y; π) = πy (1 _ π), y e {0, 1, . . .}.
(a) [4 marks] Show that the Geometric family is an exponential dispersion family. Identify the functions b(.) and c(.), as well as the dispersion and canonical parameters.
(b) [3 marks] Compute the mean and variance of the Geometric distribution and identify the mean-variance relationship.
(c) [2 marks] Identify the canonical link for a Geometric GLM. Comment on its suitability.
(d) [2 marks] Explain why the link function log(_ log(µ/(1 + µ))) may be better suited than the canonical link for this GLM.
(e) [1 mark] For what kind of data would a Geometric GLM be better suited than a Poisson GLM (Hint: look at the mean-variance relationship)?
(f) [4 marks] Derive the likelihood (score) equations for the Geometric GLM when the
link from part (d) is used. Explain how the equations simplify when the canonical link is used.
(g) [4 marks] Calculate the Deviance for a Geometric GLM. Does it depend on the link
function used? Explain.
Problem 2. Consider the following data on ear infections in swimmers from the 1990 Pilot Surf/Health Study of the New South Wales Water Board.
NumInfec |
the number of self-diagnosed ear infections |
Age |
the age of the swimmer (with levels 15-19, 20-24 and 25-29); |
Sex |
gender of the swimmer (with levels Female and Male); |
Loc |
the usual swimming location (with levels Beach and NonBeach) |
Swim |
frequency of swims in the ocean (with levels Freq(ently) and Occas(ionally)) |
(a) [5 marks] The data were first modeled with a GLM model m1 whose output is given
on page 4, lines 1–23. From this output:
– Identify the response and the predictors;
– Identify the GLM that was used and the link function;
– Identify the sample size n;
– For each main effect, write down whether it is treated as a factor or a covariate (continuous predictor).
(b) [3 marks] In model m1, quantify the effect of Age and Swim on the response.
(c) [3 marks] Fill in the values marked by XXX on line 12. Does the p-value allow you to conclude that Age is not a significant predictor? Explain.
(d) [4 marks] What is the estimated mean number of self-diagnosed ear infections of a swimmer aged 27 who swims frequently in the ocean?
(e) [5 marks] A simpler model m2 whose output is given on lines 26–46 has been fitted to the data. Test whether m2 is an adequate simplification of m1 at the 5%
significance level. Interpret the finding in terms of significance of Age and Swim. Use the R output on page 4, and the χy(2) table on page 5.
1 C a l l :
2 glm ( f o r m u l a = NumInfec ˜ Age + Swim , f a m i l y = p o i s s o n ) 3
4 D e v i a n c e R e s i d u a l s :
5 Min 1Q Median 3Q Max 6 - 2.0300 - 1.5001 - 1.2851 0 . 6 7 7 4 7 . 1 6 2 0 7
8 C o e f f i c i e n t s :
9 E s t i m a t e Std . E r r o r z v a l u e Pr ( s|z | )
10 ( I n t e r c e p t ) 0 . 1 1 7 9 4 0 . 0 9 5 3 7 1 . 2 3 7 0 . 2 1 6 2
11 Age20 -24 - 0.30947 0 . 1 2 3 9 1 - 2.497 0 . 0 1 2 5 *
12 Age25 -29 - 0.27479 0 . 1 2 8 5 9 XXX XXX
13 SwimOccas 0 . 6 0 5 0 2 0 . 1 0 4 9 8 5 . 7 6 3 8 . 2 5 e -09 *** 14 15 S i g n i f . c o d e s : 0 ’* * * ’ 0 . 0 0 1 ’* * ’ 0 . 0 1 ’* ’ 0 . 0 5 ’. ’ 0 . 1 ’ ’ 1
16 17 ( D i s p e r s i o n p a r a m e t e r f o r p o i s s o n f a m i l y t a k e n t o be 1 )
18 19 N u l l d e v i a n c e : 8 2 4 . 5 1 on 286 d e g r e e s o f f r e e d o m
20 R e s i d u a l d e v i a n c e : 7 8 1 . 2 6 on 283 d e g r e e s o f f r e e d o m
21 AIC : 1 1 6 1 . 7
22
23 Number o f F i s h e r S c o r i n g i t e r a t i o n s : 6 24
25
26 C a l l :
27 glm ( f o r m u l a = NumInfec ˜ Swim , f a m i l y = p o i s s o n ) 28
29 D e v i a n c e R e s i d u a l s :
30 Min 1Q Median 3Q Max 31 - 1.8930 - 1.3993 - 1.3993 0 . 8 2 2 3 6 . 7 8 8 6 32
33 C o e f f i c i e n t s :
34 E s t i m a t e Std . E r r o r z v a l u e Pr ( s|z | ) 35 ( I n t e r c e p t ) - 0.02120 0 . 0 8 4 5 2 - 0.251 0 . 8 0 2
36 SwimOccas 0 . 6 0 4 3 5 0 . 1 0 4 9 7 5 . 7 5 7 8 . 5 5 e -09 ***
37 38 S i g n i f . c o d e s : 0 ’* * * ’ 0 . 0 0 1 ’* * ’ 0 . 0 1 ’* ’ 0 . 0 5 ’. ’ 0 . 1 ’ ’ 1
39 40 ( D i s p e r s i o n p a r a m e t e r f o r p o i s s o n f a m i l y t a k e n t o be 1 )
41 42 N u l l d e v i a n c e : 8 2 4 . 5 1 on 286 d e g r e e s o f f r e e d o m
43 R e s i d u a l d e v i a n c e : 7 8 9 . 8 1 on 285 d e g r e e s o f f r e e d o m
44 AIC : 1 1 6 6 . 2
45 46 Number o f F i s h e r S c o r i n g i t e r a t i o n s : 6
2022-02-23