STAT7030 Generalised Linear Models

Question 1. [48 Marks]. The file Doctor Data, available on the course Wattle page in the
Datasets section, is concerned with the number of deaths by heart disease in a sample of
doctors from different age groups and with different smoking status (smoker/non-smoker).
Further details are given below.
Column 1 Observation number – [Variable V1=1 to 10]
Column 2 Age – [Variable V2= 1,2,3,4,5] {1 = 35-44 years; 2 = 45-54 years;
3 = 55-64 years; 4 = 65-74 years; 5 = 75-84 years}.
Column 3 Deaths – [Varaible V3 = Number of Deaths] {Count Variable}.
Column 4 Person-Years at Risk – [Variable V4 = Aggregate years] {Count Variable}.
Column 5 Smoker – [Variable V5 = 1,2] { Factor: 1 = Smoker; 2 = Non-Smoker}.
Further comment: do not forget to declare variable V 5 as a factor.
Please provide answers to the following.
(a) Fit a suitable set of Poission regression models to the data and select what you think is
the best model, using AIC to inform your selection. Please bear the following points
in mind:
• Treat log(V 4) as a covariate (as opposed to an offset) throughout part (a).
• Although variable V 2 could be treated as a factor, it is suggested that you treat
it as a numerical variable. If treated as a factor, Age will use up too many
parameters relative to the sample size. Also, consider the covariate V 2 ∗ V 2.
• Investigate interactions as well as main effects.
• Present the AIC values in a suitable table.
• Use suitable residual plots and other diagnostics to investigate the goodness of
fit of the model you select.
• In part (a) and in part (b) below, take the scale parameter φ to be 1.
[12 marks]
(b) Focusing on the model you selected in part (a), perform a change of deviance test to
assess whether it would be better to treat log(V 4) as an offset or as a covariate.
[6 marks]
(c) Repeat parts (a) and (b) using negative binomial regression models with a log link.
Comment on (i) similarities and (ii) differences in your results for part (c), compared
with parts (a) and (b).
[18 marks]
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(d) Returning to the best Poisson regression model that you identified in part (a), estimate
the scale parameter φ and provide a Wald test of the null hypothesis that φ = 1,
presenting your answer in the form of a p-value. As part of your answer to part (d)
you should present the R code that you used.
[12 marks]
Question 2. [12 marks]. A dataset was collected on a sample of 400 melanoma patients.
Two factors were recorded on each patient: tumour type (4 levels) and tumour site (3
levels).
The tumour types considered were: Hutchinson’s melanotic freckle (H); Superficial spreading melanoma (S); Nodular (N); and Indeterminate (I).
The tumour sites considered were: Head and neck (HN); Trunk (T); and Extremities (E).
The dataset may be found in the file Melanoma Data on the Wattle page in the Datasets
section.
(a) Treating both tumour type and tumour site as response factors, test the null hypothesis
of independence. Present your result in the form of a p-value and give a brief explanation
and interpretation of your findings.
[9 marks]
(b) You are at a conference presenting this work and a doctor says to you that it makes
more sense to treat site as an explanatory factor. You accept that there may be good reasons
for doing this. Present an appropriately modified explanation and interpretation of your
findings.
[3 marks]
Note: In this question you are expected to use log linear modelling implemented via the R
function glm(.). You should presentPthe relevant section of R output. An answer which
uses the classical chi-squared statistic (O − E)2 /E will not receive many marks.
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1 1 32 52407 1
2 2 104 43248 1
3 3 206 28612 1
4 4 186 12663 1
5 5 102 5317 1
6 1 2 18790 2
7 2 12 10673 2
8 3 28 5710 2
9 4 28 2585 2
10 5 31 1462 2Tumour Site Count
1 H HN 22
2 S HN 16
3 N HN 19
4 I HN 11
5 H T 2
6 S T 54
7 N T 33
8 I T 17
9 H E 10
10 S E 115
11 N E 73
12 I E 28