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LANCASTER UNIVERSITY
2010 EXAMINATIONS
PART II (Third or Fourth Year)
MATHEMATICS & STATISTICS 1 12 Hours
Math 352: Generalized Linear Models
You should answer ALL Section A questions and ONE Section B question. In Section A there are questions worth a total of 50 marks, but the maximum mark that you can gain there is capped at 40.
SECTION A
A1. (a) Let q be a probability density or mass function. Define its cumulant generating function (cgf) k. [3] (b) Define an exponential family (EF) distribution with canonical parameter θ based on a probability density or mass function q. [2] (c) State the expressions for mean and variance of the EF distribution in terms of the cgf k and the canonical parameter θ; you don’t need to derive these expressions. [4] (d) Explain why the mean is a monotonically increasing function of θ and hence explain the concept of variance function. [6] (e) Derive the EF based on the probability density function (pdf)
q(y) = exp(−y), y > 0. [6]
(f) Use (c) and (d) to derive the mean function and the variance function of the above EF distribution. [4]
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SECTION A continued
A2. The following data are survival times {Yi; 1 ≤ i ≤ n} in months of n = 24 patients after having an operation. The patients are in four groups giving the month the operation occurred after the diagnosis. group 0 110.1 52.2 45.1 19.9 90.0 41. 1 32.1 4.0 5.2 53.7 10.3 12. 2 4.8 3.5 1.8 0.1 9.1 5. 3 5.0 2.1 5.9 0.1 1.4 1. So, for example, 6 patients had immediate operations, 6 patients waited 1 month, and so on.
(a) Plot the survival time against the covariate month. [4] (b) Give at least two reasons why it would not be sensible to fit a simple normal linear model to this data to explain the relation between the response survival time and covariate. [4] (c) Identify mathematically a GLM that might fit this data. You should carefully state the model, link function and covariates. [8] (d) Construct the design matrix for this model. [4] (e) Write down an expression for the log-likelihood function. [5]
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SECTION B continued
B2. Consider the Poisson GLM for the aids data of your lecture note where the response is the number of deaths from AIDS at covariate time t. Consider the model Yi ∼ Poisson (μi), log(μi) = βti, 1 ≤ i ≤ n.
(a) Write down the log-likelihood function (β) based on {(Yi, ti); 1 ≤ i ≤ n}. [4] (b) Write down the equation that needs to be solved to find the maximum likelihood estimate βˆ. Is it possible to solve this equation explicitly here? [4] Next we implement Newton-Raphson algorithm with the following iterative scheme for obtaining βˆ:
Set k = 0. Choose an initial estimate β^0.
At the k + 1-th step calculate βk+1^ = βk^ − ′(βk)/′′(βk) (1) and set k to k + 1.
Repeat this step until the change in βk^ is negligible. Put βˆ = βk.
(c) Write down the (k + 1)-th step for this model. [3]
On the Aegean island of Kalythos the male inhabitants suffer from a congenital eye disease, the effects of which become more marked with increasing age. Samples of is- lander males of various ages were tested for blindness and the results recorded. The data is shown below: x = Age: 20 35 45 55 70 m = No. tested: 50 50 50 50 50 r = No. blind: 6 17 26 37 44
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SECTION B continued
B2. (d) Identify the response and covariate and give any sensible plot of the response against covariate. State your understanding of the plot. [6] (e) Identify mathematically a GLM that might fit this data. You should carefully state the model, link function and the explanatory variables. [8] (f) Briefly indicate how would you form the likelihood to estimate parameters for the model. [5]
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