ST404 - Assessed coursework 2 - Generalized Linear Models - Solved

To access the data needed for this assignment, download the file courseworkData2.rda from the ST346 Moodle web page and read it into R using the function load(). This will create a copy of 2 data frames in your R workspace: ships and esoph.

1.    The data frame ships contains data on damage incidents in ships. The ships have been aggregated into classes defined by type (5 categories), year of construction (4 categories), and period of operation (2 categories), giving 5 × 4 × 2 = 40 rows. For each combination of type, year and period, the total months of service of all ships in that class has been aggregated in the variable service, and the total number of damage incidents has been aggregated in the variable incidents.

Note that some classes are empty. For these classes, the variable service is set to zero and the variable incidents is set to the missing value NA. Rows with missing values are automatically removed by R when you fit a statistical model.

(a)     Fit an appropriate GLM to model the risk of damage incidents in terms of the other variables.

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(b)     Which combination of ship type, year of construction and period of operation is at highest risk of damage?         [2]

(c)      Calculate the expected number of damage incidents in twelve months for this class of shipping.

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2.    Consider the following scenarios. Determine the appropriate model to use in each case. You should give the exponential dispersion model (or family in R terminology), the link function, the outcome variable, and the predictor variables.

(a)     A cohort of 18 year-old school leavers were surveyed regarding their future plans, with specific interest in higher education. Several variables were measured, including age, gender, smoking status, as well as whether or not they had a place at university or other higher education institute.

We wish to investigate the variables which are associated with higher education attendance.[1]

(b)     Sixty rats of the same age were divided into 2 groups of 30. At the start of the experiment, all animals were weighed. Then one group was fed a control diet, whilst the other was fed a diet supplemented with vitamin D. After 6 weeks, all rats were weighed again. We wish to understand how vitamin D supplementation affects weight gain. [1]

(c)      A car manufacturer conducted a study to investigate the reliability of their cars. They measured the number of times that each of 1000 vehicles had broken down in the 10 years since it was

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made. They also measured the number of miles that each car had been driven as well as the model of each vehicle and the number of times that it had been serviced. [1]

3.    The data set esoph concerns a study of esophageal cancer conducted in the Ille-et-Vilaine department of France in the 1970s. Participants have been classified by age group (agegp), alcohol consumption in g/day (alcgp) and tobacco consumption in g/day (tobgp). In each class defined by these variables, ncases gives the number of participants in the study with esophageal cancer and ncontrols gives the number of healthy controls without cancer.

(a)     Fit an appropriate GLM to these data to model the risk of esophageal cancer. [2]

(b)     The variables agegp, alcgp, and tobgp are factors. They can be converted to numeric variables using the function as.numeric(). Determine whether each variable should be a factor or a numeric variable using an appropriate methodology for model choice. [2]

(c)      Using your final choice of model, estimate the odds ratio of esophageal cancer for a participant who drinks more than 120g of alcohol/day compared with one who drinks less than 40g of alcohol per day.           [2]

4. Consider an experiment in which it is known that the variance of the errors of the first two observations (i = 1,i = 2) is 3 times the variance of the errors for the next three observations (i = 3,i = 4,i = 5).

The errors are assumed to be uncorrelated. You are given the following model:

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                                                                                                      β0                             i = 1 or i = 4,

E(yi | β) = µi = β0+ β1        i = 2 or i = 5, β0+2β1 i = 3.

and you may assume that Var(yi | β) = σ2

Find the design matrix for this model and then calculate the maximum likelihood estimate of β1 to 2 decimal places when yT = c(115.0,72.5,22.5,110.0,60.0). You should do this calculation twice: once by calling the lm() function in R and once by solving the weighted normal equations (or score equations) for this model.            [5]

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