MAST90104 Lab 9 Solution

Week 9 Practical and Workshop
1 Practical questions
1. We revisit the milk data last week. We study the effect of various breeds and diets on the milk yield of cows. A study is conducted on 9 cows and the following data obtained:
Diet
Breed 1 2 3
1 18.8 16.7 19.8
21.2 23.9
2 22.3 15.9
19.2 21.8
(a) Input this data into R.
(b) Test for the presence of interaction.
(c) What is the degrees of freedom used for the interaction test?
(d) From the interaction model, what is the estimated amount of milk produced from breed 2 and diet 3?
(e) Find a 95% confidence interval under the interaction model, for the amount of milk produced from breed 2 and diet 3.
(a) This question use a data set in package faraway. Load the package and read the help file (?pima) to get a description of the predictor and response variables, then use pairs and summary to perform simple graphical and numerical summaries of the data.
There are some obvious irregularities in the data. Take appropriate steps to correct the problems.
(b) Fit a model with test as the response and all the other variables as predictors.
Can you tell whether this model fits the data?
Odds are sometimes a better scale than probability to represent chance. The odds o and probability p are related by

In a binomial regression model with a logit link we have
logit( .
That is logoj = ηj, where oj are the odds for the j-th observation.
(c) By what proportion do the odds of testing positive for diabetes change for a woman with a BMI at the first quartile compared with a woman at the third quartile, assuming that all other factors are held constant? Give a confidence interval for this difference.
(d) Do women who test positive have higher diastolic blood pressures? Is the diastolic blood pressure significant in the regression model? Explain the distinction between the two questions and discuss why the answers are only apparently contradictory.
(e) Predict the outcome for a woman with predictor values 1, 99, 64, 22, 76, 27, 0.25, 25 (same order as in the dataset). Give a confidence interval for your prediction.
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2 Workshop questions
1. Verify that for the binomial regression model with logistic link

2. Show that the Gamma density, f, in the form

is an exponential family with . Identify the functions a,b,c and find the mean and variance functions as functions of θ.
3. Show that the inverse Gaussian density, f, in the form

is an exponential family with . Identify the functions a,b,c and find the mean and variance functions as functions of µ,λ.
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