MTH6134 / MTH6134P: Statistical Modelling II Main Examination period 2022

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Main Examination period 2022 – January – Semester A

MTH6134/MTH6134P: Statistical Modelling II

Question 1  [24 marks]. Suppose that Yi N(μi ; σi2 ) for i = 1; 2; : : : ; n, all independent, where μi = β1xi+β2xi(2) , xi is a known covariate and the σi are known.

(a) Write down the likelihood for the data y1,...,yn.                        [6]

(b)  Find the maximum likelihood estimatorsβˆ1  andβˆ2  of β1 and β2 . [12]

(c)  Explain why the above is a generalised linear model. [4]

(d)  State the iterative weights and working dependent variates for Fisher’s method of scoring. [2]

Question 2  [19 marks]. The numbers of new melanoma cases (y) in 1969-1971 among white males in two areas (w) for six ages (x), in years, were recorded, where the ages are midpoints of intervals.

Below are the data.

Let Yjk denote the number of new melanoma cases for age xk in area j. Then it is assumed that

Yjk Poisson(μjk) for j = 1; 2 and k = 1; 2; : : : ; 6, all independent, where log(μjk) = aj+βjxk. This model was fitted to the data using R and the following output was obtained:

Call:

glm(formula  =  y  ~  w  +  w:x,  family  =  poisson)

Deviance Residuals:

Min 1Q Median 3Q Max

-2.29127 -1.75130 -0.07461 1.19941 2.42769

Coefficients:

Estimate Std. Error z value Pr(>|z|)

(Intercept) 4.264158 0.155300 27.458 < 2e-16 ***

w2 0.531125 0.232380 2.286 0.0223 *

w1:x 0.002206 0.002668 0.827 0.4084

w2:x -0.014209 0.003225 -4.405 1.06e-05 ***

---

Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

(Dispersion parameter for poisson family taken to be 1)

Null deviance: 74.240 on 11 degrees of freedom

Residual deviance: 29.885 on 8 degrees of freedom

AIC: 110.11

Number of Fisher Scoring iterations: 4

(a) Plot the numbers of new melanoma cases against age by area. What are your conclusions? [5]

(b) Write down the fitted Poisson regression model for each area. [5]

(c) Use the above output to assess the goodness of fit of the model. [4]

(d) Test whether the regression lines are parallel. [5]

Question 3  [21 marks]. Suppose that Yi Bin(ri ; πi) for i = 1; 2; : : : ; n, all independent, where the

ri are known, Φ- 1 (πi) = β0+β1xi , xi is a known covariate and Φ denotes the standard normal distribution function.

(a) Find the Fisher information matrix.                    [8]

(b)  Obtain the asymptotic distribution of the maximum likelihood estimatorβ(ˆ)0  of β0 . [8]

(c) Write down an approximate 100(1 - α )% confidence interval for β0 . [3]

(d)  Given that the vectors y and x in R contain the responses and the covariate values, what commands would you use to obtain the details of the fitted model? [2]

Question 4  [23 marks]. An experiment was conducted in which 141 fish were placed in a large  tank for a period of time and some are eaten by large birds of prey. The fish are categorised by their level of parasitic infection. A summary of the data is provided in the contingency table below.

Let Yjk denote the number of fish classified in row j and column k. Then it is assumed that the Yjk have a multinomial distribution with parameters n and θjk for j = 1; 2 and k = 1; 2; 3,where n = 141 and θjk is the probability that a fishis classified in row j and column k. The null hypothesis is that being eaten  and infection status are independent.

(a) State the null hypothesis in terms of E(Yjk). Express this as a log-linear model, explaining your notation and any additional constraints.                                  [6]

(b) Write down the maximal model. [4]

(c)  Obtain the expected values under the null hypothesis. Compare these with the observed values. [5]

(d)  Find the deviance and the value of Pearson’s goodness-of-fit test statistic. What is your conclusion about the independence of being eaten and infection status? [8]

Question 5  [13 marks]. Suppose that T1 ; : : : ; Tn are independent Weibull random variables with probability density function

f(t ) = 3λ t2 e-λ t3 ;

where λ > 0.

(a) Show that this distribution is a member of the exponential family.                     [4]

(b) Explain why the distribution is not in canonical form. [1]

(c)  Write down the likelihood for the data (ti ; δi) for i = 1; 2; : : : ; n, where δi is a censoring variable. [4]

(d) Find the maximum likelihood estimatorλ(ˆ) of λ . [4]