MTH6134 / MTH6134P: Statistical Modelling II

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Main Examination period 2022 – January – Semester A

MTH6134 / MTH6134P: Statistical Modelling II

Question 1 [0 marks].    Suppose that Yi ∼ N(ui , σi2 ) for i = 1, 2, . . . , n, all independent, where ui = β 1xi+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 estimators 1 and 2 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 [0 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.

x

30

40

50

60

70

80

30

40

50

60

70

80

w

1

1

1

1

1

1

2

2

2

2

2

2

y

61

76

98

104

63

80

64

75

68

63

45

27

Let Yjk denote the number of new melanoma cases for age xk in area j. Then it is assumed that Yjk ∼ Poisson(ujk) for j = 1, 2 and k = 1, 2, . . . , 6, all independent, where log(ujk) = αj+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

-2 .29127

1Q -1 .75130

Median

-0 .07461

3Q

1 .19941

Max

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 Residual deviance:  29 .885 AIC:  110 .11

on  11

on   8

degrees  of freedom

degrees  of freedom

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 [0 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.

(b) Obtain the asymptotic distribution of the maximum likelihood estimator 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 [0 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.

Uninfected

Level of Infection

Lightly Infected Highly Infected

Total

Eaten

1

10

37

48

Not Eaten

49

35

9

93

Total

50

45

46

141

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 fish is 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]

(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 [0 marks].    Suppose that T1 , . . . , Tn are independent Weibull random variables with probability density function

f (t) = 3λt2e −λt3 ,

where λ > 0.

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

(c) Write down the likelihood for the data (ti , 6i) for i = 1, 2,  , n, where 6i is a censoring variable.    [4]

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


2023-01-06