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Page 1 of 23 — add extra pages after page 23 — Page 1 of 23 Student Number Semester 1 Assessment, 2022 School of Mathematics and Statistics MAST90139 Statistical Modelling for Data Science This exam consists of 23 pages (including this page) Authorised materials: printed one-sided copy of the Exam or the Masked Exam made available earlier; any hard-copy or offline e-copy of notes, or offline electronic PDF reader; a Casio FX82 calculator; and blank A4 paper. Instructions to Students • During exam writing time you may only interact with the device running the Zoom session with supervisor permission. The screen of any other device must be visible in Zoom from the start of the session. • If you have a printer, print out the exam single-sided and hand write your solutions into the answer spaces. • If you do not have a printer, or if your printer fails on the day of the exam, (a) download the exam paper to a second device (not running Zoom), disconnect it from the internet as soon as the paper is downloaded and read the paper on the second device; (b) write your answers on the Masked Exam PDF if you were able to print it single-sided before the exam day. If you do not have the Masked Exam PDF, write single-sided on blank sheets of paper. • If you are unable to answer the whole question in the answer space provided then you can append additional handwritten solutions to the end of your exam submission. If you do this you MUST make a note in the correct answer space or page for the question, warning the marker that you have appended additional remarks at the end. • Assemble all the exam pages (or template pages) in correct page number order and the correct way up, and add any extra pages with additional working at the end. • Scan your exam submission to a single PDF file with a mobile phone or a scanner. Scan from directly above to avoid any excessive keystone effect. Check that all pages are clearly readable and cropped to the A4 borders of the original page. Poorly scanned submissions may be impossible to mark. • Upload the PDF file via the Canvas Assignments menu and submit the PDF to the GradeScope tool by first selecting your PDF file and then clicking on Upload PDF. • Confirm with your Zoom supervisor that you have GradeScope confirmation of submission before leaving Zoom supervision. • You should attempt all questions. • There are 8 questions with marks as shown. The total number of marks available is 110. Supplied by download for enrolled students only—©University of Melbourne 2022 Page 2 of 23 — add extra pages after page 23 — Page 2 of 23 MAST90139 Statistical Modelling for Data Science Semester 1, 2022 Question 1 (14 marks) In a clinical study of diabetes, each of 336 female patients was given a test to see whether the patient showed signs of diabetes, with result given in test (coded 0 if negative, and 1 if positive). Measurements of variables glucose, bmi, pedigree and age were also collected from these patients, where glucose= “plasma glucose concentration at 2 hours in an oral glucose tolerance test”, bmi= “body mass index [weight in kg/(height in metres squared)]”, pedigree= “diabetes pedigree function”, and age= “age(years)”. The main purpose of the study was to analyse the relationship between test and the other four variables. The data have been stored in the data frame Q1.dat with the following information: > head(Q1.dat) glucose bmi pedigree age test 1 89 28 0.17 21 0 2 78 31 0.25 26 1 3 197 30 0.16 53 1 4 189 30 0.40 59 1 5 166 26 0.59 51 1 6 103 43 0.18 33 0 > summary(Q1.dat) glucose bmi pedigree age test Min. : 56 Min. :18.2 Min. :0.085 Min. :21.0 Min. :0.00 1st Qu.: 99 1st Qu.:27.8 1st Qu.:0.268 1st Qu.:24.0 1st Qu.:0.00 Median :119 Median :32.8 Median :0.446 Median :28.0 Median :0.00 Mean :122 Mean :32.3 Mean :0.519 Mean :31.8 Mean :0.33 3rd Qu.:144 3rd Qu.:36.2 3rd Qu.:0.688 3rd Qu.:38.0 3rd Qu.:1.00 Max. :197 Max. :57.3 Max. :2.329 Max. :81.0 Max. :1.00 Answer the following questions in the provided boxes or on plain A4 paper: (a) The following is R output from fitting a logistic regression model. > Q1mod1 <- glm(test ~ glucose + bmi + pedigree + age, family = binomial,data = Q1.dat) > summary(Q1mod1) Coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) -10.81047 1.25381 -8.62 < 2e-16 glucose 0.03639 0.00549 6.62 3.5e-11 bmi 0.08916 0.02430 3.67 0.00024 pedigree 1.05588 0.46598 2.27 0.02346 age 0.05940 0.01451 4.09 4.3e-05 Null deviance: 426.34 on 335 degrees of freedom Residual deviance: 291.12 on 331 degrees of freedom AIC: 301.1 (i) What can be said about the adequacy of fit of the model? (ii) Use an estimate and its associated 95% confidence interval to describe the relationship between test and glucose in terms of odds ratio. (iii) For a patient having mean glucose, bmi and pedigree values, estimate the age at which or older she will have at least 0.5 probability to be tested positive. (Question 1 continued on next page) Page 2 of 23 pages Page 3 of 23 — add extra pages after page 23 — Page 3 of 23 MAST90139 Statistical Modelling for Data Science Semester 1, 2022 (Question 1 continued) Answer the questions in the provided boxes or on plain A4 paper: (b) Patient A has her BMI equal the third quartile of bmi while patient B has her BMI equal the first quartile of bmi. Values of glucose, pedigree and age do not change between A and B. Use the results from Q1mod1 to estimate the odds ratio of positive test for A versus B. Also find a 95% confidence interval for this odds ratio. Page 3 of 23 pages Page 4 of 23 — add extra pages after page 23 — Page 4 of 23 MAST90139 Statistical Modelling for Data Science Semester 1, 2022 (Question 1 continued) (c) Below is R output of variable selection by AIC using backward elimination procedure: > Q1mod2 <- glm(test ~ (glucose + bmi + pedigree + age)^2, family = binomial,data = Q1.dat) > step(Q1mod2, trace=1) Start: AIC=301; test ~ (glucose + bmi + pedigree + age)^2 Df Deviance AIC – glucose:age 1 279 299.1 – bmi:pedigree 1 279 299.2 – pedigree:age 1 279 299.4 – glucose:bmi 1 280 300.5 279 301.2 – bmi:age 1 282 302.2 – glucose:pedigree 1 285 305.0 Step: AIC=299.1 test ~ glucose + bmi + pedigree + age + glucose:bmi + glucose:pedigree + bmi:pedigree + bmi:age + pedigree:age Df Deviance AIC – bmi:pedigree 1 279 297 – pedigree:age 1 280 298 – glucose:bmi 1 280 298 279 299 – bmi:age 1 282 300 – glucose:pedigree 1 286 304 Step: AIC=297; test ~ glucose + bmi + pedigree + age + glucose:bmi + glucose:pedigree + bmi:age + pedigree:age Df Deviance AIC – pedigree:age 1 280 296.3 – glucose:bmi 1 280 296.4 279 297.0 – bmi:age 1 282 298.1 – glucose:pedigree 1 286 302.2 Step: AIC=296.3; test ~ glucose + bmi + pedigree + age + glucose:bmi + glucose:pedigree + bmi:age Df Deviance AIC – glucose:bmi 1 280 294 280 296 – bmi:age 1 283 297 – glucose:pedigree 1 287 301 Step: AIC=294; test ~ glucose + bmi + pedigree + age + glucose:pedigree + bmi:age Df Deviance AIC 280 294 – bmi:age 1 284 296 – glucose:pedigree 1 288 300 Write down the best logistic model selected by AIC by specifying which predictor terms remain in the best model together with its AIC value. Also explain how those predictor terms not in the best model are eliminated. Page 4 of 23 pages Page 5 of 23 — add extra pages after page 23 — Page 5 of 23 MAST90139 Statistical Modelling for Data Science Semester 1, 2022 Question 2 (15 marks) A study was conducted to investigate the toxicity to the tobacco budworm Heliothis virescens of doses of the pyrethroid trans-cypermethrin to which the moths were beginning to show resistance. Batches of 20 moths of each sex were exposed for three days to the pyrethroid and the number in each batch that were killed was recorded. The results are given below Dosage Number killed (out of 20) Dose (µg) (=log2(dose)) Males Females 1 0 1 0 2 1 4 2 4 2 8 6 8 3 13 10 16 4 17 12 32 5 20 16 Five (logistic regression) models (Q2mod1 to Q2mod5) were fitted and resulted in the residual deviances given in the table below, together with the R model specifications. For these models, sex is a factor with 2 levels (male; female), dose.f refers to dose treated as a factor with 6 levels, dose is the dose treated as a (continuous) variable and dosage is log2(dose) also continuous. Model formula specification residual deviance DF Q2mod1 sex + dose.f 4.75 5 Q2mod2 sex + dose + I(doseˆ2) 14.6 8 Q2mod3 sex*dosage 5.18 8 Q2mod4 sex + dosage 6.48 9 Q2mod5 sex + dosage + I(dosageˆ2) 5.78 8 The following R output is needed for answering this question. > qchisq(0.95, c(3,4,5,8,9)) [1] 7.81 9.49 11.07 15.51 16.92 Answer the following questions: (a) For each pair of models below explain whether or not they can be compared by a likelihood ratio test. If yes, explain what hypothesis is being tested and provide a conclusion. (i) model Q2mod1 and model Q2mod3; (ii) model Q2mod1 and model Q2mod5 Page 5 of 23 pages Page 6 of 23 — add extra pages after page 23 — Page 6 of 23 MAST90139 Statistical Modelling for Data Science Semester 1, 2022 (Question 2 continued) (b) Some R output of model Q2mod4 is given below > summary(Q2mod4)$coef Estimate Std. Error z value Pr(>|z|) (Intercept) -3.405 0.461 -7.39 1.5e-13 sexmale 0.970 0.350 2.77 0.0055 dosage 1.042 0.129 8.10 5.7e-16 Use the above output to answer the following questions. (i) Is the effect of sex significant in the model? Justify your answer. (ii) Quantify, in terms of odds ratios, the effects of sex and dosage on toxicity. (iii) Find an estimate of the dosage, that would kill 50% of male budworm moths (the LD(50)). State, with reasons, whether the LD(50) for female moths would be greater or less than that for males. (iv) Obtain an estimate of the probability that a female moth will die within three days if subjected to a dose of 16µg of pyrethroid (or a dosage of 4) for three days. Page 6 of 23 pages Page 7 of 23 — add extra pages after page 23 — Page 7 of 23 MAST90139 Statistical Modelling for Data Science Semester 1, 2022 (Question 2 continued) (c) Which of the 5 models Q2mod1 to Q2mod5 listed in the table above is most appropriate for these data? Give details of any tests that you carry out and clearly state the conclusions drawn from each test. (d) A total of 240 budworms were used in this study. Describe what would have been the same and what would have been different in the output (residual deviances and their degrees of freedom, differences between residual deviances and their degrees of freedom, parameter estimates and their standard errors) had the data been treated as 240 ungrouped, 0–1 observations, rather than as 12 groups of 20 moths. Page 7 of 23 pages Page 8 of 23 — add extra pages after page 23 — Page 8 of 23 MAST90139 Statistical Modelling for Data Science Semester 1, 2022 Question 3 (12 marks) The following table presents data for 174 poliomyelitis cases from the US. The individuals were categorized based on age (A; 6 levels: 2=‘0-4’ years old, 7=‘5-9’, 12=‘10-14’, 17=‘15-19’, 30=‘20- 39’, and 50=‘40+’ ), degree of debilitation (P; 2 levels: OK, Paralyzed), and whether or not they had been vaccinated (V; 2 levels: Vaccinated, Not Vaccinated). Table 1: Number of patients disabled, by age and vaccination. Ages Vaccinated Not Vaccinated OK Paralyzed OK Paralyzed 0-4 (2) 20 14 10 24 5-9 (7) 15 12 3 15 10-14 (12) 3 2 3 2 15-19 (17) 7 4 1 6 20-39 (30) 12 3 7 5 40+ (50) 1 0 3 2 From these data the following residual deviances and residual degrees of freedom were obtained, using log-linear models with Poisson error. Table 2: Log-linear Model summaries for 174 poliomyelitis cases. Model R Formula Res. Dev. Res. D.F. 1 A + V + P 34.00 16 2 A + V*P 19.17 15 3 V + A*P 25.42 11 4 P + A*V 28.88 11 5 A*V + V*P 14.05 10 6 A*P + V*P 10.59 10 7 A*V + A*P 20.29 6 8 A*V + V*P + A*P 2.71 5 The following 95 percentiles of χ 2 distributions may be required in answering questions: > qchisq(0.95, df = c(5, 6, 10, 11, 15, 16)) [1] 11.071 12.592 18.307 19.675 24.996 26.296 (Question 3 continued on next page) Page 8 of 23 pages Page 9 of 23 — add extra pages after page 23 — Page 9 of 23 MAST90139 Statistical Modelling for Data Science Semester 1, 2022 (Question 3 continued) Answer the following questions: (a) Give an interpretation to each of the following models. (i) A + V + P (ii) A*V + V*P + A*P (b) (i) Test the hypothesis that there is no association between vaccination and paralysis when age is given. State the model under test, the test statistic value, and a conclusion at the 5% significance level. (ii) Test the hypothesis that there is no association between age and paralysis when vaccination status is given. State the model under test, the test statistic value, and a conclusion at the 5% significance level. Page 9 of 23 pages Page 10 of 23 — add extra pages after page 23 — Page 10 of 23 MAST90139 Statistical Modelling for Data Science Semester 1, 2022 (Question 3 continued) (c) Find the “best” log-linear model based on the results in Table 2. Justify your finding using necessary analysis of deviance tests and model adequacy tests. Page 10 of 23 pages Page 11 of 23 — add extra pages after page 23 — Page 11 of 23 MAST90139 Statistical Modelling for Data Science Semester 1, 2022 Question 4 (12 marks) Refer to the same data in Table 1 in Question 3. Suppose we want to collapse the table over age to investigate the relation between vaccination and paralysis. The parameter estimates in Model 2 in Table 2 are given in the R output below: > summary(logMod2) Call: glm(formula = freq ~ factor(A) + V * P, family = poisson, data = polio) Deviance Residuals: Min 1Q Median 3Q Max -1.70192 -0.69864 0.04334 0.45244 1.69779 Coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) 2.3563 0.2145 10.987 < 2e-16 *** factor(A)7 -0.4128 0.1922 -2.148 0.031685 * factor(A)12 -1.9169 0.3387 -5.660 1.51e-08 *** factor(A)17 -1.3291 0.2651 -5.014 5.32e-07 *** factor(A)30 -0.9237 0.2275 -4.061 4.89e-05 *** factor(A)50 -2.4277 0.4259 -5.701 1.19e-08 *** V 0.7646 0.2330 3.282 0.001031 ** P 0.6931 0.2357 2.941 0.003274 ** V:P -1.1982 0.3184 -3.764 0.000168 *** — Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 (Dispersion parameter for poisson family taken to be 1) Null deviance: 129.139 on 23 degrees of freedom Residual deviance: 19.173 on 15 degrees of freedom AIC: 118.29 Number of Fisher Scoring iterations: 5 (Question 4 continued on next page) Page 11 of 23 pages Page 12 of 23 — add extra pages after page 23 — Page 12 of 23 MAST90139 Statistical Modelling for Data Science Semester 1, 2022 (Question 4 continued) Answer the following questions: (a) Is it reasonable to collapse the table over age? Explain based on the results in Table 2. (b) Give the value of the residual deviance that will be obtained if the ‘no association’ model is fitted to the collapsed table. Is the ‘no association’ model adequate? (c) A saturated log-linear model has been fitted to the collapsed table. Give the values of the coefficients of V, P and V:P terms in the fitted saturated model. Also give the associated standard errors of these coefficient estimates. Page 12 of 23 pages Page 13 of 23 — add extra pages after page 23 — Page 13 of 23 MAST90139 Statistical Modelling for Data Science Semester 1, 2022 Question 5 (12 marks) In this question we continue to use the data and model results provided in Questions 3 and 4. We now treat P as the response variable, and then fit the frequencies of Paralyzed and OK to V by the logistic regression model M : logit(p) = β0 + β1V, where p is the probability of Paralyzed for an individual. Use the equivalence between logistic and log-linear models for certain contingency table data to answer the following questions: (a) Give the null deviance value for model M and explain. (b) Find the residual deviance value of model M. Then test its model adequacy. (c) Give the maximum likelihood estimates of β0 and β1 together with their standard errors. Page 13 of 23 pages Page 14 of 23 — add extra pages after page 23 — Page 14 of 23 MAST90139 Statistical Modelling for Data Science Semester 1, 2022 Question 6 (15 marks) A survey of terrace house residents in Copenhagen classified the householders according to the degree of Contact they had with other residents, their feeling of Influence on apartment management and their level of Satisfaction with their housing conditions. The data are summarized in the following table and R data.frame satis. Contact Low High Influence Low Med. High Low Med. High Low 18 15 7 57 31 5 Satisfaction Medium 6 13 5 23 21 6 High 7 13 11 13 13 13 > satis Freq Satis Contact Influ 1 18 1Low Low Low 2 6 2Medium Low Low 3 7 3High Low Low 4 15 1Low Low Mid 5 13 2Medium Low Mid 6 13 3High Low Mid 7 7 1Low Low High 8 5 2Medium Low High 9 11 3High Low High 10 57 1Low High Low 11 23 2Medium High Low 12 13 3High High Low 13 31 1Low High Mid 14 21 2Medium High Mid 15 13 3High High Mid 16 5 1Low High High 17 6 2Medium High High 18 13 3High High High Treating the level of Satisfaction as a nominal categorical response variable, a multicategorical logit model has been fitted resulting in the following R output: > satis.mod <- multinom(Satis~Contact+Influ, data=satis, weights=Freq, Hess=T) > summary(satis.mod) Call: multinom(formula = Satis ~ Contact + Influ, data = satis, weights = Freq, Hess = T) Coefficients: (Intercept) ContactHigh InfluMid InfluHigh 2Medium -0.96 0.0129 0.65 0.866 3High -1.05 -0.3770 0.70 1.934 Std. Errors: (Intercept) ContactHigh InfluMid InfluHigh 2Medium 0.328 0.320 0.317 0.476 3High 0.341 0.324 0.355 0.439 Residual Deviance: 555 AIC: 571 (Question 6 continued on next page) Page 14 of 23 pages Page 15 of 23 — add extra pages after page 23 — Page 15 of 23 MAST90139 Statistical Modelling for Data Science Semester 1, 2022 (Question 6 continued) Answer the following questions: (a) Write down the model fitted in the above R output. You need to define the response variable and predictors for the model. Also, you need to specify the probability distribution of the response variable and estimates of all parameters in the model. (b) Provide an interpretation for the coefficient estimate 1.934. Then calculate an approximate 95% confidence interval for the odds ratio of high level of Satisfaction versus lower level of Satisfaction for householders having high Influence feeling against those having low Influence feeling on management. Page 15 of 23 pages Page 16 of 23 — add extra pages after page 23 — Page 16 of 23 MAST90139 Statistical Modelling for Data Science Semester 1, 2022 (Question 6 continued) (c) Estimate the probability for each of the 3 levels of Satisfaction for a householder having low degree of Contact and high Influence feeling. Page 16 of 23 pages Page 17 of 23 — add extra pages after page 23 — Page 17 of 23 MAST90139 Statistical Modelling for Data Science Semester 1, 2022 Question 7 (15 marks) Refer to the satis data in Q6. Note that the level of Satisfaction is ordinal by nature, thus we treat it as an ordinal categorical response variable. A cumulative proportional odds model is fitted to the satis data, producing the following R output (Note Coefficients Values need to change sign for being used in the model): > satis$SatisO=as.ordered(satis$Satis) > Osatis.mod=polr(SatisO~Contact + Influ, data=satis,weights=Freq, Hess=T, method=”logistic”) > summary(Osatis.mod) Call: polr(formula = SatisO ~ Contact + Influ, data = satis, weights = Freq, Hess = T, method = “logistic”) Coefficients: Value Std. Error t value ContactHigh -0.242 0.246 -0.984 InfluMid 0.603 0.256 2.357 InfluHigh 1.591 0.342 4.646 Intercepts: Value Std. Error t value 1Low|2Medium 0.234 0.259 0.905 2Medium|3High 1.496 0.275 5.434 Residual Deviance: 556.90 AIC: 566.90 > solve(Osatis.mod$Hessian) ContactHigh InfluMid InfluHigh 1Low|2Medium 2Medium|3High ContactHigh 0.06072 0.0081 0.0126 0.0457 -0.00093 InfluMid 0.00808 0.0654 0.0350 0.0374 0.00218 InfluHigh 0.01260 0.0350 0.1172 0.0396 0.00656 1Low|2Medium 0.04574 0.0374 0.0396 0.0671 -0.00365 2Medium|3High -0.00093 0.0022 0.0066 -0.0036 0.01123 (Question 7 continued on next page) Page 17 of 23 pages Page 18 of 23 — add extra pages after page 23 — Page 18 of 23 MAST90139 Statistical Modelling for Data Science Semester 1, 2022 (Question 7 continued) Answer the following questions: (a) Write down the model fitted in the above R output. You need to define the response variable and predictors for the model. Also, you need to specify the probability distribution of the response variable and estimates of all parameters in the model. (b) Estimate the odds ratio of high level of Satisfaction versus the other levels of Satisfaction for a householder who has low degree of Contact but high Influence feeling against another householder who has high degree of Contact but low Influence feeling. Also calculate an approximate 95% confidence interval for this odds ratio. Page 18 of 23 pages Page 19 of 23 — add extra pages after page 23 — Page 19 of 23 MAST90139 Statistical Modelling for Data Science Semester 1, 2022 (Question 7 continued) (c) Estimate the probability for each of the 3 levels of Satisfaction for a householder having low degree of Contact and high Influence feeling. Page 19 of 23 pages Page 20 of 23 — add extra pages after page 23 — Page 20 of 23 MAST90139 Statistical Modelling for Data Science Semester 1, 2022 Question 8 (15 marks) The toenail data comes from a multi-center study comparing two oral treatments for toenail infection. Patients were evaluated for the degree of separation of the nail. A total of 294 patients were randomised into two treatments and were followed over seven visits: four in the first year and yearly thereafter. Some of the patients did not attend all seven visits, thus only a total of 1908 visits were observed. The patients had not been treated prior to the first visit so this should be regarded as the baseline. The variables available in the data are outcome: 0 = none or mild separation, 1 = moderate or severe separation ID: ID of patient treatment: the treatment; A = 0 or B = 1 month: time of the visit, in months, from the first visit visit: the number of the visit The purpose of this study is to see how toenail infection responds to the treatments and progresses over time. Some analysis has been done to the data in R, producing the following output. > toenail[1:14,] ID outcome treatment month visit 1 1 1 1 0.000 1 2 1 1 1 0.857 2 3 1 1 1 3.536 3 4 1 0 1 4.536 4 5 1 0 1 7.536 5 6 1 0 1 10.036 6 7 1 0 1 13.071 7 8 2 0 0 0.000 1 9 2 0 0 0.964 2 10 2 1 0 2.000 3 11 2 1 0 3.036 4 12 2 0 0 6.500 5 13 2 0 0 9.000 6 14 3 0 0 0.000 1 > tail(toenail) ID outcome treatment month visit 1903 383 1 1 0.00 1 1904 383 1 1 1.04 2 1905 383 1 1 2.04 3 1906 383 1 1 3.29 4 1907 383 0 1 7.29 5 1908 383 0 1 10.79 6 > str(toenail) ‘data.frame’: 1908 obs. of 5 variables: $ ID : int 1 1 1 1 1 1 1 2 2 2 … $ outcome : int 1 1 1 0 0 0 0 0 0 1 … $ treatment: int 1 1 1 1 1 1 1 0 0 0 … $ month : num 0 0.857 3.536 4.536 7.536 … $ visit : int 1 2 3 4 5 6 7 1 2 3 … Page 20 of 23 pages Page 21 of 23 — add extra pages after page 23 — Page 21 of 23 MAST90139 Statistical Modelling for Data Science Semester 1, 2022 > library(geepack} fit.exch <- geeglm(outcome~treatment+month, family=binomial(link=”logit”), data=toenail, id=ID, corstr = “exchangeable”, std.err=”san.se”); summary(fit.exch) Call: geeglm(formula = outcome ~ treatment + month, family = binomial(link = “logit”), data = toenail, id = ID, corstr = “exchangeable”, std.err = “san.se”) Coefficients: Estimate Std.err Wald Pr(>|W|) (Intercept) -0.6104 0.1777 11.80 0.00059 *** treatment 0.0402 0.2532 0.03 0.87388 month -0.2051 0.0259 62.66 2.4e-15 *** — Estimated Scale Parameters: Estimate Std.err (Intercept) 1.09 0.423 Correlation: Structure = exchangeable Link = identity Estimated Correlation Parameters: Estimate Std.err alpha 0.424 0.182 Number of clusters: 294 Maximum cluster size: 7 > summary(fit.exch)$cov.unscaled [,1] [,2] [,3] [1,] 0.03159 -0.031374 -0.001395 [2,] -0.03137 0.064120 -0.000546 [3,] -0.00139 -0.000546 0.000671 Use the above output to answer the following questions. (Question 8 continued on next page) Page 21 of 23 pages Page 22 of 23 — add extra pages after page 23 — Page 22 of 23 MAST90139 Statistical Modelling for Data Science Semester 1, 2022 (Question 8 continued) Answer the following questions: (a) Let yit be the response value outcome of patient i during visit t. Write down the model involved in the analysis, including the mean, variance and correlation coefficient of yit’s. Give the estimates of the parameters appearing in the model. (b) Write down the model’s design matrix for data where ID=383. Page 22 of 23 pages Page 23 of 23 — add extra pages after page 23 — Page 23 of 23 MAST90139 Statistical Modelling for Data Science Semester 1, 2022 (Question 8 continued) (c) Estimate the odds ratio of toe infection of a patient with treatment B versus with treatment A at a given value of month. Calculate an approximate 95% confidence interval for this odds ratio. (d) Estimate the probability of toe infection in the first month from the first visit for a patient using treatment A. Also compute an approximate 95% confidence interval for this probability. End of Exam—Total Available Marks = 110 Page 23 of 23 pages
