Semester 1 Assessment, 2021 School of Mathematics and Statistics MAST30025 Linear Statistical Models 

Reading time: 30 minutes 

 Writing time: 3 hours

 Upload time: 30 minutes This exam consists of 23 pages (including this page) Permitted Materials This exam and/or an offline electronic PDF reader, one or more copies of the masked exam template made available earlier, blank loose-leaf paper and a Casio FX-82 calculator. One double sided A4 page of notes (handwritten only). Instructions to Students If you have a printer, print the exam one-sided. If you cannot print, download the exam to a second device and disconnect that device from the internet. Ask the supervisor if you want to use the device running Zoom. Check your scanned PDF before submitting. You must submit while in the Zoom room and no submissions will be accepted after you have left the Zoom room. Writing There are 7 questions with marks as shown. The total number of marks available is 90. Write your answers in the boxes provided on the exam that you have printed or the masked exam template that has been previously made available. If you need more space, you can use blank paper. Note this in the answer box, so the marker knows. The extra pages can be added to the end of the exam to scan. If you have been unable to print the exam and do not have the masked template write your answers on A4 paper. The first page should contain only your student number, the subject code and the subject name. Write on one side of each sheet only. Start each question on a new page and include the question number at the top of each page. Scanning Put the pages in number order and the correct way up. Add any extra pages to the end. Use a scanning app to scan all pages to PDF. Scan directly from above. Crop pages to A4. Check PDF is readable. Submitting You must submit while in the Zoom room. No submissions will be accepted after you have left the Zoom room. Go to the Gradescope window. Choose the Canvas assignment for this exam. Submit your file as a single PDF document only. Get Gradescope confirmation on email. Tell your supervisor it is submitted. University of Melbourne 2021 Page 1 of 23 pages Can be placed in Baillieu Library Student number Page 1 of 23 add any extra pages after page 23 Page 1 of 23 MAST30025 Linear Statistical Models Semester 1, 2021 Question 1 (10 marks) 54 54 LetA= 45 andB= 108. (a) [3 marks] Show that A is positive definite. (b) [3 marks] Find all possible values of c such that c(A I) is idempotent. Page 2 of 23 pages Page 2 of 23 add any extra pages after page 23 Page 2 of 23 MAST30025 Linear Statistical Models Semester 1, 2021 (c) [2 marks] Find a conditional inverse for B. (d) [2marks]Lety=(y1,y2)T. Find yTBy intermsofy1 andy2. y Page 3 of 23 pages Page 3 of 23 add any extra pages after page 23 Page 3 of 23 MAST30025 Linear Statistical Models Semester 1, 2021 Question 2 (13 marks) Let y1 4 2 1 0 y=y2 MVN2, 1 2 0, y3 0002 andz=(y1,y2)T MVN(,V). (a) [2 marks] Find and V . y1+y2 (b) [3 marks] Describe the distribution of y 2y . 12 Page 4 of 23 pages Page 4 of 23 add any extra pages after page 23 Page 4 of 23 MAST30025 Linear Statistical Models Semester 1, 2021 (c) [4 marks] Describe the distribution of 23 (y12 + y2 + y1y2). (d) [4 marks] Describe the distribution of 23 (y12 + y2 + y1y2) + 12 y32. Page 5 of 23 pages Page 5 of 23 add any extra pages after page 23 Page 5 of 23 MAST30025 Linear Statistical Models Semester 1, 2021 Question 3 (17 marks) In this question, we study the number of sales of hot dogs in a stadium over 20 days. This dataset contains the variables: sale (y): the number of hot dogs sold on the day temp (x1): the maximum temperature on the day nppl (x2): the number of people in the stadium on the day (in thousands). The data are stored in the saledata data frame and the following R calculations performed: > X = cbind(rep(1,n), saledata$temp, saledata$nppl) > y = saledata$sales > > t(X)% %X [,1] [,2] [,3] [1,] 20.0 -4.8 33.8 [2,] -4.8 1104.8 6.6 [3,] 33.8 6.6 70.0 > solve(t(X)% %X) [,1] [,2] [,3] [1,] 0.278 0.00202 -0.1345 [2,] 0.002 0.00092 -0.0011 [3,] -0.134 -0.00106 0.0794 > t(X)% %y [,1] [1,] 319 [2,] 2381 [3,] 661 > t(y)% %y [,1] [1,] 11271 > qt(0.975,15:20) [1] 2.131 2.120 2.110 2.101 2.093 2.086 > qf(0.95,1,15:20) [1] 4.543 4.494 4.451 4.414 4.381 4.351 > qf(0.95,2,15:20) [1] 3.682 3.634 3.592 3.555 3.522 3.493 > qf(0.95,3,15:20) [1] 3.287 3.239 3.197 3.160 3.127 3.098 Page 6 of 23 pages Page 6 of 23 add any extra pages after page 23 Page 6 of 23 MAST30025 Linear Statistical Models Semester 1, 2021 (a) [2 marks] Calculate the least squares estimates of the parameters. (b) [2 marks] Calculate the sample variance s2. Page 7 of 23 pages Page 7 of 23 add any extra pages after page 23 Page 7 of 23 MAST30025 Linear Statistical Models Semester 1, 2021 (c) [2 marks] Calculate a 95% confidence interval for the parameter corresponding to temp. (d) [3 marks] Test for model relevance at the 5% significance level. Page 8 of 23 pages Page 8 of 23 add any extra pages after page 23 Page 8 of 23 MAST30025 Linear Statistical Models Semester 1, 2021 (e) [4 marks] Consider the linear model without an intercept, y = 1x1 + 2x2 + . Calculate the least squares estimates of the parameters. (f) [4 marks] Test the hypothesis H0 : 0 = 0 in the full model (including the intercept), using an F -test at the 5% significance level. Page 9 of 23 pages Page 9 of 23 add any extra pages after page 23 Page 9 of 23 MAST30025 Linear Statistical Models Semester 1, 2021 Question 4 (8 marks) Consider the general full rank linear model y = X+. For given inputs x, we wish to predict the difference of two responses with these inputs, y 1 = ( x ) T + 1 , y 2 = ( x ) T + 2 , where 1 N(0,2), 2 N(0,2) and is independent of 1. Let b be the least squares estimator of and s2 be the sample variance. (a) [1 mark] Write down the predictor of y1 y2. (b) [2 marks] Find the distribution of the prediction error. Page 10 of 23 pages Page 10 of 23 add any extra pages after page 23 Page 10 of 23 MAST30025 Linear Statistical Models Semester 1, 2021 (c) [3 marks] Formulate a t-distributed random variable based on the prediction error, and specify its degrees of freedom. (d) [2 marks] Based on (c), construct a 100(1 )% prediction interval for y1 y2. Page 11 of 23 pages Page 11 of 23 add any extra pages after page 23 Page 11 of 23 MAST30025 Linear Statistical Models Semester 1, 2021 Question 5 (16 marks) A small university collected data on the salary of its faculty members in the early 1980s. The dataset contains the following variables: degree: Highest qualification (Masters or PhD) rank: Faculty position (Asst, Assoc, or Prof) sex: Gender (Male or Female) year: Years in current rank ysdeg: Years since highest degree salary: Current salary ($k/yr) We wish to model the salary of the faculty members in terms of the other variables. The following R calculations are produced: > model <- lm(salary ~ ., data=salary) > model2 <- lm(salary ~ . sex, data=salary) > anova(model, model2) Analysis of Variance Table Model 1: salary ~ degree + rank + sex + year + ysdeg Model 2: salary ~ (degree + rank + sex + year + ysdeg) sex Res.Df RSS Df Sum of Sq F Pr(>F) 1 45 258.86 2 40 241.45 5 17.406 0.5767 0.7174 > summary(model) Call: lm(formula = salary ~ ., data = salary) Residuals: Min 1Q Median 3Q Max -4.0452 -1.0947 -0.3615 0.8132 9.1931 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 15.74605 degreePhD 1.38861 rankAssoc 5.29236 rankProf 11.11876 sexFemale 1.16637 year 0.47631 ysdeg -0.12457 --- 0.80018 19.678 < 2e-16 1.01875 1.363 0.180 1.14540 4.621 3.22e-05 1.35177 8.225 1.62e-10 0.92557 1.260 0.214 0.09491 5.018 8.65e-06 0.07749 -1.608 0.115 Signif. codes: 0 0.001 0.01 0.05 . 0.1 1 Residual standard error: 2.398 on 45 degrees of freedom Multiple R-squared: 0.855, Adjusted R-squared: 0.8357 F-statistic: 44.24 on 6 and 45 DF, p-value: < 2.2e-16 Page 12 of 23 pages Page 12 of 23 add any extra pages after page 23 Page 12 of 23 MAST30025 Linear Statistical Models Semester 1, 2021 > par(mfrow=c(2,2)) > plot(model) Residuals vs Fitted 20 25 30 35 Normal QQ 2 1 0 1 2 Theoretical Quantiles Residuals vs Leverage 24 2 20 Fitted values ScaleLocation 20 25 30 Fitted values 1 0.5 24 29 Cook's distance 20 24 2 20 20 24 2 35 0.00 0.10 0.20 0.30 Leverage Page 13 of 23 pages Page 13 of 23 add any extra pages after page 23 Page 13 of 23 Standardized residuals 0.0 0.5 1.0 1.5 2.0 Residuals Standardized residuals Standardized residuals 2 01234 2 01234 5 0 5 10 MAST30025 Linear Statistical Models Semester 1, 2021 > model3 <- step(model) Start: AIC=97.46 salary ~ degree + rank + sex + year + ysdeg Df Sum of Sq RSS AIC - sex 1 9.13 267.99 97.265 <none> 258.86 97.462 - degree 1 - ysdeg 1 - year 1 - rank 2 10.69 269.55 97.566 14.87 273.73 98.366 144.87 403.73 118.574 399.79 658.65 142.025 Step: AIC=97.27 salary ~ degree + rank + year + ysdeg Df Sum of Sq RSS AIC - degree 1 - ysdeg 1 <none> - year 1 - rank 2 6.68 274.68 96.547 7.87 275.87 96.771 267.99 97.265 147.64 415.64 118.085 404.11 672.10 141.077 Step: AIC=96.55 salary ~ rank + year + ysdeg DfSumofSq RSS AIC - ysdeg 1 2.31 276.99 94.983 <none> - year - rank 274.68 96.547 1 141.11 415.78 116.104 2 478.54 753.22 145.002 Step: AIC=94.98 salary ~ rank + year 161.95 438.95 116.923 632.06 909.05 152.780 <none> - year 1 - rank 2 Df Sum of Sq RSS AIC 276.99 94.983 Page 14 of 23 pages Page 14 of 23 add any extra pages after page 23 Page 14 of 23 MAST30025 Linear Statistical Models Semester 1, 2021 > summary(model3) Call: lm(formula = salary ~ rank + year, data = salary) Residuals: Min 1Q Median 3Q Max -3.4620 -1.3028 -0.2992 0.7835 9.3816 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 16.20327 rankAssoc 4.26228 rankProf 9.45452 year 0.37570 --- 0.63868 25.370 < 2e-16 0.88289 4.828 1.45e-05 0.90583 10.437 6.12e-14 0.07092 5.298 2.90e-06 Signif. codes: 0 0.001 0.01 0.05 . 0.1 1 Residual standard error: 2.402 on 48 degrees of freedom Multiple R-squared: 0.8449, Adjusted R-squared: 0.8352 F-statistic: 87.15 on 3 and 48 DF, p-value: < 2.2e-16 > linearHypothesis(model3, c(0,-1,1,-10), 0) Linear hypothesis test Hypothesis: - rankAssoc + rankProf - 10 year = 0 Model 1: restricted model Model 2: salary ~ rank + year 1 2 > > Res.Df RSS Df Sum of Sq F Pr(>F) 49 284.38 48 276.99 1 7.3901 1.2806 0.2634 person <- data.frame(degree='PhD', rank='Asst', sex='Female', year=4, ysdeg=10) predict(model3, person, interval='confidence', level=0.95) fit lwr upr 1 17.70605 16.56736 18.84474 > qt(0.975,45:50) [1] 2.014103 2.012896 2.011741 2.010635 2.009575 2.008559 > qf(0.95,2,45:50) [1] 3.204317 3.199582 3.195056 3.190727 3.186582 3.182610 Page 15 of 23 pages Page 15 of 23 add any extra pages after page 23 Page 15 of 23 MAST30025 Linear Statistical Models Semester 1, 2021 (a) [4 marks] Based on the diagnostic plots, comment on the fit of the linear model. Suggest a possible transformation for consideration that might improve the fit. (b) [2 marks] Interpret the output of the anova function in the context of the study. Page 16 of 23 pages Page 16 of 23 add any extra pages after page 23 Page 16 of 23 MAST30025 Linear Statistical Models Semester 1, 2021 (c) [2 marks] Calculate a confidence interval for the average difference in salary between females and males. (d) [2 marks] Interpret the output of the step function in the context of the study. Page 17 of 23 pages Page 17 of 23 add any extra pages after page 23 Page 17 of 23 MAST30025 Linear Statistical Models Semester 1, 2021 (e) [2 marks] Interpret the output of the linearHypothesis function in the context of the study. (f) [4 marks] Based on model3, calculate a 95% prediction interval for the salary of a female assistant professor with a PhD for 10 years, who has been in her position for 4 years. Page 18 of 23 pages Page 18 of 23 add any extra pages after page 23 Page 18 of 23 MAST30025 Linear Statistical Models Semester 1, 2021 Question 6 (12 marks) Consider the general linear model, y = X + . This model may be of full or less than full rank. (a) [2 marks] State the distributional assumption used in fitting a linear model and explain how it may be verified. (b) [2 marks] Define the hat matrix and explain its importance in the theory of linear models. (c) [2 marks] State the correction factor for a corrected sum of squares, and explain its use in testing for model relevance. Page 19 of 23 pages Page 19 of 23 add any extra pages after page 23 Page 19 of 23 MAST30025 Linear Statistical Models Semester 1, 2021 (d) [2 marks] Outline the similarities and differences between Akaikes information criterion and the Bayesian information criterion. (e) [2 marks] State two differences between a conditional inverse and a regular inverse. (f) [2 marks] Define interaction between two categorical predictors, and explain how to model it. Page 20 of 23 pages Page 20 of 23 add any extra pages after page 23 Page 20 of 23 MAST30025 Linear Statistical Models Semester 1, 2021 Question 7 (14 marks) In this question, we consider the balanced incomplete block design. We use the standard notation that In is the n n identity matrix and Jn is the n n matrix with all elements equal to 1. (a) [2 marks] Under what circumstances should one prefer a balanced incomplete block design to a complete block design, and vice versa? (b) [2 marks] Write down a design matrix and parameter vector for a balanced incomplete block design with 3 treatments and 3 blocks, each of size 2. Ensure that subjects in the same block are placed consecutively. Page 21 of 23 pages Page 21 of 23 add any extra pages after page 23 Page 21 of 23 MAST30025 Linear Statistical Models Semester 1, 2021 (c) [3 marks] Calculate the reduced design matrix X2|1 for your design. You are given that the hat matrix corresponding to the nuisance parameters has the block structure . H 1 = X 1 ( X 1T X ) c X 1T = 1 0 J 2 0 0 0 J2 J2 0 0 2 Page 22 of 23 pages Page 22 of 23 add any extra pages after page 23 Page 22 of 23 MAST30025 Linear Statistical Models Semester 1, 2021 (d) [4 marks] It can be shown that the reduced design matrix X2|1 for this model satisfies T31 Show that for some constant c, and find the value of c. X2|1X2|1=2 I33J3 . (XTX )c=cI 2|1 2|1 (e) [3 marks] Using your answers above, show directly that the reduced normal equations for this design does not have the same solution as the normal equations for a completely randomised design of 6 experimental units over 3 treatments. End of Exam Total Available Marks = 90 Page 23 of 23 pages Page 23 of 23 add any extra pages after page 23 Page 23 of 23