Instructions
Midterm Exam WS 2022
Midterm Exam
MATH 523: Generalized Linear Models
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Question 1.
Midterm Exam WS 2022
Consider the family of distributions with density
fpy; θq “ exprθy ` logtcospθqus ˆ 2 cosh`πy ̆,
(c) The mean-variance relationship is
A) 1 ` μ2 B) μ C) ttanpμqu2 D)1
and parameter θ P p ́π{2, π{2q.
(a) This family is an exponential family with known dispersion φ “ 2 cosh` π ̆.
A) TRUE B) FALSE
Do not submit any justifications or calculations, only the answer will be graded.
(b) If Y is a response with the above density, then EpY q equals A) sinpθq B) tanpθq C) cospθq D) coshpθq
Do not submit any justifications or calculations, only the answer will be graded.
Do not submit any justifications or calculations, only the answer will be graded.
(d) Find the canonical link and comment on its suitability. Propose an alternative link function that may be better suited for a GLM with the above response distribution and explain why.
(e) For which data may a GLM with the above response distribution be well-suited? Give two characteristics.
MATH 523 Midterm Exam WS 2022
Question 2.
Recall the Gamma GLM, as we covered it in Lecture 3a.
(a) Calculate the Hessian when the reciprocal link is used. Justify your calculations. You can use any result from the lecture notes.
(b) Calculate the Anscombe residuals for a Gamma GLM.
(c) Do the values of the residuals calculated in part (b) depend on the link function? Explain. 2 MARK
MATH 523 Midterm Exam WS 2022
Question 3.
As part of a study of the learning ability of tropical birds, a researcher at the University of Texas at Austin collected data on the response of a rufous-tailed jacamar Galbula ruficauda, to butterflies. The underside of the wings of eight species of butterflies was painted using marker pens, and each butterfly was then released in the cage where the bird was confined. The bird responded in three ways: by not attacking the butterfly (N), by attacking the butterfly, then sampling but not eating it (S); or by attacking and eating the butterfly (E).
The butterfly species investigated were: Aphrissa boisduvalli (Ab), Phoebis argante (Pa), Dryas iulia (Di), Pierella luna (Pl), and Siproeta stelenes (Ss). The butterfly wings were painted in colour as follows: Unpainted, Brown, Yellow, Blue, Green, Red, Orange, Black. In what follows, N denotes the number of butterflies that were not attacked, S denotes the number of butterflies attacked but not eaten, and E denotes the number of butterflies eaten.
The output of the analysis in R appears on Page 6 and 7.
(a) Consider the model mod1 whose summary is given on Page 6, lines 1–31. From this output, identify: (i) the GLM that was used; (ii) the response; (iii) the predictors; (iv) the link function; (v) the sample size.
(b) Fill in the missing values on line 13 of the output on Page 6 (i.e. the statistic and the corresponding p-value). Clearly state the hypothesis the p-value corresponds to. Can you conclude that species is not a significant predictor? Justify your answer.
(c) Using the output of mod1, calculate the estimated probability that a butterfly of the species Aphrissa boisduvalli will be eaten if its wings are unpainted, along with an approximate two-sided 95% confidence interval.
The question continues overleaf.
MATH 523 Midterm Exam WS 2022 (d) Fill in the blanks in the following sentence:
“Butterflies whose wings are painted red, orange, or black are the
(choose between ‘least’ or ‘most’) palatable. The effect of painting the wings in red, orange, or black on the linear predictor (choose between ‘differs’ or ‘does not differ’) from one species to another.”
Do not submit any justifications or calculations, only the answer will be graded.
(e) Another model has been fitted to the same data, using the same GLM as mod1, but this time with the intercept and colour as predictors. The deviance of this new model is 81.474. Decide, using an appropriate test at the 5% level, whether the simpler model is a reasonable simplification of mod1. What can you conclude concerning the significance of the predictor species? 4 MARKS
(f) Another model mod2 has been fitted to the same data, its summary appears on Page 6, lines 34–44, and on Page 7, lines 45–64. Explain what is the difference between mod1 and mod2. Decide which one seems to provide a better fit to the data and explain why.
2 glm(formula = cbind(E, N + S) ̃ species + colour,
4 Deviance Residuals:
5 Min 1Q Median 3Q
6 -2.1510 -0.8729 0.0000 0.6284
8 Coefficients:
Midterm Exam WS 2022
family = binomial(link = “cloglog”))
10 (Intercept)
11 speciesPa
12 speciesDi
13 speciesPl
14 speciesSs
15 colourBrown
16 colourYellow
17 colourBlue
18 colourGreen
19 colourRed
20 colourOrange
21 colourBlack
23 Signif. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’.’ 0.1 ’ ’ 1
25 (Dispersion parameter for binomial family taken to be 1) 26
27 Null deviance: 107.866 on 36 degrees of freedom
28 Residual deviance: 51.939 on 25 degrees of freedom
29 AIC: 115.84
31 Number of Fisher Scoring iterations: 8 32
35 glm(formula = cbind(E, N + S) ̃ species + colour,
37 Deviance Residuals:
38 Min 1Q Median 3Q
39 -2.1683 -0.7150 -0.0600 0.5831
41 Coefficients:
42 Estimate Std. Error z
43 (Intercept) 0.3558 0.2690
44 speciesPa -1.0749 0.4469 -2.405 0.016165
Std. Error
value Pr(>|z|)
Max 3.2361
0.537 0.591146
-2.426 0.015270
-0.205 0.837552
3.766 0.000166
-2.565 0.010322
-2.392 0.016750
-3.017 0.002554
-3.179 0.001479
-3.920 8.87e-05
-4.074 4.62e-05
-3.179 0.001477
Max 3.5410
value Pr(>|z|)
1.323 0.185933
family = binomial(link = “probit”))
45 speciesDi 0.1216
46 speciesPl 0.5082
47 speciesSs 1.4228
48 colourBrown
49 colourYellow
50 colourBlue
51 colourGreen
52 colourRed
0.356 0.721995
1.915 0.055508 .
3.603 0.000314 ***
-2.632 0.008493 **
-2.251 0.024409 *
-3.046 0.002323 **
-3.442 0.000577 ***
-4.298 1.73e-05 ***
-4.538 5.67e-06 ***
-3.512 0.000444 ***
53 colourOrange
54 colourBlack
56 Signif. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’.’ 0.1 ’ ’ 1
58 (Dispersion parameter for binomial family taken to be 1) 59
60 Null deviance: 107.866 on 36 degrees of freedom
61 Residual deviance: 52.592 on 25 degrees of freedom
62 AIC: 116.5
64 Number of Fisher Scoring iterations: 5
Table of the Normal distribution
Entries in the table are the values of the cumulative distribution function Φ of the Normalp0, 1q distribution, evaluated at z.
0.00 0.01 0.02 0.03 0.04
0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359
0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753
0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141
0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517
0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879
0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224
0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549
0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852
0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133
0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389
1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621
1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830
1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015
1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177
1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319
1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441
1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545
1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633
1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706
1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767
2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817
2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857
2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890
2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916
2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936
2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952
2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964
2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974
2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981
2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986
3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990
3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993
3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995
3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997
3.4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998
0.05 0.06 0.07 0.08 0.09
10 2.15586
11 2.60322
12 3.07382
13 3.56503
14 4.07467
15 4.60092
16 5.14221
17 5.69722
18 6.26480
19 6.84397
20 7.43384
21 8.03365
22 8.64272
23 9.26042
24 9.88623
25 10.51965
26 11.16024
27 11.80759
28 12.46134
29 13.12115
30 13.78672
40 20.70654 50 27.99075 60 35.53449 70 43.27518 80 51.17193 90 59.19630 100 67.32756
Table of the Chi-squared distribution
Entries in the table are χ2αpνq: the α tail quantile of the Chi-squaredpνq distribution α given in columns, ν given in rows.
7.87944 10.59663 12.83816 14.86026 16.74960 18.54758 20.27774 21.95495 23.58935 25.18818 26.75685 28.29952 29.81947 31.31935 32.80132 34.26719 35.71847 37.15645 38.58226 39.99685 41.40106 42.79565 44.18128 45.55851 46.92789 48.28988 49.64492 50.99338 52.33562 53.67196 66.76596 79.48998 91.95170
104.21490 116.32106 128.29894 140.16949
0.99000 0.97500 0.00016 0.00098 0.02010 0.05064 0.11483 0.21580 0.29711 0.48442 0.55430 0.83121 0.87209 1.23734 1.23904 1.68987 1.64650 2.17973 2.08790 2.70039 2.55821 3.24697 3.05348 3.81575 3.57057 4.40379 4.10692 5.00875 4.66043 5.62873 5.22935 6.26214 5.81221 6.90766 6.40776 7.56419 7.01491 8.23075 7.63273 8.90652 8.26040 9.59078 8.89720 10.28290 9.54249 10.98232
10.19572 11.68855 10.85636 12.40115 11.52398 13.11972 12.19815 13.84390 12.87850 14.57338 13.56471 15.30786 14.25645 16.04707 14.95346 16.79077 22.16426 24.43304 29.70668 32.35736 37.48485 40.48175 45.44172 48.75756 53.54008 57.15317 61.75408 65.64662 70.06489 74.22193
0.95000 0.00393 0.10259 0.35185 0.71072 1.14548 1.63538 2.16735 2.73264 3.32511 3.94030 4.57481 5.22603 5.89186 6.57063 7.26094 7.96165 8.67176 9.39046
10.11701 10.85081 11.59131 12.33801 13.09051 13.84843 14.61141 15.37916 16.15140 16.92788 17.70837 18.49266 26.50930 34.76425 43.18796 51.73928 60.39148 69.12603 77.92947
0.90000 0.01579 0.21072 0.58437 1.06362 1.61031 2.20413 2.83311 3.48954 4.16816 4.86518 5.57778 6.30380 7.04150 7.78953 8.54676 9.31224
10.08519 10.86494 11.65091 12.44261 13.23960 14.04149 14.84796 15.65868 16.47341 17.29188 18.11390 18.93924 19.76774 20.59923 29.05052 37.68865 46.45889 55.32894 64.27784 73.29109 82.35814
0.10000 2.70554 4.60517 6.25139 7.77944 9.23636
10.64464 12.01704 13.36157 14.68366 15.98718 17.27501 18.54935 19.81193 21.06414 22.30713 23.54183 24.76904 25.98942 27.20357 28.41198 29.61509 30.81328 32.00690 33.19624 34.38159 35.56317 36.74122 37.91592 39.08747 40.25602 51.80506 63.16712 74.39701 85.52704 96.57820
107.56501 118.49800
0.05000 3.84146 5.99146 7.81473 9.48773
11.07050 12.59159 14.06714 15.50731 16.91898 18.30704 19.67514 21.02607 22.36203 23.68479 24.99579 26.29623 27.58711 28.86930 30.14353 31.41043 32.67057 33.92444 35.17246 36.41503 37.65248 38.88514 40.11327 41.33714 42.55697 43.77297 55.75848 67.50481 79.08194 90.53123
101.87947 113.14527 124.34211
Right-tail
0.02500 5.02389 7.37776 9.34840
11.14329 12.83250 14.44938 16.01276 17.53455 19.02277 20.48318 21.92005 23.33666 24.73560 26.11895 27.48839 28.84535 30.19101 31.52638 32.85233 34.16961 35.47888 36.78071 38.07563 39.36408 40.64647 41.92317 43.19451 44.46079 45.72229 46.97924 59.34171 71.42020 83.29767 95.02318
106.62857 118.13589 129.56120
0.01000 6.63490 9.21034
11.34487 13.27670 15.08627 16.81189 18.47531 20.09024 21.66599 23.20925 24.72497 26.21697 27.68825 29.14124 30.57791 31.99993 33.40866 34.80531 36.19087 37.56623 38.93217 40.28936 41.63840 42.97982 44.31410 45.64168 46.96294 48.27824 49.58788 50.89218 63.69074 76.15389 88.37942
100.42518 112.32879 124.11632 135.80672
Tables Page 10
2 3 199.50 215.71 19.00 19.16 9.55 9.28 6.94 6.59 5.79 5.41 5.14 4.76 4.74 4.35 4.46 4.07 4.26 3.86 4.10 3.71 3.98 3.59 3.89 3.49 3.81 3.41 3.74 3.34 3.68 3.29 3.63 3.24 3.59 3.20 3.55 3.16 3.52 3.13 3.49 3.10 3.47 3.07 3.44 3.05 3.42 3.03 3.40 3.01 3.39 2.99 3.37 2.98 3.35 2.96 3.34 2.95 3.33 2.93 3.32 2.92 3.30 2.91 3.29 2.90
4 5 224.58 230.16 19.25 19.30 9.12 9.01 6.39 6.26 5.19 5.05 4.53 4.39 4.12 3.97 3.84 3.69 3.63 3.48 3.48 3.33 3.36 3.20 3.26 3.11 3.18 3.03 3.11 2.96 3.06 2.90 3.01 2.85 2.96 2.81 2.93 2.77 2.90 2.74 2.87 2.71 2.84 2.68 2.82 2.66 2.80 2.64 2.78 2.62 2.76 2.60 2.74 2.59 2.73 2.57 2.71 2.56 2.70 2.55 2.69 2.53 2.68 2.52 2.67 2.51
6 7 233.99 236.77 19.33 19.35 8.94 8.89 6.16 6.09 4.95 4.88 4.28 4.21 3.87 3.79 3.58 3.50 3.37 3.29 3.22 3.14 3.09 3.01 3.00 2.91 2.92 2.83 2.85 2.76 2.79 2.71 2.74 2.66 2.70 2.61 2.66 2.58 2.63 2.54 2.60 2.51 2.57 2.49 2.55 2.46 2.53 2.44 2.51 2.42 2.49 2.40 2.47 2.39 2.46 2.37 2.45 2.36 2.43 2.35 2.42 2.33 2.41 2.32 2.40 2.31
8 238.88 19.37 8.85 6.04 4.82 4.15 3.73 3.44 3.23 3.07 2.95 2.85 2.77 2.70 2.64 2.59 2.55 2.51 2.48 2.45 2.42 2.40 2.37 2.36 2.34 2.32 2.31 2.29 2.28 2.27 2.25 2.24
9 240.54 19.38 8.81 6.00 4.77 4.10 3.68 3.39 3.18 3.02 2.90 2.80 2.71 2.65 2.59 2.54 2.49 2.46 2.42 2.39 2.37 2.34 2.32 2.30 2.28 2.27 2.25 2.24 2.22 2.21 2.20 2.19
10 11 241.88 242.98 19.40 19.40 8.79 8.76 5.96 5.94 4.74 4.70 4.06 4.03 3.64 3.60 3.35 3.31 3.14 3.10 2.98 2.94 2.85 2.82 2.75 2.72 2.67 2.63 2.60 2.57 2.54 2.51 2.49 2.46 2.45 2.41 2.41 2.37 2.38 2.34 2.35 2.31 2.32 2.28 2.30 2.26 2.27 2.24 2.25 2.22 2.24 2.20 2.22 2.18 2.20 2.17 2.19 2.15 2.18 2.14 2.16 2.13 2.15 2.11 2.14 2.10
12 ν2zν1 1 2 3 4 5 6 7 8 9 10 11 12 243.91 33 4.14 3.28 2.89 2.66 2.50 2.39 2.30 2.23 2.18 2.13 2.09 2.06 19.41 34 4.13 3.28 2.88 2.65 2.49 2.38 2.29 2.23 2.17 2.12 2.08 2.05 8.74 35 4.12 3.27 2.87 2.64 2.49 2.37 2.29 2.22 2.16 2.11 2.07 2.04 5.91 36 4.11 3.26 2.87 2.63 2.48 2.36 2.28 2.21 2.15 2.11 2.07 2.03 4.68 37 4.11 3.25 2.86 2.63 2.47 2.36 2.27 2.20 2.14 2.10 2.06 2.02 4.00 38 4.10 3.24 2.85 2.62 2.46 2.35 2.26 2.19 2.14 2.09 2.05 2.02 3.57 39 4.09 3.24 2.85 2.61 2.46 2.34 2.26 2.19 2.13 2.08 2.04 2.01 3.28 40 4.08 3.23 2.84 2.61 2.45 2.34 2.25 2.18 2.12 2.08 2.04 2.00 3.07 41 4.08 3.23 2.83 2.60 2.44 2.33 2.24 2.17 2.12 2.07 2.03 2.00 2.91 42 4.07 3.22 2.83 2.59 2.44 2.32 2.24 2.17 2.11 2.06 2.03 1.99 2.79 43 4.07 3.21 2.82 2.59 2.43 2.32 2.23 2.16 2.11 2.06 2.02 1.99 2.69 44 4.06 3.21 2.82 2.58 2.43 2.31 2.23 2.16 2.10 2.05 2.01 1.98 2.60 45 4.06 3.20 2.81 2.58 2.42 2.31 2.22 2.15 2.10 2.05 2.01 1.97 2.53 46 4.05 3.20 2.81 2.57 2.42 2.30 2.22 2.15 2.09 2.04 2.00 1.97 2.48 47 4.05 3.20 2.80 2.57 2.41 2.30 2.21 2.14 2.09 2.04 2.00 1.96 2.42 48 4.04 3.19 2.80 2.57 2.41 2.29 2.21 2.14 2.08 2.03 1.99 1.96 2.38 49 4.04 3.19 2.79 2.56 2.40 2.29 2.20 2.13 2.08 2.03 1.99 1.96 2.34 50 4.03 3.18 2.79 2.56 2.40 2.29 2.20 2.13 2.07 2.03 1.99 1.95 2.31 51 4.03 3.18 2.79 2.55 2.40 2.28 2.20 2.13 2.07 2.02 1.98 1.95 2.28 52 4.03 3.18 2.78 2.55 2.39 2.28 2.19 2.12 2.07 2.02 1.98 1.94 2.25 53 4.02 3.17 2.78 2.55 2.39 2.28 2.19 2.12 2.06 2.01 1.97 1.94 2.23 54 4.02 3.17 2.78 2.54 2.39 2.27 2.18 2.12 2.06 2.01 1.97 1.94 2.20 55 4.02 3.16 2.77 2.54 2.38 2.27 2.18 2.11 2.06 2.01 1.97 1.93 2.18 56 4.01 3.16 2.77 2.54 2.38 2.27 2.18 2.11 2.05 2.00 1.96 1.93 2.16 57 4.01 3.16 2.77 2.53 2.38 2.26 2.18 2.11 2.05 2.00 1.96 1.93 2.15 58 4.01 3.16 2.76 2.53 2.37 2.26 2.17 2.10 2.05 2.00 1.96 1.92 2.13 59 4.00 3.15 2.76 2.53 2.37 2.26 2.17 2.10 2.04 2.00 1.96 1.92 2.12 60 4.00 3.15 2.76 2.53 2.37 2.25 2.17 2.10 2.04 1.99 1.95 1.92 2.10 61 4.00 3.15 2.76 2.52 2.37 2.25 2.16 2.09 2.04 1.99 1.95 1.91 2.09 62 4.00 3.15 2.75 2.52 2.36 2.25 2.16 2.09 2.03 1.99 1.95 1.91 2.08 63 3.99 3.14 2.75 2.52 2.36 2.25 2.16 2.09 2.03 1.98 1.94 1.91 2.07 64 3.99 3.14 2.75 2.52 2.36 2.24 2.16 2.09 2.03 1.98 1.94 1.91
Entries in the
table are the α “ 0.05 tail quantile of Fisher-Fpν1,ν2q distribution
Table of the Fisher-F distribution
in columns, ν2 given in rows.
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