STAT2008 REGRESSION MODELLING R Output for the 2017 Final Examination Page 1 of 20 Question 1 > library(UsingR) > library(faraway) > > help(cheddar) > attach(cheddar) > > summary(cheddar) taste Acetic H2S Lactic Min. : 0.70 Min. :4.477 Min. : 2.996 Min. :0.860 1st Qu.:13.55 1st Qu.:5.237 1st Qu.: 3.978 1st Qu.:1.250 Median :20.95 Median :5.425 Median : 5.329 Median :1.450 Mean :24.53 Mean :5.498 Mean : 5.942 Mean :1.442 3rd Qu.:36.70 3rd Qu.:5.883 3rd Qu.: 7.575 3rd Qu.:1.667 Max. :57.20 Max. :6.458 Max. :10.199 Max. :2.010 > > cor(cheddar) taste Acetic H2S Lactic taste 1.0000000 0.5495393 0.7557523 0.7042362 Acetic 0.5495393 1.0000000 0.6179559 0.6037826 H2S 0.7557523 0.6179559 1.0000000 0.6448123 Lactic 0.7042362 0.6037826 0.6448123 1.0000000 > > plot(cheddar) STAT2008 REGRESSION MODELLING R Output for the 2017 Final Examination Page 2 of 20 Question 1 continued > summary(cheddar.lm) Call: lm(formula = taste ~ H2S + Lactic) Residuals: Min 1Q Median 3Q Max -17.343 -6.530 -1.164 4.844 25.618 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -27.592 8.982 -3.072 0.00481 ** H2S 3.946 1.136 3.475 0.00174 ** Lactic 19.887 7.959 2.499 0.01885 * — Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 9.942 on 27 degrees of freedom Multiple R-squared: 0.6517, Adjusted R-squared: 0.6259 F-statistic: 25.26 on 2 and 27 DF, p-value: 6.551e-07 > > plot(cheddar.lm, which=c(1,2,4)) STAT2008 REGRESSION MODELLING R Output for the 2017 Final Examination Page 3 of 20 Question 1 continued STAT2008 REGRESSION MODELLING R Output for the 2017 Final Examination Page 4 of 20 Question 1 continued > cbind(hii=hatvalues(cheddar.lm), ti=rstudent(cheddar.lm), DFFITS=dffits(cheddar.lm), COVRATIO=covratio(cheddar.lm)) hii ti DFFITS COVRATIO 1 0.16055269 1.15041237 0.50311309 1.1494533 2 0.05816258 -0.18691381 -0.04644888 1.1842565 3 0.05474793 1.46948593 0.35365147 0.9325540 4 0.08418765 1.04377653 0.31646702 1.0811434 5 0.10993173 -0.15909118 -0.05591084 1.2545277 6 0.21752949 0.20314321 0.10710935 1.4244164 7 0.19919204 0.53253269 0.26559388 1.3536575 8 0.07932505 -1.90436363 -0.55898723 0.8221281 9 0.06773616 0.49753633 0.13411139 1.1675796 10 0.11507987 0.07327532 0.02642441 1.2647309 11 0.06453556 0.49658794 0.13043137 1.1637103 12 0.11200441 1.74947037 0.62132490 0.9031599 13 0.10737069 -0.48392942 -0.16783762 1.2212857 14 0.04255541 0.83151141 0.17530276 1.0810929 15 0.03834518 2.98866976 0.59679303 0.4801714 16 0.13557969 -0.42061322 -0.16657805 1.2694363 17 0.07055189 0.51291069 0.14131345 1.1690398 18 0.06195838 -1.50769212 -0.38748209 0.9284259 19 0.05153776 -1.16303642 -0.27111033 1.0140815 20 0.15120439 -0.96523689 -0.40739393 1.1871285 21 0.06664757 -0.08261418 -0.02207618 1.1989018 22 0.03863636 -0.43011462 -0.08622605 1.1403704 23 0.17622646 -0.60416330 -0.27943833 1.3037655 24 0.19140592 0.44851508 0.21821743 1.3533150 25 0.07673890 0.39982180 0.11526886 1.1908598 26 0.13605080 -0.27079011 -0.10745812 1.2853273 27 0.06299378 -0.78277052 -0.20296074 1.1145012 28 0.06475665 -1.26220126 -0.33213028 1.0017578 29 0.15624474 -0.74572696 -0.32090338 1.2456083 30 0.04821029 -1.10319703 -0.24828612 1.0257167 > > dfbetas(cheddar.lm) (Intercept) H2S Lactic 1 0.4811881189 -0.025353883 -0.325413289 2 0.0050935953 0.028521553 -0.026317517 3 -0.0734147314 -0.186837458 0.210953564 4 -0.2005979232 -0.013544907 0.196447544 5 -0.0516418001 0.001776529 0.034500947 6 0.0629737292 0.085252473 -0.092777315 7 0.0931507811 0.235950450 -0.194446572 8 0.3194683253 -0.112639847 -0.241099206 9 0.0598127714 -0.082693326 0.016691188 10 -0.0050951615 -0.021277671 0.018747330 11 -0.0578075841 -0.051332423 0.090248058 12 -0.4412151544 -0.021931464 0.411800257 13 -0.1306430438 0.071353862 0.045496982 14 0.1009420291 -0.034759048 -0.034021275 15 -0.0180293815 0.159359222 0.008418083 16 0.0552392628 -0.118835068 0.013576993 17 0.1059522789 -0.045909248 -0.040562386 18 0.0060543584 0.259376544 -0.202194302 19 0.0962191535 0.084209897 -0.159298496 20 0.2988241399 -0.077295941 -0.218666237 21 -0.0183087878 -0.003191663 0.013735751 22 -0.0447616300 0.010805953 0.016009652 23 -0.0436345950 -0.251625015 0.162873522 24 -0.1439021726 0.096316150 0.070390476 25 0.0423720868 -0.079912289 0.025840926 26 -0.0008375856 0.093315603 -0.062623524 27 0.1060876444 0.021801131 -0.119193700 28 -0.1853243688 0.173258991 0.005476049 29 -0.2216048754 -0.220622636 0.279703575 30 -0.1687739108 0.037089185 0.077622179 > STAT2008 REGRESSION MODELLING R Output for the 2017 Final Examination Page 5 of 20 Question 1 continued > cheddar.lm2 <- lm(taste ~ Acetic + H2S + Lactic) > > anova(cheddar.lm2) Analysis of Variance Table Response: taste Df Sum Sq Mean Sq F value Pr(>F) Acetic 1 2314.14 2314.14 22.5481 6.528e-05 *** H2S 1 2147.02 2147.02 20.9197 0.0001035 *** Lactic 1 533.32 533.32 5.1964 0.0310795 * Residuals 26 2668.41 102.63 — Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > > summary(cheddar.lm2) Call: lm(formula = taste ~ Acetic + H2S + Lactic) Residuals: Min 1Q Median 3Q Max -17.390 -6.612 -1.009 4.908 25.449 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -28.8768 19.7354 -1.463 0.15540 Acetic 0.3277 4.4598 0.073 0.94198 H2S 3.9118 1.2484 3.133 0.00425 ** Lactic 19.6705 8.6291 2.280 0.03108 * — Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 10.13 on 26 degrees of freedom Multiple R-squared: 0.6518, Adjusted R-squared: 0.6116 F-statistic: 16.22 on 3 and 26 DF, p-value: 3.81e-06 > STAT2008 REGRESSION MODELLING R Output for the 2017 Final Examination Page 6 of 20 Question 2 > attach(kid.weights) > help(kid.weights) > summary(kid.weights) age weight height gender Min. : 3.00 Min. : 10.00 Min. :12.00 F:129 1st Qu.: 12.25 1st Qu.: 22.00 1st Qu.:28.00 M:121 Median : 39.00 Median : 32.00 Median :36.00 Mean : 47.95 Mean : 38.38 Mean :36.52 3rd Qu.: 69.75 3rd Qu.: 45.00 3rd Qu.:43.00 Max. :144.00 Max. :150.00 Max. :67.00 > > # pairs(kid.weights) > > log_weight <- log(weight) > # pairs(cbind(age, height, log_weight)) > > log_age <- log(age) > pairs(cbind(log_age, height, log_weight)) > # growth.lm <- lm(log_weight ~ height) > # plot(growth.lm, which=c(1,2,4)) > > # anova(growth.lm) > > # growth.lm1 <- lm(log_weight ~ height + log_age) > # plot(growth.lm1, which=c(1,2,4)) > # anova(growth.lm1) > # vif(growth.lm1) > > log_age2 <- log_age^2 > # growth.lm2 <- lm(log_weight ~ height + log_age + log_age2) > # plot(growth.lm2, which=c(1,2,4)) > # anova(growth.lm2) > # vif(growth.lm2) > > boy <- ifelse(gender=="M",1,0) > growth.lm3 <- lm(log_weight ~ boy*(height + log_age + log_age2)) > plot(growth.lm3, which=c(1,2,4)) STAT2008 REGRESSION MODELLING R Output for the 2017 Final Examination Page 7 of 20 Question 2 continued STAT2008 REGRESSION MODELLING R Output for the 2017 Final Examination Page 8 of 20 > anova(growth.lm3) Analysis of Variance Table Response: log_weight Df Sum Sq Mean Sq F value Pr(>F) boy 1 0.169 0.169 4.7657 0.02999 * height 1 60.139 60.139 1693.5977 < 2.2e-16 *** log_age 1 9.532 9.532 268.4233 < 2.2e-16 *** log_age2 1 0.619 0.619 17.4430 4.135e-05 *** boy:height 1 0.314 0.314 8.8428 0.00324 ** boy:log_age 1 0.143 0.143 4.0289 0.04584 * boy:log_age2 1 0.003 0.003 0.0796 0.77804 Residuals 242 8.593 0.036 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > > summary(growth.lm3) Call: lm(formula = log_weight ~ boy * (height + log_age + log_age2)) Residuals: Min 1Q Median 3Q Max -0.41286 -0.12511 -0.04004 0.11195 0.61353 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 2.274074 0.164990 13.783 < 2e-16 *** boy -0.175592 0.256450 -0.685 0.49419 height 0.006419 0.003254 1.973 0.04965 * log_age 0.122682 0.095786 1.281 0.20149 log_age2 0.044346 0.016929 2.620 0.00936 ** boy:height 0.005029 2.535 0.01188 * boy:log_age -0.118486 0.142527 -0.831 0.40661 boy:log_age2 0.007300 0.025871 0.282 0.77804 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: on degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: on and DF, p-value: < 2.2e-16 > > vif(growth.lm3) boy height log_age log_age2 boy:height boy:log_age boy:log_age2 115.635862 8.506479 79.005792 97.722824 71.804918 511.926528 313.935801 > > range(rstandard(growth.lm3)) [1] -2.211860 3.328865 > > rstandard(growth.lm3)[c(228,9,158)] 228 9 158 3.328865 3.230874 2.856645 > > range(rstudent(growth.lm3)) [1] -2.229941 3.400754 > > rstudent(growth.lm3)[c(228,9,158)] 228 9 158 3.400754 3.296063 2.900052 > STAT2008 REGRESSION MODELLING R Output for the 2017 Final Examination Page 9 of 20 Question 3 > cement x1 x2 x3 x4 y 1 7 26 6 60 78.5 2 1 29 15 52 74.3 3 11 56 8 20 104.3 4 11 31 8 47 87.6 5 7 52 6 33 95.9 6 11 55 9 22 109.2 7 3 71 17 6 102.7 8 1 31 22 44 72.5 9 2 54 18 22 93.1 10 21 47 4 26 115.9 11 1 40 23 34 83.8 12 11 66 9 12 113.3 13 10 68 8 12 109.4 > attach(cement) > > x1 + x2 + x3 + x4 [1] 99 97 95 97 98 97 97 98 96 98 98 98 98 > x1 + x3 [1] 13 16 19 19 13 20 20 23 20 25 24 20 18 > x2 + x4 [1] 86 81 76 78 85 77 77 75 76 73 74 78 80 > > pairs(cement) STAT2008 REGRESSION MODELLING R Output for the 2017 Final Examination Page 10 of 20 Question 3 continued > cor(cement) x1 x2 x3 x4 y x1 1.0000000 0.2285795 -0.8241338 -0.2454451 0.7307175 x2 0.2285795 1.0000000 -0.1392424 -0.9729550 0.8162526 x3 -0.8241338 -0.1392424 1.0000000 0.0295370 -0.5346707 x4 -0.2454451 -0.9729550 0.0295370 1.0000000 -0.8213050 y 0.7307175 0.8162526 -0.5346707 -0.8213050 1.0000000 > > cement_all.lm <- lm(y ~ x1 + x2 + x3 + x4) > anova(cement_all.lm) Analysis of Variance Table Response: y Df Sum Sq Mean Sq F value Pr(>F) x1 1 1450.08 1450.08 242.3679 2.888e-07 *** x2 1 1207.78 1207.78 201.8705 5.863e-07 *** x3 1 9.79 9.79 1.6370 0.2366 x4 1 0.25 0.25 0.0413 0.8441 Residuals 8 47.86 5.98 — Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > > cement_all.lm2 <- lm(y ~ x4 + x1 + x2 + x3) > anova(cement_all.lm2) Analysis of Variance Table Response: y Df Sum Sq Mean Sq F value Pr(>F) x4 1 1831.90 1831.90 306.1859 1.161e-07 *** x1 1 809.10 809.10 135.2350 2.722e-06 *** x2 1 26.79 26.79 4.4776 0.06724 . x3 1 0.11 0.11 0.0182 0.89592 Residuals 8 47.86 5.98 — Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > > summary(cement_all.lm) Call: lm(formula = y ~ x1 + x2 + x3 + x4) Residuals: Min 1Q Median 3Q Max -3.1750 -1.6709 0.2508 1.3783 3.9254 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 62.4054 70.0710 0.891 0.3991 x1 1.5511 0.7448 2.083 0.0708 . x2 0.5102 0.7238 0.705 0.5009 x3 0.1019 0.7547 0.135 0.8959 x4 -0.1441 0.7091 -0.203 0.8441 — Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 2.446 on 8 degrees of freedom Multiple R-squared: 0.9824, Adjusted R-squared: 0.9736 F-statistic: 111.5 on 4 and 8 DF, p-value: 4.756e-07 > vif(cement_all.lm) x1 x2 x3 x4 38.49621 254.42317 46.86839 282.51286 > > plot(cement_all.lm, which=c(1,2,4)) STAT2008 REGRESSION MODELLING R Output for the 2017 Final Examination Page 11 of 20 Question 3 continued STAT2008 REGRESSION MODELLING R Output for the 2017 Final Examination Page 12 of 20 Question 3 continued > cement.lm <- lm(y ~ x4 + x1) > anova(cement.lm) Analysis of Variance Table Response: y Df Sum Sq Mean Sq F value Pr(>F) x4 1 1831.90 1831.90 245.03 2.319e-08 *** x1 1 809.10 809.10 108.22 1.105e-06 *** Residuals 10 74.76 7.48 — Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > > summary(cement.lm) Call: lm(formula = y ~ x4 + x1) Residuals: Min 1Q Median 3Q Max -5.0234 -1.4737 0.1371 1.7305 3.7701 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 103.09738 2.12398 48.54 3.32e-13 *** x4 -0.61395 0.04864 -12.62 1.81e-07 *** x1 1.43996 0.13842 10.40 1.11e-06 *** — Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 2.734 on 10 degrees of freedom Multiple R-squared: 0.9725, Adjusted R-squared: 0.967 F-statistic: 176.6 on 2 and 10 DF, p-value: 1.581e-08 > vif(cement.lm) x4 x1 1.064105 1.064105 > > # plot(cement.lm, which=c(1,2,4)) STAT2008 REGRESSION MODELLING R Output for the 2017 Final Examination Page 13 of 20 Question 4 > temp <- mammalsleep[,c("brain","body","lifespan","gestation","danger","sleep")] > temp$danger <- ifelse(temp$danger<4,exp(0),exp(1)) > temp <- temp[!is.na(temp$sleep),] > temp <- temp[!is.na(temp$lifespan),] > temp <- temp[!is.na(temp$gestation),] > pairs(temp) STAT2008 REGRESSION MODELLING R Output for the 2017 Final Examination Page 14 of 20 Question 4 continued > msleep <- log(temp) > names(msleep) <- c("log_brain","log_body","log_lifespn","log_gestn","danger", "log_sleep") > pairs(msleep) > > cor(msleep) log_brain log_body log_lifespn log_gestn danger log_sleep log_brain 1.0000000 0.9587409 0.8044310 0.7924933 0.3218992 -0.6204359 log_body 0.9587409 1.0000000 0.7300478 0.7352547 0.3455494 -0.6349638 log_lifespn 0.8044310 0.7300478 1.0000000 0.6698078 0.2926964 -0.4360847 log_gestn 0.7924933 0.7352547 0.6698078 1.0000000 0.3715447 -0.6685500 danger 0.3218992 0.3455494 0.2926964 0.3715447 1.0000000 -0.5509559 log_sleep -0.6204359 -0.6349638 -0.4360847 -0.6685500 -0.5509559 1.0000000 > > attach(msleep) > STAT2008 REGRESSION MODELLING R Output for the 2017 Final Examination Page 15 of 20 Question 4 continued > species_name <- row.names(msleep) > data.frame(species_name, msleep, row.names=1:51) species_name log_brain log_body log_lifespn log_gestn danger log_sleep 1 African.elephant 8.6503245 8.80297346 3.6532523 6.469250 0 1.1939225 2 African.giant.pouched.rat 1.8870696 0.00000000 1.5040774 3.737670 0 2.1162555 3 Arctic.Fox 3.7954892 1.21935391 2.6390573 4.094345 0 2.5257286 4 Asian.elephant 8.4344635 7.84267147 4.2341065 6.436150 1 1.3609766 5 Baboon 5.1901752 2.35612586 3.2958369 5.192957 1 2.2823824 6 Big.brown.bat -1.20
39728 -3.77226106 2.9444390 3.555348 0 2.9806186 7 Brazilian.tapir 5.1298987 5.07517382 3.4144426 5.971262 1 1.8245493 8 Cat 3.2425924 1.19392247 3.3322045 4.143135 0 2.6741486 9 Chimpanzee 6.0867747 3.95431592 3.9120230 5.438079 0 2.2721259 10 Chinchilla 1.8562980 -0.85566611 1.9459101 4.718499 1 2.5257286 11 Cow 6.0473722 6.14203741 3.4011974 5.638355 1 1.3609766 12 Donkey 6.0378709 5.23164323 3.6888795 5.899897 1 1.1314021 13 Eastern.American.mole 0.1823216 -2.59026717 1.2527630 3.737670 0 2.1282317 14 Echidna 3.2188758 1.09861229 3.9120230 3.332205 0 2.1517622 15 European.hedgehog 1.2527630 -0.24207156 1.7917595 3.737670 0 2.3702437 16 Galago 1.6094379 -1.60943791 2.3418058 4.787492 0 2.3702437 17 Goat 4.7449321 3.31998733 2.9957323 4.997212 1 1.3350011 18 Golden.hamster 0.0000000 -2.12026354 1.3609766 2.772589 0 2.6672282 19 Gorilla 6.0063532 5.33271879 3.6712245 5.529429 0 2.4849066 20 Gray.seal 5.7838252 4.44265126 3.7135721 5.736572 0 1.8245493 21 Gray.wolf 4.7833164 3.59264385 2.7850112 4.143135 0 2.5649494 22 Ground.squirrel 1.3862944 -2.29263476 2.1972246 3.332205 0 2.6246686 23 Guinea.pig 1.7047481 0.03922071 2.0281482 4.219508 1 2.1041342 24 Horse 6.4846352 6.25575004 3.8286414 5.817111 1 1.0647107 25 Jaguar 5.0562458 4.60517019 3.1090610 4.605170 0 2.3795461 26 Lesser.short-tailed.shrew -1.9661129 -5.29831737 0.9555114 3.068053 1 2.2082744 27 Little.brown.bat -1.3862944 -4.60517019 3.1780538 3.912023 0 2.9907197 28 Man 7.1853870 4.12713439 4.6051702 5.587249 0 2.0794415 29 Mouse -0.9162907 -3.77226106 1.1631508 2.944439 0 2.5802168 30 Musk.shrew -1.1086626 -3.03655427 0.6931472 3.401197 0 2.5494452 31 N.American.opossum 1.8405496 0.53062825 1.6094379 2.484907 0 2.9652731 32 Nine-banded.armadillo 2.3795461 1.25276297 1.8718022 4.787492 0 2.8564702 33 Owl.monkey 2.7408400 -0.73396918 2.4849066 4.941642 0 2.8332133 34 Patas.monkey 4.7449321 2.30258509 3.0056826 5.135798 1 2.3887628 35 Phanlanger 2.4336134 0.48242615 2.5649494 2.833213 0 2.6173958 36 Pig 5.1929569 5.25749537 3.2958369 4.744932 1 2.1282317 37 Rabbit 2.4932055 0.91629073 2.8903718 3.433987 1 2.1282317 38 Raccoon 3.6686767 1.45582042 2.6173958 4.143135 0 2.5257286 39 Rat 0.6418539 -1.27296568 1.5475625 3.044522 0 2.5802168 40 Red.fox 3.9199912 1.44338333 2.2823824 3.951244 0 2.2823824 41 Rhesus.monkey 5.1873858 1.91692261 3.3672958 5.099866 0 2.2617631 42 Rock.hyrax.Hetero 2.5095993 -0.28768207 1.9459101 5.416100 0 1.8870696 43 Rock.hyrax.Procavia 3.0445224 1.28093385 1.7917595 5.416100 0 1.6863990 44 Roe.deer 4.5870062 2.69665216 2.8332133 5.010635 1 0.9555114 45 Sheep 5.1647860 4.01638302 2.9957323 5.017280 1 1.3350011 46 Slow.loris 2.5257286 0.33647224 2.5416020 4.499810 0 2.3978953 47 Tenrec 0.9555114 -0.10536052 1.5040774 4.094345 0 2.5877640 48 Tree.hyrax 2.5095993 0.69314718 2.0149030 5.298317 0 1.6863990 49 Tree.shrew 0.9162907 -2.26336438 0.8329091 3.828641 0 2.7600099 50 Vervet 4.0604430 1.43270073 3.1780538 5.347108 1 2.3321439 51 Water.opossum 1.3609766 1.25276297 1.0986123 2.639057 0 2.9652731 > STAT2008 REGRESSION MODELLING R Output for the 2017 Final Examination Page 16 of 20 Question 4 continued > msleep.lm <- step(lm(log_sleep ~ 1), scope=list(upper=as.formula("log_sleep ~ log_brain + log_body + log_lifespn + log_gestn + danger")), direction="both") Start: AIC=-62.17 log_sleep ~ 1 Df Sum of Sq RSS AIC + log_gestn 1 6.4770 8.0142 -90.381 + log_body 1 5.8426 8.6487 -86.495 + log_brain 1 5.5783 8.9130 -84.960 + danger 1 4.3988 10.0924 -78.622 + log_lifespn 1 2.7558 11.7354 -70.930 14.4912 -62.172 Step: AIC=-90.38 log_sleep ~ log_gestn Df Sum of Sq RSS AIC + danger 1 1.5390 6.4752 -99.256 + log_body 1 0.6487 7.3655 -92.686 + log_brain 1 0.3199 7.6943 -90.458 8.0142 -90.381 + log_lifespn 1 0.0036 8.0106 -88.404 - log_gestn 1 6.4770 14.4912 -62.172 Step: AIC=-99.26 log_sleep ~ log_gestn + danger Df Sum of Sq RSS AIC + log_body 1 0.4452 6.0301 -100.888 + log_brain 1 0.2561 6.2191 -99.314 6.4752 -99.256 + log_lifespn 1 0.0194 6.4558 -97.409 - danger 1 1.5390 8.0142 -90.381 - log_gestn 1 3.6172 10.0924 -78.622 Step: AIC=-100.89 log_sleep ~ log_gestn + danger + log_body Df Sum of Sq RSS AIC + log_lifespn 1 0.26183 5.7682 -101.152 6.0301 -100.888 + log_brain 1 0.06129 5.9688 -99.409 - log_body 1 0.44515 6.4752 -99.256 - log_gestn 1 0.80970 6.8398 -96.463 - danger 1 1.33543 7.3655 -92.686 Step: AIC=-101.15 log_sleep ~ log_gestn + danger + log_body + log_lifespn Df Sum of Sq RSS AIC 5.7682 -101.152 - log_lifespn 1 0.26183 6.0301 -100.888 + log_brain 1 0.00000 5.7682 -99.152 - log_body 1 0.68760 6.4558 -97.409 - log_gestn 1 1.01476 6.7830 -94.888 - danger 1 1.34786 7.1161 -92.443 > > plot(msleep.lm, which=c(1,2,4)) STAT2008 REGRESSION MODELLING R Output for the 2017 Final Examination Page 17 of 20 Question 4 continued STAT2008 REGRESSION MODELLING R Output for the 2017 Final Examination Page 18 of 20 Question 4 continued > anova(msleep.lm) Analysis of Variance Table Response: log_sleep Df Sum Sq Mean Sq F value Pr(>F) log_gestn 1 6.4770 6.4770 51.6520 4.781e-09 *** danger 1 1.5390 1.5390 12.2732 0.001035 ** log_body 1 0.4452 0.4452 3.5499 0.065876 . log_lifespn 1 0.2618 0.2618 2.0880 0.155236 Residuals 46 5.7682 0.1254 — Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > > summary(msleep.lm) Call: lm(formula = log_sleep ~ log_gestn + danger + log_body + log_lifespn) Residuals: Min 1Q Median 3Q Max -0.82494 -0.25085 -0.04913 0.24363 0.66865 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 3.07444 0.31460 9.772 8.42e-13 *** log_gestn -0.21364 0.07510 -2.845 0.00661 ** danger -0.37992 0.11588 -3.279 0.00199 ** log_body -0.06212 0.02653 -2.342 0.02358 * log_lifespn 0.11434 0.07913 1.445 0.15524 — Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.3541 on 46 degrees of freedom Multiple R-squared: 0.6019, Adjusted R-squared: 0.5673 F-statistic: 17.39 on 4 and 46 DF, p-value: 9.336e-09 > vif(msleep.lm) log_gestn danger log_body log_lifespn 2.446765 1.175837 2.827213 2.333961 STAT2008 REGRESSION MODELLING R Output for the 2017 Final Examination Page 19 of 20 Statistical Tables > # Selected quantiles of the F distribution: > NumeratorDF <- c(1:10) > DenominatorDF <- c(1:50,60,70,80,90,100) > Fquantiles0.95 <- matrix(0,ncol=length(NumeratorDF),nrow=length(DenominatorDF)) > for(i in 1:length(NumeratorDF)){for(j in 1:length(DenominatorDF)){ + Fquantiles0.95[j,i]=round(qf(0.95,i,DenominatorDF[j]),3)}} > dimnames(Fquantiles0.95) <- list(DenominatorDF, NumeratorDF) > Fquantiles0.95 1 2 3 4 5 6 7 8 9 10 1 161.448 199.500 215.707 224.583 230.162 233.986 236.768 238.883 240.543 241.882 2 18.513 19.000 19.164 19.247 19.296 19.330 19.353 19.371 19.385 19.396 3 10.128 9.552 9.277 9.117 9.013 8.941 8.887 8.845 8.812 8.786 4 7.709 6.944 6.591 6.388 6.256 6.163 6.094 6.041 5.999 5.964 5 6.608 5.786 5.409 5.192 5.050 4.950 4.876 4.818 4.772 4.735 6 5.987 5.143 4.757 4.534 4.387 4.284 4.207 4.147 4.099 4.060 7 5.591 4.737 4.347 4.120 3.972 3.866 3.787 3.726 3.677 3.637 8 5.318 4.459 4.066 3.838 3.687 3.581 3.500 3.438 3.388 3.347 9 5.117 4.256 3.863 3.633 3.482 3.374 3.293 3.230 3.179 3.137 10 4.965 4.103 3.708 3.478 3.326 3.217 3.135 3.072 3.020 2.978 11 4.844 3.982 3.587 3.357 3.204 3.095 3.012 2.948 2.896 2.854 12 4.747 3.885 3.490 3.259 3.106 2.996 2.913 2.849 2.796 2.753 13 4.667 3.806 3.411 3.179 3.025 2.915 2.832 2.767 2.714 2.671 14 4.600 3.739 3.344 3.112 2.958 2.848 2.764 2.699 2.646 2.602 15 4.543 3.682 3.287 3.056 2.901 2.790 2.707 2.641 2.588 2.544 16 4.494 3.634 3.239 3.007 2.852 2.741 2.657 2.591 2.538 2.494 17 4.451 3.592 3.197 2.965 2.810 2.699 2.614 2.548 2.494 2.450 18 4.414 3.555 3.160 2.928 2.773 2.661 2.577 2.510 2.456 2.412 19 4.381 3.522 3.127 2.895 2.740 2.628 2.544 2.477 2.423 2.378 20 4.351 3.493 3.098 2.866 2.711 2.599 2.514 2.447 2.393 2.348 21 4.325 3.467 3.072 2.840 2.685 2.573 2.488 2.420 2.366 2.321 22 4.301 3.443 3.049 2.817 2.661 2.549 2.464 2.397 2.342 2.297 23 4.279 3.422 3.028 2.796 2.640 2.528 2.442 2.375 2.320 2.275 24 4.260 3.403 3.009 2.776 2.621 2.508 2.423 2.355 2.300 2.255 25 4.242 3.385 2.991 2.759 2.603 2.490 2.405 2.337 2.282 2.236 26 4.225 3.369 2.975 2.743 2.587 2.474 2.388 2.321 2.265 2.220 27 4.210 3.354 2.960 2.728 2.572 2.459 2.373 2.305 2.250 2.204 28 4.196 3.340 2.947 2.714 2.558 2.445 2.359 2.291 2.236 2.190 29 4.183 3.328 2.934 2.701 2.545 2.432 2.346 2.278 2.223 2.177 30 4.171 3.316 2.922 2.690 2.534 2.421 2.334 2.266 2.211 2.165 31 4.160 3.305 2.911 2.679 2.523 2.409 2.323 2.255 2.199 2.153 32 4.149 3.295 2.901 2.668 2.512 2.399 2.313 2.244 2.189 2.142 33 4.139 3.285 2.892 2.659 2.503 2.389 2.303 2.235 2.179 2.133 34 4.130 3.276 2.883 2.650 2.494 2.380 2.294 2.225 2.170 2.123 35 4.121 3.267 2.874 2.641 2.485 2.372 2.285 2.217 2.161 2.114 36 4.113 3.259 2.866 2.634 2.477 2.364 2.277 2.209 2.153 2.106 37 4.105 3.252 2.859 2.626 2.470 2.356 2.270 2.201 2.145 2.098 38 4.098 3.245 2.852 2.619 2.463 2.349 2.262 2.194 2.138 2.091 39 4.091 3.238 2.845 2.612 2.456 2.342 2.255 2.187 2.131 2.084 40 4.085 3.232 2.839 2.606 2.449 2.336 2.249 2.180 2.124 2.077 41 4.079 3.226 2.833 2.600 2.443 2.330 2.243 2.174 2.118 2.071 42 4.073 3.220 2.827 2.594 2.438 2.324 2.237 2.168 2.112 2.065 43 4.067 3.214 2.822 2.589 2.432 2.318 2.232 2.163 2.106 2.059 44 4.062 3.209 2.816 2.584 2.427 2.313 2.226 2.157 2.101 2.054 45 4.057 3.204 2.812 2.579 2.422 2.308 2.221 2.152 2.096 2.049 46 4.052 3.200 2.807 2.574 2.417 2.304 2.216 2.147 2.091 2.044 47 4.047 3.195 2.802 2.570 2.413 2.299 2.212 2.143 2.086 2.039 48 4.043 3.191 2.798 2.565 2.409 2.295 2.207 2.138 2.082 2.035 49 4.038 3.187 2.794 2.561 2.404 2.290 2.203 2.134 2.077 2.030 50 4.034 3.183 2.790 2.557 2.400 2.286 2.199 2.130 2.073 2.026 60 4.001 3.150 2.758 2.525 2.368 2.254 2.167 2.097 2.040 1.993 70 3.978 3.128 2.736 2.503 2.346 2.231 2.143 2.074 2.017 1.969 80 3.960 3.111 2.719 2.486 2.329 2.214 2.126 2.056 1.999 1.951 90 3.947 3.098 2.706 2.473 2.316 2.201 2.113 2.043 1.986 1.938 100 3.936 3.087 2.696 2.463 2.305 2.191 2.103 2.032 1.975 1.927 > # Columns are numerator df, rows are denominator df > STAT2008 REGRESSION MODELLING R Output for the 2017 Final Examination Page 20 of 20 Statistical Tables continued > # Selected quantiles of Student’s t distribution: > DF <- c(1:100, 150, 200, 250, 300, 400, 500, 750, 1000, 10000, 1000000) > tquantiles0.025 <- round(qt(0.025,DF),4) > tquantiles0.05 <- round(qt(0.05,DF),4) > tquantiles0.95 <- round(qt(0.95,DF),4) > tquantiles0.975 <- round(qt(0.975,DF),4) > tquantiles <- rbind(tquantiles0.025, tquantiles0.05, tquantiles0.95, tquantiles0.975) > dimnames(tquantiles) <- list(c("t 0.025", "t 0.05 ", "t 0.95 ", "t 0.975"), DF) > tquantiles 1 2 3 4 5 6 7 8 9 10 t 0.025 -12.7062 -4.3027 -3.1824 -2.7764 -2.5706 -2.4469 -2.3646 -2.3060 -2.2622 -2.2281 t 0.05 -6.3138 -2.9200 -2.3534 -2.1318 -2.0150 -1.9432 -1.8946 -1.8595 -1.8331 -1.8125 t 0.95 6.3138 2.9200 2.3534 2.1318 2.0150 1.9432 1.8946 1.8595 1.8331 1.8125 t 0.975 12.7062 4.3027 3.1824 2.7764 2.5706 2.4469 2.3646 2.3060 2.2622 2.2281 11 12 13 14 15 16 17 18 19 20 t 0.025 -2.2010 -2.1788 -2.1604 -2.1448 -2.1314 -2.1199 -2.1098 -2.1009 -2.0930 -2.0860 t 0.05 -1.7959 -1.7823 -1.7709 -1.7613 -1.7531 -1.7459 -1.7396 -1.7341 -1.7291 -1.7247 t 0.95 1.7959 1.7823 1.7709 1.7613 1.7531 1.7459 1.7396 1.7341 1.7291 1.7247 t 0.975 2.2010 2.1788 2.1604 2.1448 2.1314 2.1199 2.1098 2.1009 2.0930 2.0860 21 22 23 24 25 26 27 28 29 30 t 0.025 -2.0796 -2.0739 -2.0687 -2.0639 -2.0595 -2.0555 -2.0518 -2.0484 -2.0452 -2.0423 t 0.05 -1.7207 -1.7171 -1.7139 -1.7109 -1.7081 -1.7056 -1.7033 -1.7011 -1.6991 -1.6973 t 0.95 1.7207 1.7171 1.7139 1.7109 1.7081 1.7056 1.7033 1.7011 1.6991 1.6973 t 0.975 2.0796 2.0739 2.0687 2.0639 2.0595 2.0555 2.0518 2.0484 2.0452 2.0423 31 32 33 34 35 36 37 38 39 40 t 0.025 -2.0395 -2.0369 -2.0345 -2.0322 -2.0301 -2.0281 -2.0262 -2.0244 -2.0227 -2.0211 t 0.05 -1.6955 -1.6939 -1.6924 -1.6909 -1.6896 -1.6883 -1.6871 -1.6860 -1.6849 -1.6839 t 0.95 1.6955 1.6939 1.6924 1.6909 1.6896 1.6883 1.6871 1.6860 1.6849 1.6839 t 0.975 2.0395 2.0369 2.0345 2.0322 2.0301 2.0281 2.0262 2.0244 2.0227 2.0211 41 42 43 44 45 46 47 48 49 50 t 0.025 -2.0195 -2.0181 -2.0167 -2.0154 -2.0141 -2.0129 -2.0117 -2.0106 -2.0096 -2.0086 t 0.05 -1.6829 -1.6820 -1.6811 -1.6802 -1.6794 -1.6787 -1.6779 -1.6772 -1.6766 -1.6759 t 0.95 1.6829 1.6820 1.6811 1.6802 1.6794 1.6787 1.6779 1.6772 1.6766 1.6759 t 0.975 2.0195 2.0181 2.0167 2.0154 2.0141 2.0129 2.0117 2.0106 2.0096 2.0086 51 52 53 54 55 56 57 58 59 60 t 0.025 -2.0076 -2.0066 -2.0057 -2.0049 -2.0040 -2.0032 -2.0025 -2.0017 -2.0010 -2.0003 t 0.05 -1.6753 -1.6747 -1.6741 -1.6736 -1.6730 -1.6725 -1.6720 -1.6716 -1.6711 -1.6706 t 0.95 1.6753 1.6747 1.6741 1.6736 1.6730 1