NOTE: For your homework download and use the template (https://math.dartmouth.edu/~m50f17/HW7.Rmd)
Read the green comments in the rmd file to see where your answers should go.
[,1] sr numeric aggregate personal savings [,2] pop15 numeric % of population under 15 [,3] pop75 numeric % of population over 75 [,4] dpi numeric real per-capita disposable income [,5] ddpi numeric % growth rate of dpi
data(LifeCycleSavings)
lm.SR <- lm(sr ~ pop15 + pop75 + dpi + ddpi, data = LifeCycleSavings)
summary(inflm.SR <- influence.measures(lm.SR))
## Potentially influential observations of
## lm(formula = sr ~ pop15 + pop75 + dpi + ddpi, data = LifeCycleSavings) :
##
## dfb.1_ dfb.pp15 dfb.pp75 dfb.dpi dfb.ddpi dffit cov.r
## Chile -0.20 0.13 0.22 -0.02 0.12 -0.46 0.65_*
## United States 0.07 -0.07 0.04 -0.23 -0.03 -0.25 1.66_*
## Zambia 0.16 -0.08 -0.34 0.09 0.23 0.75 0.51_*
## Libya 0.55 -0.48 -0.38 -0.02 -1.02_* -1.16_* 2.09_*
## cook.d hat
## Chile 0.04 0.04
## United States 0.01 0.33_*
## Zambia 0.10 0.06
## Libya 0.27 0.53_*
inflm.SR
## Influence measures of
## lm(formula = sr ~ pop15 + pop75 + dpi + ddpi, data = LifeCycleSavings) :
##
## dfb.1_ dfb.pp15 dfb.pp75 dfb.dpi dfb.ddpi dffit cov.r
## Australia 0.01232 -0.01044 -0.02653 0.04534 -0.000159 0.0627 1.193
## Austria -0.01005 0.00594 0.04084 -0.03672 -0.008182 0.0632 1.268
## Belgium -0.06416 0.05150 0.12070 -0.03472 -0.007265 0.1878 1.176
## Bolivia 0.00578 -0.01270 -0.02253 0.03185 0.040642 -0.0597 1.224
## Brazil 0.08973 -0.06163 -0.17907 0.11997 0.068457 0.2646 1.082
## Canada 0.00541 -0.00675 0.01021 -0.03531 -0.002649 -0.0390 1.328
## Chile -0.19941 0.13265 0.21979 -0.01998 0.120007 -0.4554 0.655
## China 0.02112 -0.00573 -0.08311 0.05180 0.110627 0.2008 1.150
## Colombia 0.03910 -0.05226 -0.02464 0.00168 0.009084 -0.0960 1.167
## Costa Rica -0.23367 0.28428 0.14243 0.05638 -0.032824 0.4049 0.968
## Denmark -0.04051 0.02093 0.04653 0.15220 0.048854 0.3845 0.934
## Ecuador 0.07176 -0.09524 -0.06067 0.01950 0.047786 -0.1695 1.139
## Finland -0.11350 0.11133 0.11695 -0.04364 -0.017132 -0.1464 1.203
## France -0.16600 0.14705 0.21900 -0.02942 0.023952 0.2765 1.226
## Germany -0.00802 0.00822 0.00835 -0.00697 -0.000293 -0.0152 1.226
## Greece -0.14820 0.16394 0.02861 0.15713 -0.059599 -0.2811 1.140
## Guatamala 0.01552 -0.05485 0.00614 0.00585 0.097217 -0.2305 1.085
## Honduras -0.00226 0.00984 -0.01020 0.00812 -0.001887 0.0482 1.186
## Iceland 0.24789 -0.27355 -0.23265 -0.12555 0.184698 -0.4768 0.866
## India 0.02105 -0.01577 -0.01439 -0.01374 -0.018958 0.0381 1.202
## Ireland -0.31001 0.29624 0.48156 -0.25733 -0.093317 0.5216 1.268
## Italy 0.06619 -0.07097 0.00307 -0.06999 -0.028648 0.1388 1.162
## Japan 0.63987 -0.65614 -0.67390 0.14610 0.388603 0.8597 1.085
## Korea -0.16897 0.13509 0.21895 0.00511 -0.169492 -0.4303 0.870
## Luxembourg -0.06827 0.06888 0.04380 -0.02797 0.049134 -0.1401 1.196
## Malta 0.03652 -0.04876 0.00791 -0.08659 0.153014 0.2386 1.128
## Norway 0.00222 -0.00035 -0.00611 -0.01594 -0.001462 -0.0522 1.168
## Netherlands 0.01395 -0.01674 -0.01186 0.00433 0.022591 0.0366 1.229
## New Zealand -0.06002 0.06510 0.09412 -0.02638 -0.064740 0.1469 1.134
## Nicaragua -0.01209 0.01790 0.00972 -0.00474 -0.010467 0.0397 1.174
## Panama 0.02828 -0.05334 0.01446 -0.03467 -0.007889 -0.1775 1.067
## Paraguay -0.23227 0.16416 0.15826 0.14361 0.270478 -0.4655 0.873
## Peru -0.07182 0.14669 0.09148 -0.08585 -0.287184 0.4811 0.831
## Philippines -0.15707 0.22681 0.15743 -0.11140 -0.170674 0.4884 0.818
## Portugal -0.02140 0.02551 -0.00380 0.03991 -0.028011 -0.0690 1.233
## South Africa 0.02218 -0.02030 -0.00672 -0.02049 -0.016326 0.0343 1.195
## South Rhodesia 0.14390 -0.13472 -0.09245 -0.06956 -0.057920 0.1607 1.313
## Spain -0.03035 0.03131 0.00394 0.03512 0.005340 -0.0526 1.208
## Sweden 0.10098 -0.08162 -0.06166 -0.25528 -0.013316 -0.4526 1.086
## Switzerland 0.04323 -0.04649 -0.04364 0.09093 -0.018828 0.1903 1.147
## Turkey -0.01092 -0.01198 0.02645 0.00161 0.025138 -0.1445 1.100
## Tunisia 0.07377 -0.10500 -0.07727 0.04439 0.103058 -0.2177 1.131
## United Kingdom 0.04671 -0.03584 -0.17129 0.12554 0.100314 -0.2722 1.189
## United States 0.06910 -0.07289 0.03745 -0.23312 -0.032729 -0.2510 1.655
## Venezuela -0.05083 0.10080 -0.03366 0.11366 -0.124486 0.3071 1.095
## Zambia 0.16361 -0.07917 -0.33899 0.09406 0.228232 0.7482 0.512
## Jamaica 0.10958 -0.10022 -0.05722 -0.00703 -0.295461 -0.3456 1.200
## Uruguay -0.13403 0.12880 0.02953 0.13132 0.099591 -0.2051 1.187
## Libya 0.55074 -0.48324 -0.37974 -0.01937 -1.024477 -1.1601 2.091
## Malaysia 0.03684 -0.06113 0.03235 -0.04956 -0.072294 -0.2126 1.113
## cook.d hat inf
## Australia 8.04e-04 0.0677
## Austria 8.18e-04 0.1204
## Belgium 7.15e-03 0.0875
## Bolivia 7.28e-04 0.0895
## Brazil 1.40e-02 0.0696
## Canada 3.11e-04 0.1584
## Chile 3.78e-02 0.0373 *
## China 8.16e-03 0.0780
## Colombia 1.88e-03 0.0573
## Costa Rica 3.21e-02 0.0755
## Denmark 2.88e-02 0.0627
## Ecuador 5.82e-03 0.0637
## Finland 4.36e-03 0.0920
## France 1.55e-02 0.1362
## Germany 4.74e-05 0.0874
## Greece 1.59e-02 0.0966
## Guatamala 1.07e-02 0.0605
## Honduras 4.74e-04 0.0601
## Iceland 4.35e-02 0.0705
## India 2.97e-04 0.0715
## Ireland 5.44e-02 0.2122
## Italy 3.92e-03 0.0665
## Japan 1.43e-01 0.2233
## Korea 3.56e-02 0.0608
## Luxembourg 3.99e-03 0.0863
## Malta 1.15e-02 0.0794
## Norway 5.56e-04 0.0479
## Netherlands 2.74e-04 0.0906
## New Zealand 4.38e-03 0.0542
## Nicaragua 3.23e-04 0.0504
## Panama 6.33e-03 0.0390
## Paraguay 4.16e-02 0.0694
## Peru 4.40e-02 0.0650
## Philippines 4.52e-02 0.0643
## Portugal 9.73e-04 0.0971
## South Africa 2.41e-04 0.0651
## South Rhodesia 5.27e-03 0.1608
## Spain 5.66e-04 0.0773
## Sweden 4.06e-02 0.1240
## Switzerland 7.33e-03 0.0736
## Turkey 4.22e-03 0.0396
## Tunisia 9.56e-03 0.0746
## United Kingdom 1.50e-02 0.1165
## United States 1.28e-02 0.3337 *
## Venezuela 1.89e-02 0.0863
## Zambia 9.66e-02 0.0643 *
## Jamaica 2.40e-02 0.1408
## Uruguay 8.53e-03 0.0979
## Libya 2.68e-01 0.5315 *
## Malaysia 9.11e-03 0.0652
which(apply(inflm.SR$is.inf, 1, any))
## Chile United States Zambia Libya
## 7 44 46 49
rstandard(lm.SR)
## Australia Austria Belgium Bolivia Brazil
## 0.23520105 0.17282943 0.61085760 -0.19245030 0.96858807
## Canada Chile China Colombia Costa Rica
## -0.09083873 -2.20907436 0.69453131 -0.39319153 1.40168682
## Denmark Ecuador Finland France Germany
## 1.46686216 -0.65379142 -0.46394723 0.70042898 -0.04974135
## Greece Guatamala Honduras Iceland India
## -0.86217889 -0.91031261 0.19259259 -1.69401854 0.13881900
## Ireland Italy Japan Korea Luxembourg
## 1.00475012 0.52442520 1.57595468 -1.65713877 -0.45967116
## Malta Norway Netherlands New Zealand Nicaragua
## 0.81536209 -0.23495632 0.11735008 0.61802723 0.17443311
## Panama Paraguay Peru Philippines Portugal
## -0.88366877 -1.66987256 1.77851567 1.81461452 -0.21267488
## South Africa South Rhodesia Spain Sweden Switzerland
## 0.13140922 0.37072635 -0.18374340 -1.19700295 0.67944806
## Turkey Tunisia United Kingdom United States Venezuela
## -0.71532499 -0.77031393 -0.75327449 -0.35811077 0.99934066
## Zambia Jamaica Uruguay Libya Malaysia
## 2.65091534 -0.85634746 -0.62681420 -1.08705199 -0.80805950
rstudent(lm.SR)
## Australia Austria Belgium Bolivia Brazil
## 0.23271611 0.17095506 0.60655220 -0.19037831 0.96790816
## Canada Chile China Colombia Costa Rica
## -0.08983197 -2.31342946 0.69048169 -0.38946778 1.41731062
## Denmark Ecuador Finland France Germany
## 1.48644473 -0.64957871 -0.45986445 0.69640933 -0.04918692
## Greece Guatamala Honduras Iceland India
## -0.85967533 -0.90854545 0.19051919 -1.73119989 0.13729730
## Ireland Italy Japan Korea Luxembourg
## 1.00485886 0.52015744 1.60321582 -1.69103214 -0.45560591
## Malta Norway Netherlands New Zealand Nicaragua
## 0.81227407 -0.23247367 0.11605663 0.61373189 0.17254242
## Panama Paraguay Peru Philippines Portugal
## -0.88147653 -1.70488128 1.82391409 1.86382587 -0.21040432
## South Africa South Rhodesia Spain Sweden Switzerland
## 0.12996586 0.36714512 -0.18175853 -1.20293404 0.67532922
## Turkey Tunisia United Kingdom United States Venezuela
## -0.71138840 -0.76677907 -0.74959873 -0.35461507 0.99932569
## Zambia Jamaica Uruguay Libya Malaysia
## 2.85355834 -0.85376418 -0.62253411 -1.08930326 -0.80489153
# dfbetas(lm.SR)
dffits(lm.SR)
## Australia Austria Belgium Bolivia Brazil
## 0.06271756 0.06324405 0.18780542 -0.05967770 0.26464755
## Canada Chile China Colombia Costa Rica
## -0.03897262 -0.45535788 0.20077524 -0.09602160 0.40493458
## Denmark Ecuador Finland France Germany
## 0.38451126 -0.16946909 -0.14641688 0.27653834 -0.01521770
## Greece Guatamala Honduras Iceland India
## -0.28114772 -0.23053977 0.04816829 -0.47676403 0.03808618
## Ireland Italy Japan Korea Luxembourg
## 0.52157524 0.13884474 0.85965081 -0.43025048 -0.14006342
## Malta Norway Netherlands New Zealand Nicaragua
## 0.23855360 -0.05216187 0.03663477 0.14694487 0.03972980
## Panama Paraguay Peru Philippines Portugal
## -0.17751461 -0.46547654 0.48109398 0.48840149 -0.06901872
## South Africa South Rhodesia Spain Sweden Switzerland
## 0.03429664 0.16071740 -0.05261883 -0.45256252 0.19034296
## Turkey Tunisia United Kingdom United States Venezuela
## -0.14453378 -0.21765669 -0.27221843 -0.25095085 0.30708996
## Zambia Jamaica Uruguay Libya Malaysia
## 0.74823509 -0.34555773 -0.20513659 -1.16013341 -0.21262745
covratio(lm.SR)
## Australia Austria Belgium Bolivia Brazil
## 1.1928303 1.2678392 1.1761879 1.2238199 1.0823332
## Canada Chile China Colombia Costa Rica
## 1.3283009 0.6547098 1.1498637 1.1666845 0.9681384
## Denmark Ecuador Finland France Germany
## 0.9344047 1.1393880 1.2031561 1.2262654 1.2256855
## Greece Guatamala Honduras Iceland India
## 1.1396174 1.0852720 1.1855450 0.8658808 1.2024438
## Ireland Italy Japan Korea Luxembourg
## 1.2680432 1.1624611 1.0845999 0.8695843 1.1961844
## Malta Norway Netherlands New Zealand Nicaragua
## 1.1282611 1.1680616 1.2285315 1.1336998 1.1742677
## Panama Paraguay Peru Philippines Portugal
## 1.0667255 0.8732040 0.8312741 0.8177726 1.2331038
## South Africa South Rhodesia Spain Sweden Switzerland
## 1.1945449 1.3130954 1.2081541 1.0864869 1.1471125
## Turkey Tunisia United Kingdom United States Venezuela
## 1.1003557 1.1314365 1.1886236 1.6554816 1.0945955
## Zambia Jamaica Uruguay Libya Malaysia
## 0.5116454 1.1995171 1.1872025 2.0905736 1.1126445
Chapter 6, Problem 15.
First check the following page from R project documentation (for various plots to visualize the influence measures):
https://cran.r-project.org/web/packages/olsrr/vignettes/influence_measures.html
Note: You might need libraries such as olsrr for some of the plots below.
Plot : Cook’s D chart, DFBETAs Panel, DFFITS Plot and Standardized Residual Chart that are shown in the above link.
Find the points with high leverage and Cook’s distance.
Plot “Studentized Residuals vs Leverage Plot” that you see in the above link. Which regions in this plot corresponds to leverage points, pure leverage and influential regions. Detect the points in each region.
What do you think are the most influential points? (You can use the stats shown above or plots in previous parts.)
Comment about the normality assumption using probability plot. Remove the most influential points (that you suggested in part-d) and discuss the change/improvements on normality assumption (comparing probability plots).