Savings Data Analysis
This is a homework assignment from Math 50 at Dartmouth College.
An example from Regression Diagnostics: Identifying Influential Data and Sources of Collinearity (Belsley, Kuh and Welsch)
[,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 cook.d
## Chile -0.20 0.13 0.22 -0.02 0.12 -0.46 0.65_* 0.04
## United States 0.07 -0.07 0.04 -0.23 -0.03 -0.25 1.66_* 0.01
## Zambia 0.16 -0.08 -0.34 0.09 0.23 0.75 0.51_* 0.10
## Libya 0.55 -0.48 -0.38 -0.02 -1.02_* -1.16_* 2.09_* 0.27
## hat
## Chile 0.04
## United States 0.33_*
## Zambia 0.06
## Libya 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
Question-1
- 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.
Part (a)
- Plot : Cook’s D chart, DFBETAs Panel, DFFITS Plot and Standardized Residual Chart that are shown in the above link.
library(olsrr)
## Attaching package: 'olsrr'
## The following object is masked from 'package:datasets':
##
## rivers
library(MPV)
## Loading required package: lattice
## Loading required package: KernSmooth
## KernSmooth 2.23 loaded
## Copyright M. P. Wand 1997-2009
##
## Attaching package: 'MPV'
## The following object is masked from 'package:olsrr':
##
## cement
data(table.b14)
y = table.b14$y
x1 = table.b14$x1
x2 = table.b14$x2
x3 = table.b14$x3
x4 = table.b14$x4
model = lm(y ~ x1 + x2+ x3+x4)
ols_cooksd_chart(model)
## Warning: 'ols_cooksd_chart' is deprecated.
## Use 'ols_plot_cooksd_chart()' instead.
## See help("Deprecated")
ols_dfbetas_panel(model)
## Warning: 'ols_dfbetas_panel' is deprecated.
## Use 'ols_plot_dfbetas()' instead.
## See help("Deprecated")
ols_dffits_plot(model)
## Warning: 'ols_dffits_plot' is deprecated.
## Use 'ols_plot_dffits()' instead.
## See help("Deprecated")
ols_srsd_chart(model)
## Warning: 'ols_srsd_chart' is deprecated.
## Use 'ols_plot_resid_stand()' instead.
## See help("Deprecated")
hatVals = hatvalues(model)
meanHatVal = mean(hatVals)
hatVals > 2 * meanHatVal
## 1 2 3 4 5 6 7 8 9 10 11 12 13
## FALSE TRUE FALSE TRUE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE
## 14 15 16 17 18 19 20 21 22 23 24 25
## FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
Part (b)
- Find the points with high leverage and Cook’s distance.
Observations 2, 4, 8, and 9 have large cook’s distances, however the cook’s distances for observations 2 and 4 are far higher than those of 8 and 9. This means that observations 2 and 4 are more likely influential, as they have considerable influence on the least-squares estimates (upon removal from the data set each would cause a large difference in squared distance between original β̂ and β̂i. Furthermore, only the hat values for observations 2, 4, and 9 exceed twice the mean hat value, indicating those as points with high leverage.
Part (c)
- 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.
ols_rsdlev_plot(model)
## Warning: 'ols_rsdlev_plot' is deprecated.
## Use 'ols_plot_resid_lev()' instead.
## See help("Deprecated")
Based on the Studentized Residuals vs. Leverage Plot, the middle-right region contains pure leverage point. Here, residuals are small and leverage is high, so there is remoteness in x but consistency with the model’s predicted values. In general, influential points lie in the top-right and bottom-right regions because they have high leverage but are NOT consistent with the model’s predicted values, ie. high-magnitude standard residuals (observation 4). The top-left and bottom-left regions describe contain points with low leverage and inconsistency with model, or simply outliers (observations 2 and 8, although observation 2 is fairly close to being considered influential and observation 8 is close to not being an outlier)
Part (d)
- What do you think are the most influential points? (You can use the stats shown above or plots in previous parts.)
Based on the considerably large cook’s distances and high leverages for observations 2 and 4 from Parts (a/b), as well as observation 4’s lying in the influential region and observation 2’s relative closeness to the influential region from Part (c), I think observations 2 and 4 are influential points.”
Part (e)
- 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).
rStuRes = rstudent(model)
qqnorm(rStuRes, datax = TRUE, main="Normal Probability Plot")
qqline(rStuRes, datax = TRUE)
yNew = y[-4]
x1New = x1[-4]
x2New = x2[-4]
x3New = x3[-4]
x4New = x4[-4]
yNew = yNew[-2]
x1New = x1New[-2]
x2New = x2New[-2]
x3New = x3New[-2]
x4New = x4New[-2]
newModel = lm(yNew ~ x1New + x2New + x3New + x4New)
rStuRes = rstudent(newModel)
qqnorm(rStuRes, datax = TRUE, main="Normal Probability Plot w/out Influential Points")
qqline(rStuRes, datax = TRUE)
Initially the normality assumption is moderately fulfilled as most of the points follow the line in the Normal Probability Plot, however there is clearly an outlier to the left of the data. Upon removal of the influential points from part (d), the normality assumption is even better fulfilled and it is safer to conclude that the assumption has been met.