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.

ILKER :

In part c Check The regions you mentioned.




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  
## 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



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.

  1. Plot : Cook’s D chart, DFBETAs Panel, DFFITS Plot and Standardized Residual Chart that are shown in the above link.

  2. Find the points with high leverage and Cook’s distance.

  3. 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.

  4. What do you think are the most influential points? (You can use the stats shown above or plots in previous parts.)

  5. 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).

Answer:

library(olsrr)
## 
## Attaching package: 'olsrr'
## The following object is masked from 'package:datasets':
## 
##     rivers
library(MPV)
## 
## Attaching package: 'MPV'
## The following object is masked from 'package:olsrr':
## 
##     cement
## The following object is masked from 'package:datasets':
## 
##     stackloss
data(table.b14)
fitted=lm(y~x1+x2+x3+x4, data=table.b14)


cat ("Part a.")
## Part a.
# Cooks D 
ols_cooksd_chart(fitted)

#   DFBETAs Panel
ols_dfbetas_panel(fitted)

# DFFITS Plot  
ols_dffits_plot(fitted)