data605hw12
Aaron Zalki
4/26/2020
Data Summary
The below dataset contains real-world data from 2008. The variables included:
| variable | description |
|---|---|
Country |
name of the country. |
LifeExp |
average life expectancy for the country in years. |
InfantSurvival |
proportion of those surviving to one year or more. |
Under5Survival |
proportion of those surviving to five years or more. |
TBFree |
proportion of the population without TB. |
PropMD |
proportion of the population who are MDs |
PropRN |
proportion of the population who are RNs. |
PersExp |
mean personal expenditures on healthcare in US dollars at average exchange rate. |
GovtExp |
mean government expenditures per capita on healthcare, US dollars at average exchange rate. |
TotExp |
sum of personal and government expenditures. |
df <- read.csv("/Users/aaronzalki/Desktop/who.csv")
df## Country LifeExp InfantSurvival
## 1 Afghanistan 42 0.835
## 2 Albania 71 0.985
## 3 Algeria 71 0.967
## 4 Andorra 82 0.997
## 5 Angola 41 0.846
## 6 Antigua and Barbuda 73 0.990
## 7 Argentina 75 0.986
## 8 Armenia 69 0.979
## 9 Australia 82 0.995
## 10 Austria 80 0.996
## 11 Azerbaijan 64 0.927
## 12 Bahamas 74 0.987
## 13 Bahrain 75 0.991
## 14 Bangladesh 63 0.948
## 15 Barbados 75 0.989
## 16 Belarus 69 0.994
## 17 Belgium 79 0.996
## 18 Belize 69 0.986
## 19 Benin 55 0.912
## 20 Bhutan 64 0.937
## 21 Bolivia 66 0.950
## 22 Bosnia and Herzegovina 75 0.987
## 23 Botswana 52 0.910
## 24 Brazil 72 0.981
## 25 Brunei Darussalam 77 0.992
## 26 Bulgaria 73 0.990
## 27 Burkina Faso 47 0.878
## 28 Burundi 49 0.891
## 29 Cambodia 62 0.935
## 30 Cameroon 51 0.913
## 31 Canada 81 0.995
## 32 Cape Verde 70 0.975
## 33 Central African Republic 48 0.886
## 34 Chad 46 0.876
## 35 Chile 78 0.992
## 36 China 73 0.980
## 37 Colombia 74 0.983
## 38 Comoros 65 0.949
## 39 Congo 54 0.921
## 40 Cook Islands 73 0.984
## 41 Costa Rica 78 0.989
## 42 C\xf4te d'Ivoire 53 0.910
## 43 Croatia 76 0.995
## 44 Cuba 78 0.995
## 45 Cyprus 80 0.997
## 46 Czech Republic 77 0.997
## 47 Democratic Republic of the Congo 47 0.871
## 48 Denmark 79 0.997
## 49 Djibouti 56 0.914
## 50 Dominica 74 0.987
## 51 Dominican Republic 70 0.975
## 52 Ecuador 73 0.979
## 53 Egypt 68 0.971
## 54 El Salvador 71 0.978
## 55 Equatorial Guinea 46 0.876
## 56 Eritrea 63 0.952
## 57 Estonia 73 0.995
## 58 Ethiopia 56 0.923
## 59 Fiji 69 0.984
## 60 Finland 79 0.997
## 61 France 81 0.996
## 62 Gabon 58 0.940
## 63 Gambia 59 0.916
## 64 Georgia 70 0.972
## 65 Germany 80 0.996
## 66 Ghana 57 0.924
## 67 Greece 80 0.996
## 68 Grenada 68 0.983
## 69 Guatemala 68 0.969
## 70 Guinea 53 0.902
## 71 Guinea-Bissau 48 0.881
## 72 Guyana 64 0.954
## 73 Haiti 61 0.940
## 74 Honduras 70 0.977
## 75 Hungary 73 0.994
## 76 Iceland 81 0.998
## 77 India 63 0.943
## 78 Indonesia 68 0.974
## 79 Iran (Islamic Republic of) 71 0.970
## 80 Iraq 56 0.963
## 81 Ireland 80 0.996
## 82 Israel 81 0.996
## 83 Italy 81 0.997
## 84 Jamaica 72 0.974
## 85 Japan 83 0.997
## 86 Jordan 71 0.979
## 87 Kazakhstan 64 0.974
## 88 Kenya 53 0.921
## 89 Kiribati 65 0.953
## 90 Kuwait 78 0.991
## 91 Kyrgyzstan 66 0.964
## 92 Lao People's Democratic Republic 60 0.941
## 93 Latvia 71 0.992
## 94 Lebanon 70 0.973
## 95 Lesotho 42 0.898
## 96 Liberia 44 0.843
## 97 Libyan Arab Jamahiriya 72 0.983
## 98 Lithuania 71 0.993
## 99 Luxembourg 80 0.997
## 100 Madagascar 59 0.928
## 101 Malawi 50 0.924
## 102 Malaysia 72 0.990
## 103 Maldives 72 0.974
## 104 Mali 46 0.881
## 105 Malta 79 0.995
## 106 Marshall Islands 63 0.950
## 107 Mauritania 58 0.922
## 108 Mauritius 73 0.988
## 109 Mexico 74 0.971
## 110 Micronesia (Federated States of) 69 0.967
## 111 Monaco 82 0.997
## 112 Mongolia 66 0.965
## 113 Montenegro 74 0.991
## 114 Morocco 72 0.966
## 115 Mozambique 50 0.904
## 116 Namibia 61 0.955
## 117 Nauru 61 0.975
## 118 Nepal 62 0.954
## 119 Netherlands 80 0.996
## 120 New Zealand 80 0.995
## 121 Nicaragua 71 0.971
## 122 Niger 42 0.852
## 123 Nigeria 48 0.901
## 124 Niue 70 0.966
## 125 Norway 80 0.997
## 126 Oman 74 0.990
## 127 Pakistan 63 0.922
## 128 Palau 69 0.990
## 129 Panama 76 0.982
## 130 Papua New Guinea 62 0.946
## 131 Paraguay 75 0.981
## 132 Peru 73 0.979
## 133 Philippines 68 0.976
## 134 Poland 75 0.994
## 135 Portugal 79 0.997
## 136 Qatar 77 0.991
## 137 Republic of Korea 79 0.995
## 138 Republic of Moldova 68 0.984
## 139 Romania 73 0.986
## 140 Russian Federation 66 0.990
## 141 Rwanda 52 0.903
## 142 Saint Kitts and Nevis 71 0.983
## 143 Saint Lucia 75 0.988
## 144 Saint Vincent and the Grenadines 70 0.983
## 145 Samoa 68 0.977
## 146 San Marino 82 0.997
## 147 Sao Tome and Principe 61 0.937
## 148 Saudi Arabia 70 0.979
## 149 Senegal 59 0.940
## 150 Serbia 73 0.993
## 151 Seychelles 72 0.988
## 152 Sierra Leone 40 0.841
## 153 Singapore 80 0.997
## 154 Slovakia 74 0.993
## 155 Slovenia 78 0.997
## 156 Solomon Islands 67 0.945
## 157 South Africa 51 0.944
## 158 Spain 81 0.996
## 159 Sri Lanka 72 0.989
## 160 Sudan 60 0.938
## 161 Suriname 68 0.971
## 162 Swaziland 42 0.888
## 163 Sweden 81 0.997
## 164 Switzerland 82 0.996
## 165 Syrian Arab Republic 72 0.988
## 166 Tajikistan 64 0.944
## 167 Thailand 72 0.993
## 168 The former Yugoslav Republic of Macedonia 73 0.985
## 169 Timor-Leste 66 0.953
## 170 Togo 57 0.931
## 171 Tonga 71 0.980
## 172 Trinidad and Tobago 69 0.967
## 173 Tunisia 72 0.981
## 174 Turkey 73 0.976
## 175 Turkmenistan 63 0.955
## 176 Tuvalu 65 0.969
## 177 Uganda 50 0.922
## 178 Ukraine 67 0.980
## 179 United Arab Emirates 78 0.992
## 180 United Kingdom 79 0.995
## 181 United Republic of Tanzania 50 0.926
## 182 United States of America 78 0.993
## 183 Uruguay 75 0.987
## 184 Uzbekistan 68 0.962
## 185 Vanuatu 69 0.970
## 186 Venezuela (Bolivarian Republic of) 74 0.982
## 187 Viet Nam 72 0.985
## 188 Yemen 61 0.925
## 189 Zambia 43 0.898
## 190 Zimbabwe 43 0.945
## Under5Survival TBFree PropMD PropRN PersExp GovtExp TotExp
## 1 0.743 0.99769 0.000228841 0.000572294 20 92 112
## 2 0.983 0.99974 0.001143127 0.004614439 169 3128 3297
## 3 0.962 0.99944 0.001060478 0.002091362 108 5184 5292
## 4 0.996 0.99983 0.003297297 0.003500000 2589 169725 172314
## 5 0.740 0.99656 0.000070400 0.001146162 36 1620 1656
## 6 0.989 0.99991 0.000142857 0.002773810 503 12543 13046
## 7 0.983 0.99952 0.002780191 0.000741044 484 19170 19654
## 8 0.976 0.99920 0.003698671 0.004918937 88 1856 1944
## 9 0.994 0.99993 0.002331953 0.009149391 3181 187616 190797
## 10 0.996 0.99990 0.003610904 0.006458749 3788 189354 193142
## 11 0.911 0.99913 0.003660005 0.008477873 62 780 842
## 12 0.986 0.99960 0.000954128 0.004045872 1224 55783 57007
## 13 0.990 0.99955 0.002679296 0.005967524 710 45784 46494
## 14 0.931 0.99609 0.000274894 0.000253034 12 75 87
## 15 0.988 0.99989 0.001098976 0.003372014 725 24433 25158
## 16 0.992 0.99929 0.004758674 0.012457093 204 11315 11519
## 17 0.995 0.99989 0.004230489 0.014079195 3451 239105 242556
## 18 0.984 0.99944 0.000890071 0.001074468 198 5376 5574
## 19 0.852 0.99865 0.000035500 0.000660845 28 600 628
## 20 0.930 0.99904 0.000080100 0.001124807 52 407 459
## 21 0.939 0.99734 0.001104233 0.001934039 71 2860 2931
## 22 0.985 0.99943 0.001411105 0.004669384 243 6578 6821
## 23 0.876 0.99546 0.000384822 0.002558127 431 19604 20035
## 24 0.980 0.99945 0.001046640 0.003481410 371 13940 14311
## 25 0.991 0.99901 0.001047120 0.005549738 519 30562 31081
## 26 0.988 0.99959 0.000253477 0.004553230 272 11550 11822
## 27 0.796 0.99524 0.000049300 0.000456647 27 304 331
## 28 0.819 0.99286 0.000024500 0.000164933 3 10 13
## 29 0.918 0.99335 0.000144185 0.000783616 29 140 169
## 30 0.851 0.99763 0.000171884 0.001432847 49 784 833
## 31 0.994 0.99996 0.001912607 0.010044633 3430 192800 196230
## 32 0.966 0.99676 0.000445087 0.000789981 114 5394 5508
## 33 0.826 0.99472 0.000077600 0.000378195 13 190 203
## 34 0.791 0.99430 0.000033000 0.000238728 22 234 256
## 35 0.991 0.99984 0.001047677 0.000607349 397 17952 18349
## 36 0.976 0.99799 0.001402082 0.000979500 81 1302 1383
## 37 0.979 0.99941 0.001289806 0.000525484 201 12410 12611
## 38 0.932 0.99914 0.000140587 0.000718826 14 304 318
## 39 0.874 0.99434 0.000204934 0.000995392 31 915 946
## 40 0.981 0.99976 0.001428571 0.005714286 466 27264 27730
## 41 0.988 0.99983 0.001182996 0.000830416 327 15376 15703
## 42 0.873 0.99253 0.000110024 0.000538226 34 315 349
## 43 0.994 0.99936 0.002469271 0.005459175 651 30210 30861
## 44 0.993 0.99990 0.005908139 0.007444750 310 21075 21385
## 45 0.996 0.99994 0.033228132 0.003972813 1350 39399 40749
## 46 0.996 0.99990 0.003591618 0.008942978 868 56137 57005
## 47 0.795 0.99355 0.000096100 0.000474721 5 66 71
## 48 0.996 0.99993 0.003551934 0.009958195 4350 314588 318938
## 49 0.870 0.98700 0.000170940 0.000361416 61 4002 4063
## 50 0.985 0.99984 0.000558824 0.004661765 288 13206 13494
## 51 0.971 0.99882 0.001629745 0.001596672 197 4148 4345
## 52 0.976 0.99805 0.001388805 0.001559309 147 3717 3864
## 53 0.965 0.99969 0.002425640 0.003367945 78 1290 1368
## 54 0.975 0.99936 0.001173913 0.000754658 177 5700 5877
## 55 0.794 0.99596 0.000308468 0.000546371 211 6474 6685
## 56 0.926 0.99782 0.000045800 0.000533887 8 80 88
## 57 0.994 0.99960 0.003294030 0.006900746 516 27393 27909
## 58 0.877 0.99359 0.000023900 0.000191851 6 64 70
## 59 0.982 0.99970 0.000456182 0.001992797 148 5355 5503
## 60 0.997 0.99996 0.003299183 0.008920357 2824 133956 136780
## 61 0.995 0.99989 0.003379700 0.007924442 3819 234850 238669
## 62 0.909 0.99572 0.000301297 0.005170099 276 17220 17496
## 63 0.886 0.99577 0.000093800 0.001131088 15 550 565
## 64 0.968 0.99916 0.004646289 0.004031356 123 1248 1371
## 65 0.995 0.99995 0.003441718 0.008010552 3628 209250 212878
## 66 0.880 0.99621 0.000140821 0.000856528 30 490 520
## 67 0.996 0.99984 0.004994696 0.003596152 2580 65195 67775
## 68 0.980 0.99992 0.000754717 0.003075472 342 6944 7286
## 69 0.959 0.99897 0.000764832 0.003452759 132 2400 2532
## 70 0.839 0.99534 0.000107505 0.000480122 21 66 87
## 71 0.800 0.99687 0.000114216 0.000651276 10 90 100
## 72 0.938 0.99785 0.000495264 0.002351827 60 1400 1460
## 73 0.920 0.99598 0.000206331 0.000088300 28 546 574
## 74 0.973 0.99905 0.000527479 0.001223705 91 2162 2253
## 75 0.993 0.99979 0.003039869 0.009163949 855 40602 41457
## 76 0.997 0.99997 0.003758389 0.009932886 5154 395622 400776
## 77 0.924 0.99701 0.000560733 0.001191281 36 203 239
## 78 0.966 0.99747 0.000128893 0.000786314 26 588 614
## 79 0.965 0.99972 0.000880461 0.001581144 212 7973 8185
## 80 0.953 0.99922 0.000666877 0.001333088 59 2948 3007
## 81 0.996 0.99989 0.002936271 0.019403222 3993 193553 197546
## 82 0.995 0.99994 0.003691336 0.006256828 1533 93748 95281
## 83 0.996 0.99994 0.003657769 0.007137294 2692 140148 142840
## 84 0.968 0.99992 0.000834754 0.001620600 170 4399 4569
## 85 0.996 0.99971 0.002113049 0.009461544 2936 159192 162128
## 86 0.975 0.99994 0.002349450 0.003218537 241 9047 9288
## 87 0.971 0.99858 0.003755648 0.007385268 148 5510 5658
## 88 0.879 0.99666 0.000123273 0.001015320 24 231 255
## 89 0.936 0.99598 0.000212766 0.002765957 118 4578 4696
## 90 0.989 0.99975 0.001741634 0.003576826 687 51940 52627
## 91 0.959 0.99863 0.002416809 0.005861190 28 396 424
## 92 0.925 0.99708 0.000347283 0.000972391 18 84 102
## 93 0.991 0.99940 0.003145478 0.005609436 443 18224 18667
## 94 0.969 0.99988 0.002081381 0.001163995 460 17400 17860
## 95 0.868 0.99487 0.000044600 0.000562907 41 437 478
## 96 0.765 0.99422 0.000028800 0.000289187 10 413 423
## 97 0.982 0.99982 0.001170724 0.004497433 223 13175 13398
## 98 0.991 0.99939 0.003964202 0.007670188 448 19932 20380
## 99 0.996 0.99990 0.002722343 0.009583514 6330 476420 482750
## 100 0.885 0.99585 0.000271465 0.000295475 9 162 171
## 101 0.880 0.99678 0.000019600 0.000535259 19 252 271
## 102 0.988 0.99875 0.000651758 0.001661178 222 6732 6954
## 103 0.970 0.99946 0.001006667 0.002953333 316 8100 8416
## 104 0.783 0.99422 0.000088000 0.000696691 28 434 462
## 105 0.994 0.99995 0.003861728 0.005953086 1235 91776 93011
## 106 0.944 0.99759 0.000413793 0.002620690 294 18876 19170
## 107 0.875 0.99394 0.000102825 0.000621879 17 451 468
## 108 0.985 0.99960 0.001040735 0.003677316 218 4704 4922
## 109 0.965 0.99975 0.001859629 0.000841810 474 16340 16814
## 110 0.959 0.99891 0.000540541 0.002252252 290 5830 6120
## 111 0.996 0.99998 0.005636364 0.014060606 6128 458700 464828
## 112 0.958 0.99809 0.002584261 0.003388100 35 1539 1574
## 113 0.990 0.99951 0.002051581 0.005717138 299 13725 14024
## 114 0.963 0.99921 0.000518296 0.000788513 89 1947 2036
## 115 0.862 0.99376 0.000024500 0.000294836 14 315 329
## 116 0.939 0.99342 0.000292135 0.003001954 165 3888 4053
## 117 0.970 0.99866 0.001000000 0.006300000 567 30200 30767
## 118 0.941 0.99756 0.000194783 0.000427807 16 64 80
## 119 0.995 0.99994 0.003694914 0.014602357 3560 187191 190751
## 120 0.994 0.99991 0.001978261 0.008342512 2403 159960 162363
## 121 0.964 0.99926 0.000369667 0.001059653 75 2183 2258
## 122 0.747 0.99686 0.000021500 0.000205139 9 85 94
## 123 0.809 0.99385 0.000241314 0.001453192 27 392 419
## 124 0.958 0.99915 0.002000000 0.011000000 1082 35211 36293
## 125 0.996 0.99996 0.003753052 0.016133219 5910 380380 386290
## 126 0.989 0.99986 0.001684996 0.003737628 312 18886 19198
## 127 0.903 0.99737 0.000785061 0.000439274 15 105 120
## 128 0.989 0.99949 0.001500000 0.006050000 690 43890 44580
## 129 0.977 0.99957 0.001347628 0.002481144 351 17424 17775
## 130 0.927 0.99487 0.000044300 0.000458078 34 390 424
## 131 0.978 0.99900 0.001056350 0.001705618 92 2006 2098
## 132 0.975 0.99813 0.001080104 0.000620102 125 4453 4578
## 133 0.968 0.99568 0.001047598 0.005574863 37 882 919
## 134 0.993 0.99973 0.001993865 0.005233928 495 21266 21761
## 135 0.996 0.99976 0.003416013 0.004629833 1800 75458 77258
## 136 0.989 0.99927 0.002618758 0.005943971 2186 163680 165866
## 137 0.995 0.99877 0.001561811 0.001915942 973 41715 42688
## 138 0.981 0.99846 0.002909731 0.006790764 58 1504 1562
## 139 0.984 0.99860 0.001925274 0.004212242 250 9504 9754
## 140 0.987 0.99875 0.004288359 0.008478449 277 12483 12760
## 141 0.840 0.99438 0.000045600 0.000385355 19 220 239
## 142 0.981 0.99983 0.000920000 0.003960000 478 9933 10411
## 143 0.986 0.99978 0.004595092 0.002030675 323 5068 5391
## 144 0.980 0.99953 0.000741667 0.003725000 218 6302 6520
## 145 0.972 0.99975 0.000270270 0.001675676 113 2093 2206
## 146 0.997 0.99995 0.035129032 0.070838710 3490 278163 281653
## 147 0.904 0.99748 0.000522581 0.001987097 49 2419 2468
## 148 0.974 0.99938 0.001417208 0.003065729 448 27621 28069
## 149 0.884 0.99496 0.000049200 0.000272283 38 504 542
## 150 0.992 0.99959 0.001987717 0.004287280 212 7956 8168
## 151 0.987 0.99944 0.001406977 0.007372093 557 20502 21059
## 152 0.731 0.99023 0.000029300 0.000437054 8 164 172
## 153 0.997 0.99975 0.001455956 0.004356458 944 30100 31044
## 154 0.992 0.99982 0.003130661 0.006636414 626 26096 26722
## 155 0.996 0.99985 0.002360320 0.007851574 1495 55233 56728
## 156 0.928 0.99806 0.000123967 0.001349174 28 442 470
## 157 0.931 0.99002 0.000721366 0.003820451 437 10920 11357
## 158 0.996 0.99976 0.003082917 0.007494042 2152 118426 120578
## 159 0.987 0.99920 0.000545582 0.001730255 51 360 411
## 160 0.911 0.99581 0.000293924 0.000884557 29 462 491
## 161 0.961 0.99905 0.000419780 0.001512088 209 7326 7535
## 162 0.836 0.98916 0.000150794 0.006021164 146 2256 2402
## 163 0.996 0.99995 0.003215466 0.010685724 3727 255696 259423
## 164 0.995 0.99995 0.003864789 0.010617438 5694 258248 263942
## 165 0.987 0.99960 0.000532873 0.001406018 61 1581 1642
## 166 0.932 0.99702 0.001998042 0.004994729 18 100 118
## 167 0.992 0.99803 0.000353619 0.002718571 98 2079 2177
## 168 0.983 0.99967 0.002547642 0.004338409 224 11060 11284
## 169 0.945 0.99211 0.000070900 0.001611311 45 1053 1098
## 170 0.893 0.99213 0.000035100 0.000302184 18 205 223
## 171 0.976 0.99966 0.000300000 0.003500000 104 1896 2000
## 172 0.962 0.99990 0.000756024 0.002750753 513 3575 4088
## 173 0.977 0.99972 0.001304944 0.002793637 158 4620 4778
## 174 0.974 0.99968 0.001569411 0.002944793 383 18632 19015
## 175 0.949 0.99922 0.002492345 0.004700143 156 4888 5044
## 176 0.962 0.99496 0.001000000 0.005000000 212 8786 8998
## 177 0.866 0.99439 0.000073900 0.000634436 22 78 100
## 178 0.976 0.99886 0.003087140 0.008343407 128 4624 4752
## 179 0.992 0.99976 0.001167608 0.002434087 833 45969 46802
## 180 0.994 0.99988 0.002208504 0.012241060 3064 240120 243184
## 181 0.882 0.99541 0.000020800 0.000336856 17 225 242
## 182 0.992 0.99997 0.002413151 0.008815197 6350 231822 238172
## 183 0.985 0.99969 0.003717802 0.000864605 404 15824 16228
## 184 0.956 0.99855 0.002615322 0.010754309 26 444 470
## 185 0.964 0.99935 0.000135747 0.001628959 67 1056 1123
## 186 0.979 0.99948 0.001765290 0.001029752 247 10528 10775
## 187 0.983 0.99775 0.000521541 0.000717003 37 270 307
## 188 0.900 0.99868 0.000310096 0.000632523 39 448 487
## 189 0.818 0.99432 0.000108071 0.001881840 36 595 631
## 190 0.915 0.99403 0.000157696 0.000707363 21 324 345Problem 1
Provide a scatterplot of LifeExp~TotExp, and run simple linear regression. Do not transform the variables. Provide and interpret the F statistics, \(R^2\), standard error,and p-values only. Discuss whether the assumptions of simple linear regression met.
Linear Model
lm1 = lm(df$LifeExp ~ df$TotExp)
summary(lm1)##
## Call:
## lm(formula = df$LifeExp ~ df$TotExp)
##
## Residuals:
## Min 1Q Median 3Q Max
## -24.764 -4.778 3.154 7.116 13.292
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.475e+01 7.535e-01 85.933 < 2e-16 ***
## df$TotExp 6.297e-05 7.795e-06 8.079 7.71e-14 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 9.371 on 188 degrees of freedom
## Multiple R-squared: 0.2577, Adjusted R-squared: 0.2537
## F-statistic: 65.26 on 1 and 188 DF, p-value: 7.714e-14Although the variables are significant with low p-values, the model is not good because the \(R^2\) is low (~26%). The \(R^2\) value represents the proportion of variability in the response variable that is explained by the explanatory variable. For this model, 26% of the variability in average life expectancy for the country is explained by the sum of personal and government expenditures. The F-statistic is 65 (we would like this to be higher) and this metric is used to compare different models.
Scatterplot
plot(df$LifeExp, df$TotExp)
abline(lm1, col = "green")
Diagnostic Plots
plot(lm1)
Based on the diagnostic plots, we can see that some residuals are very high. The Normal Q-Q plot shows the residuals following a slighly skewed distribution. In the Scale-Location plot, the residuals appear to not be following the model very well, since there are a lot of residuals that do not follow the line. Lastly, the Residuals vs Leverage plot, shows that a few residuals have higher leverage than our other models and some have very high influence.
Problem 2
Raise life expectancy to the 4.6 power (i.e., LifeExp^4.6). Raise total expenditures to the 0.06 power (nearly a log transform, TotExp^.06). Plot LifeExp^4.6 as a function of TotExp^.06, and re-run the simple regression model using the transformed variables. Provide and interpret the F statistics, \(R^2\), standard error, and p-values. Which model is “better?”
#transform data
df2 <- df
df2$LifeExp <- df2$LifeExp^4.6
df2$TotExp <- df2$TotExp^0.06Linear Model
lm2 <-lm(df2$LifeExp ~ df2$TotExp)
summary(lm2)##
## Call:
## lm(formula = df2$LifeExp ~ df2$TotExp)
##
## Residuals:
## Min 1Q Median 3Q Max
## -308616089 -53978977 13697187 59139231 211951764
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -736527910 46817945 -15.73 <2e-16 ***
## df2$TotExp 620060216 27518940 22.53 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 90490000 on 188 degrees of freedom
## Multiple R-squared: 0.7298, Adjusted R-squared: 0.7283
## F-statistic: 507.7 on 1 and 188 DF, p-value: < 2.2e-16The model has improved significantly. The low p-values demonstrate that the variables are still statistically significant.The \(R^2\) is much higher with an explained variance of 0.7298. The F-statistic (507.7) is much higher than the previous model, which is better for rejecting the null hypothesis that there is no relationship between the variables LifeExp and TotExp.
Scatterplot
plot(df2$LifeExp, df2$TotExp)
abline(lm2)
Diagnostic Plots
plot(lm2)
Overall the residuals are more normal than the original model. The Normal Q-Q plot shows that the residuals follow a less skewed distribution. In the Scale-Location plot, the residuals appear to be more evenly distributed. In the Residuals vs Leverage plot, no residuals have a very high leverage/influence.
Problem 3
Using the results from 2, forecast life expectancy when TotExp^.06 =1.5. Then forecast life expectancy when TotExp^.06=2.5.
\[ \hat{y} = -736527910 + 620060216 * {expenditures} \]
When TotExp^.06 =1.5, life expectancy is:
x_1.5 <- 1.5
y <- -736527910 + 620060216 * {x_1.5}
ylog <- y^(1/4.6)
ylog## [1] 63.31153When TotExp^.06 =2.5, life expectancy is:
x_2.5 <- 2.5
y2 <- -736527910 + 620060216 * {x_2.5}
ylog2 <- y2^(1/4.6)
ylog2## [1] 86.50645Problem 4
Build the following multiple regression model and interpret the F Statistics, \(R^2\), standard error, and p-values. How good is the model?
LifeExp = b0+b1 x PropMd + b2 x TotExp +b3 x PropMD x TotExp
Multiple Regression Model
df$md_exp <- df$PropMD * df$TotExp
lm4 = lm(df$LifeExp ~ df$PropMD + df$TotExp + df$md_exp, data = df)
summary(lm4)##
## Call:
## lm(formula = df$LifeExp ~ df$PropMD + df$TotExp + df$md_exp,
## data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -27.320 -4.132 2.098 6.540 13.074
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.277e+01 7.956e-01 78.899 < 2e-16 ***
## df$PropMD 1.497e+03 2.788e+02 5.371 2.32e-07 ***
## df$TotExp 7.233e-05 8.982e-06 8.053 9.39e-14 ***
## df$md_exp -6.026e-03 1.472e-03 -4.093 6.35e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 8.765 on 186 degrees of freedom
## Multiple R-squared: 0.3574, Adjusted R-squared: 0.3471
## F-statistic: 34.49 on 3 and 186 DF, p-value: < 2.2e-16The p-values demonstrate that all variables are statistically significant. The standard error is only high for PropMD (proportion of the population who are MDs). Similar to Problem 1, the \(R^2\) is low with an explained variance of .3574. The F-statistic is 34 (we would like this to be higher). I think this is lower compared to the previous two models because this metric penalizes the user for utilizing more variables.
Problem 5
Forecast LifeExp when PropMD=.03 and TotExp = 14. Does this forecast seem realistic? Why or why not?
Coefficients
options(scipen=999)
#display coefficients without scientific notation
coef(lm4)## (Intercept) df$PropMD df$TotExp df$md_exp
## 62.77270325541 1497.49395251893 0.00007233324 -0.00602568644\[ \hat{y} = 62.77 + 1497.49 * {MDproportion} + 0.00007 *expenditures -0.006*{MDproportion}*expenditures \]
md <- .03
exp <- 14
y5 <- 62.77 + 1497.49 * md + 0.00007 *exp -0.006*md*exp
y5## [1] 107.6932The forecast does not seem realistic. The prediction of 107 years old is very high for most countries. Therefore, this model must not be good at analyzing life expectancy.