Introduction to Linear Regression
Al Haque
4/4/2022
library(tidyverse)## -- Attaching packages --------------------------------------- tidyverse 1.3.1 --## v ggplot2 3.3.5 v purrr 0.3.4
## v tibble 3.1.6 v dplyr 1.0.7
## v tidyr 1.2.0 v stringr 1.4.0
## v readr 2.1.1 v forcats 0.5.1## -- Conflicts ------------------------------------------ tidyverse_conflicts() --
## x dplyr::filter() masks stats::filter()
## x dplyr::lag() masks stats::lag()library(openintro)## Loading required package: airports## Loading required package: cherryblossom## Loading required package: usdatadata('hfi', package='openintro')Exercise 1
What are the dimensions of the dataset?
The dataset contains 1,448 observations with 123 variables.
hfi## # A tibble: 1,458 x 123
## year ISO_code countries region pf_rol_procedur~ pf_rol_civil pf_rol_criminal
## <dbl> <chr> <chr> <chr> <dbl> <dbl> <dbl>
## 1 2016 ALB Albania Easte~ 6.66 4.55 4.67
## 2 2016 DZA Algeria Middl~ NA NA NA
## 3 2016 AGO Angola Sub-S~ NA NA NA
## 4 2016 ARG Argentina Latin~ 7.10 5.79 4.34
## 5 2016 ARM Armenia Cauca~ NA NA NA
## 6 2016 AUS Australia Ocean~ 8.44 7.53 7.36
## 7 2016 AUT Austria Weste~ 8.97 7.87 7.67
## 8 2016 AZE Azerbaij~ Cauca~ NA NA NA
## 9 2016 BHS Bahamas Latin~ 6.93 6.01 6.26
## 10 2016 BHR Bahrain Middl~ NA NA NA
## # ... with 1,448 more rows, and 116 more variables: pf_rol <dbl>,
## # pf_ss_homicide <dbl>, pf_ss_disappearances_disap <dbl>,
## # pf_ss_disappearances_violent <dbl>, pf_ss_disappearances_organized <dbl>,
## # pf_ss_disappearances_fatalities <dbl>, pf_ss_disappearances_injuries <dbl>,
## # pf_ss_disappearances <dbl>, pf_ss_women_fgm <dbl>,
## # pf_ss_women_missing <dbl>, pf_ss_women_inheritance_widows <dbl>,
## # pf_ss_women_inheritance_daughters <dbl>, pf_ss_women_inheritance <dbl>, ...Exercise 2
What type of plot would you use to display the relationship between the personal freedom score, pf_score
Since we are plotting 2 numerical variables it would be best to use a scatter-plot to properly illsurate the relationship. The relationship in the graph looks linear.
hfi$pf_score## [1] 7.596281 5.281772 6.111324 8.099696 6.912804 9.184438 9.246948 5.676553
## [9] 7.454538 6.136070 5.302600 7.706894 6.059028 8.987179 7.430864 7.496976
## [17] 6.600973 7.206770 7.861447 6.876334 6.665977 4.663700 8.155539 7.455340
## [25] 4.414134 7.238448 5.330774 9.151727 7.986583 5.465632 5.509541 8.216035
## [33] 5.350820 7.021513 4.947257 6.777868 8.165429 7.062542 8.456176 8.514394
## [41] 9.029761 9.325640 6.942575 7.550555 3.894554 6.917902 9.013701 5.064090
## [49] 7.792204 9.294368 8.766803 5.315575 5.302685 7.576536 9.235191 7.872178
## [57] 7.946768 6.548246 5.371202 6.859722 7.041796 7.184443 6.386483 8.583680
## [65] 8.258946 9.083506 6.195871 6.381312 4.532449 3.116028 8.939129 7.544916
## [73] 8.690446 7.268416 8.733771 6.238667 6.375676 6.445537 8.766350 5.630574
## [81] 6.247013 5.863586 8.851068 6.429994 6.688444 6.403402 3.880566 8.821339
## [89] 9.257402 7.413658 6.835124 7.445214 5.902980 6.060929 8.977997 4.987750
## [97] 7.717466 6.796020 7.059030 7.998313 7.630104 5.986757 6.659207 5.463361
## [105] 7.394916 6.911611 9.398842 9.284819 6.432495 5.714520 5.823617 9.342481
## [113] 5.509055 5.324592 7.715240 7.254949 6.974598 7.720787 6.497527 8.353120
## [121] 9.043712 5.525553 8.653232 5.714382 6.467739 4.438732 6.774188 7.847943
## [129] 7.372524 7.044659 7.479424 8.536500 8.822791 7.695144 8.758214 6.040722
## [137] 4.246047 7.788563 6.020329 9.334750 9.185518 2.511654 9.041394 5.652325
## [145] 6.126419 6.399563 6.186450 6.729938 6.920446 6.577472 6.092465 6.128889
## [153] 6.586270 5.073350 8.995836 8.747310 8.296142 5.521449 5.968008 2.166555
## [161] 6.007699 5.170726 7.587078 5.335310 6.132958 8.025096 6.871773 9.214745
## [169] 9.379794 5.552377 7.502983 6.118902 5.292209 7.821851 NA 9.185713
## [177] 7.094584 7.581713 6.662791 7.351932 8.003267 6.790132 6.704300 4.609116
## [185] 8.238405 7.367716 4.241065 7.314548 5.127940 9.130112 8.361806 5.619156
## [193] 5.634996 8.265002 5.374700 6.905774 4.819612 6.862956 8.255308 6.952270
## [201] 8.521920 8.669154 9.038707 9.336046 7.023368 7.452677 3.852272 7.119972
## [209] 8.963080 5.156317 7.701291 9.407869 8.658781 5.318685 5.460142 7.776962
## [217] 9.290914 7.711281 8.071835 6.530605 5.560062 6.831116 6.949567 7.210987
## [225] 6.397629 8.745162 8.442994 9.149527 6.204729 6.428120 4.492074 NA
## [233] 9.041050 7.822677 8.717216 7.236812 8.706370 6.293274 6.421995 6.323389
## [241] 8.796927 5.455369 6.211922 5.916014 8.816254 6.117908 6.681331 6.370389
## [249] 3.974236 8.839368 9.329334 7.470899 6.885250 7.456857 5.970135 5.971989
## [257] 9.041407 5.041825 7.781033 6.912513 7.101188 8.038873 7.982973 5.980245
## [265] 6.779523 5.582742 7.331469 6.951082 9.366518 9.273780 6.362252 5.604470
## [273] 5.630883 9.388992 5.592812 5.304687 7.706969 7.194145 6.999922 7.760073
## [281] 6.675529 8.620769 9.041336 5.594931 8.714023 5.686519 6.551657 4.404408
## [289] 6.842267 7.980960 7.746148 7.176240 7.613531 8.604980 8.955761 7.673184
## [297] 8.821933 6.037550 NA 7.750407 6.129929 9.319149 9.258916 2.861653
## [305] 9.053254 5.762052 6.191475 6.453236 5.610710 6.700934 7.039935 6.601278
## [313] 6.473634 6.119589 6.288703 5.100455 9.047408 8.827570 8.390694 5.523500
## [321] 6.014983 2.336884 6.128980 5.155279 7.750166 5.378657 6.107928 7.924290
## [329] 6.915972 9.324684 9.380256 5.823368 7.422161 5.988579 5.494260 7.880064
## [337] NA 9.101087 6.983818 7.562763 6.722924 7.342117 8.042957 6.802452
## [345] 6.959999 4.678174 8.327566 7.314257 4.638075 7.325478 5.543957 9.219105
## [353] 8.006755 5.453477 5.994091 8.195178 5.519759 6.947464 4.827701 6.765420
## [361] 8.617715 6.971078 8.478890 8.701781 9.019314 9.405681 7.291729 7.387582
## [369] 4.301382 7.491165 9.000505 5.179382 7.477389 9.486705 8.934746 5.229237
## [377] 5.777930 7.771882 9.302399 7.870900 8.143171 6.954735 5.368115 6.657558
## [385] 6.945261 7.474011 6.553164 8.784911 8.567299 9.246634 6.279888 6.398716
## [393] 4.231371 NA 8.795600 7.461034 8.703630 7.358998 8.771444 6.298162
## [401] 6.469997 6.211527 8.855441 5.754312 6.254568 5.971988 8.711074 6.175405
## [409] 6.563231 6.457114 4.216496 8.729086 9.362281 7.144729 7.221221 7.408412
## [417] 5.839001 5.853400 8.871254 5.129175 7.844266 6.992210 7.126151 8.097245
## [425] 8.101436 6.086609 6.709491 5.796951 7.258532 7.086550 9.211358 9.362772
## [433] 6.649156 5.878961 5.723368 9.526564 5.705957 5.045778 7.687835 6.951363
## [441] 7.180369 7.565960 6.608041 8.833323 9.029433 5.684795 8.902438 5.972597
## [449] 6.479574 4.723066 6.809732 8.060048 6.650740 7.287357 7.589062 8.656840
## [457] 8.936067 7.499558 8.944538 5.754403 NA 7.263678 5.954273 9.253566
## [465] 9.303864 3.077190 8.785285 5.968551 6.359931 6.567701 6.315424 6.671558
## [473] 7.057789 6.665387 6.642302 6.159892 6.160398 5.278573 9.155805 8.850570
## [481] 8.333961 5.867460 6.020682 2.519069 5.672964 5.305365 7.509193 5.146842
## [489] 6.739678 8.261689 7.103881 9.167331 9.320219 5.972940 7.417698 5.742265
## [497] 5.292128 7.901771 NA 9.041067 7.140413 7.375620 6.819091 7.647877
## [505] 8.012689 6.516080 7.404761 5.940890 8.347520 7.700455 6.357713 7.192898
## [513] 6.222504 9.225504 8.006589 4.890353 5.560528 8.108076 5.737562 6.470821
## [521] 4.455993 6.779646 8.233029 6.680421 8.481888 8.634588 9.105149 9.519350
## [529] 7.111907 7.626139 4.699170 7.207116 8.971011 5.857614 7.284248 9.456231
## [537] 8.769471 6.189146 5.929838 7.758326 9.220923 7.918135 8.093893 6.617232
## [545] 5.882375 6.185571 7.633296 7.118479 5.955376 8.983629 8.557804 9.249103
## [553] 6.965229 6.931741 4.087617 NA 9.015572 7.864782 8.760080 7.166775
## [561] 8.852600 6.120075 6.348148 6.349646 8.666065 6.187585 6.145123 NA
## [569] 8.752911 6.065344 6.436335 NA 4.995930 8.672215 9.188029 7.943612
## [577] 7.068669 7.513179 6.134829 5.974948 8.930327 5.348084 8.343368 6.790791
## [585] 7.453997 8.105190 8.051799 6.243817 7.047283 5.031095 7.376982 7.414771
## [593] 9.186037 9.257169 6.879121 6.216772 5.445457 9.560169 5.793055 4.790842
## [601] 7.718085 6.895944 7.289304 7.453131 6.067960 8.891007 8.929406 5.689483
## [609] 8.615619 6.269067 6.522739 4.361519 6.735443 7.626731 6.812436 6.378484
## [617] 7.296985 8.757185 8.920481 7.203877 8.661795 5.834280 NA 7.285546
## [625] 5.839567 9.271152 9.210524 4.120057 8.678233 6.625175 6.672414 6.880345
## [633] 6.494581 5.939951 7.065859 5.926390 7.315975 6.095985 7.491269 5.236195
## [641] 9.065507 8.634731 8.493607 6.707714 6.248651 3.295410 6.440869 5.395069
## [649] 7.670382 5.244807 6.384360 8.098573 7.368595 9.155873 9.268544 6.001311
## [657] 7.851875 5.848527 5.477917 8.037217 NA 9.033781 7.225155 7.102856
## [665] NA 7.602511 7.874977 6.408080 7.420760 6.590635 8.266662 7.441721
## [673] 6.027682 7.343626 6.130546 9.206788 8.198471 4.728113 5.545431 8.629822
## [681] 5.636779 6.395642 4.421329 6.279521 8.251476 6.270804 8.422989 8.507043
## [689] 9.030405 9.487167 7.032943 7.332286 5.202573 7.196339 8.967745 5.507089
## [697] 7.349026 9.440905 8.752234 6.092342 6.080054 7.378219 9.206540 7.627061
## [705] 8.595543 6.583208 NA 5.950756 7.712620 7.048531 5.877339 9.038171
## [713] 8.528692 9.270262 7.204515 6.921252 3.923565 NA 8.985665 7.770850
## [721] 8.725522 6.980532 8.886950 6.213783 6.375663 6.344946 8.663341 6.097100
## [729] 6.269650 NA 8.723644 6.375842 6.473314 NA NA 8.683015
## [737] 9.169140 8.261328 7.010602 7.604950 6.278362 5.943164 8.844048 5.827101
## [745] 8.582331 6.635227 7.460099 8.023393 8.077698 6.413817 6.742088 5.100395
## [753] 7.147315 7.271136 9.158976 9.099278 6.638626 6.565336 5.227435 9.568154
## [761] 5.805576 4.926955 7.728516 6.275805 7.336954 7.691982 6.900288 8.910093
## [769] 8.949400 5.254456 8.522985 6.303112 6.320318 4.380158 6.431042 7.472898
## [777] NA 6.024225 7.163451 8.758805 8.911970 7.157849 8.683587 5.721443
## [785] NA 7.623914 5.414325 9.448184 9.320210 4.463307 8.522833 6.736704
## [793] 6.540723 7.186706 6.644455 5.501812 6.989158 6.140156 7.189370 5.762716
## [801] 7.613971 4.865616 9.034968 8.622446 8.578156 6.631301 6.291366 3.381012
## [809] 6.577187 5.109619 7.764088 5.362339 5.549902 8.130653 7.410680 9.182501
## [817] 9.176560 6.085698 7.942748 6.101845 5.616966 8.028648 NA 9.071863
## [825] 7.395807 7.173281 NA 7.560832 7.993419 6.544255 7.631088 6.655830
## [833] 8.359973 7.547067 6.113624 7.508288 6.128264 9.265986 8.251562 4.879459
## [841] 5.655279 8.670305 5.738971 6.528019 4.796654 6.300028 8.269709 6.026988
## [849] 8.477837 8.657318 9.019294 9.474775 6.971973 7.476097 5.291870 6.835688
## [857] 8.956664 5.618283 7.355811 9.479371 8.764232 6.159744 6.193399 7.456488
## [865] 9.244978 7.647471 8.727118 6.571394 NA 6.251189 7.851696 7.189368
## [873] 6.074197 9.065782 8.715524 9.303933 7.287817 7.008588 4.054964 NA
## [881] 9.061253 7.935495 8.746320 7.000108 8.896011 6.201937 6.386743 6.553068
## [889] 8.630075 6.226252 6.190304 NA 8.802701 6.687906 6.571596 NA
## [897] NA 8.735453 9.204896 8.351364 7.368192 7.467499 6.485079 6.268739
## [905] 9.004442 5.879231 8.581384 6.763573 7.463974 7.981010 8.149663 6.459090
## [913] 6.880181 5.118472 7.317269 7.480462 9.217586 9.193461 6.737904 6.686819
## [921] 5.340308 9.365398 5.866622 5.087155 7.726443 6.358829 7.268874 7.746133
## [929] 7.026289 9.098542 9.000258 5.232194 8.576851 6.381338 6.349864 4.435566
## [937] 6.458420 7.491859 NA 6.011410 7.211082 8.803042 8.870551 7.243796
## [945] 8.800843 5.948040 NA 7.730232 5.471813 9.442889 9.324332 5.078535
## [953] 8.605639 6.818178 6.675361 7.423516 6.800981 5.613306 7.013735 6.153702
## [961] 7.407545 5.832225 7.701820 4.839355 9.088762 8.712934 8.692857 6.754916
## [969] 6.415083 3.231188 6.640517 5.171471 7.757250 5.208712 5.175390 8.181316
## [977] 7.416263 9.265386 9.023048 6.652655 7.894137 6.321461 5.538410 8.029702
## [985] NA 9.097983 7.379314 6.779762 NA 7.384301 8.204545 6.799652
## [993] 7.768350 6.377936 8.299025 7.584140 6.352196 7.746637 5.698524 9.270230
## [1001] 8.312047 5.387513 5.699936 8.652734 5.563644 6.883967 5.515803 6.665292
## [1009] 8.084008 5.816050 8.148327 8.938169 9.091980 9.488762 6.971664 7.411652
## [1017] 5.538519 7.228299 9.066637 5.347053 7.563436 9.500482 8.784171 6.060273
## [1025] 5.576208 8.136609 9.106744 7.581635 8.619477 7.273042 NA 6.331051
## [1033] 7.194589 7.021834 6.488554 9.036896 8.882872 9.193639 6.574172 7.162843
## [1041] 4.489005 NA 9.069072 8.476937 8.762426 7.224410 8.789731 5.935704
## [1049] 6.910899 6.464775 8.833318 6.244275 6.188584 NA 8.657641 6.453066
## [1057] 5.687221 NA NA 8.664016 9.202756 8.165920 6.822954 7.271030
## [1065] 6.304836 6.526071 9.032154 5.798658 8.506716 6.777854 7.450517 7.579833
## [1073] 8.208571 6.393932 6.399148 6.007950 7.242411 7.347654 9.197257 9.428153
## [1081] 7.400991 7.194329 5.881826 9.561851 5.433485 4.833934 7.683680 7.021329
## [1089] 7.318168 7.599118 7.319929 8.929675 8.999700 4.960796 8.749530 6.493733
## [1097] 6.710543 4.606060 6.460770 7.728383 NA 6.475251 7.445439 8.776370
## [1105] 8.620039 6.798161 9.172400 6.570650 NA 7.852177 5.852716 9.438605
## [1113] 9.336283 5.148592 8.294907 7.535268 6.479170 7.748105 7.487092 5.799787
## [1121] 6.840402 5.687479 7.152000 5.606187 7.696264 5.861769 9.012806 8.711692
## [1129] 8.817575 6.353238 6.679871 4.730448 6.087383 5.358830 7.860267 5.139881
## [1137] 5.273795 8.175462 7.408322 9.290626 9.031761 6.293471 8.000515 6.751321
## [1145] 5.533021 8.178830 NA 9.110864 7.359349 6.872947 NA 7.540680
## [1153] 8.226588 6.838069 7.751098 NA 8.406205 7.432165 6.804418 NA
## [1161] 5.737553 9.267999 NA 5.610116 5.545487 8.628272 5.565873 6.795143
## [1169] 5.608591 6.399806 8.265653 5.899114 8.409353 8.921275 9.153649 9.484657
## [1177] 7.038389 7.379937 5.551570 6.922049 9.057999 5.216424 7.556942 9.510538
## [1185] 8.999746 6.084612 NA 8.145426 9.142769 7.628736 8.766923 7.184947
## [1193] NA 6.461521 7.335984 7.073770 6.556598 9.015306 8.898743 9.245503
## [1201] 6.591647 7.198304 4.353862 NA 9.045519 7.851206 8.792230 7.294271
## [1209] 8.862743 6.019213 6.768123 6.420323 8.753504 6.266073 6.641372 NA
## [1217] 8.451635 NA 5.705277 NA NA 8.692407 8.982547 8.077733
## [1225] 6.819999 7.336294 6.548766 6.330020 9.080232 5.869709 8.360905 7.135360
## [1233] 7.727687 7.561840 8.000304 6.463867 6.243995 6.502495 7.187442 7.567099
## [1241] 9.222489 9.325434 7.308308 7.051861 5.888859 9.420202 5.585305 5.070443
## [1249] 7.787263 6.779836 7.140153 7.573309 7.240387 8.990015 8.860807 NA
## [1257] 8.668488 6.465113 7.574418 NA 6.616525 7.878666 NA 6.374351
## [1265] 7.619521 8.817191 8.632398 6.806230 8.932053 5.914972 NA NA
## [1273] NA 9.441831 9.324517 5.269188 8.236077 NA 6.455916 7.835613
## [1281] NA 5.793766 6.919811 5.820344 7.137345 5.460617 7.771996 5.563103
## [1289] 9.100353 8.571688 8.800819 6.207029 6.680303 NA 6.110227 5.355347
## [1297] 7.777698 5.146131 5.237056 8.190240 7.320561 9.292040 9.027407 6.593344
## [1305] 8.146710 6.780238 5.526272 8.135015 NA 9.064172 7.350668 6.990275
## [1313] NA 7.367289 8.004875 6.821652 7.718818 NA 8.242904 7.568461
## [1321] 6.282285 NA 5.804578 9.266003 NA 5.783164 5.585024 8.654430
## [1329] 5.721288 6.925207 5.543686 6.570636 8.295889 5.893142 8.076770 8.948103
## [1337] 9.131337 9.494348 6.878054 7.416109 5.579720 7.268472 9.039304 5.367709
## [1345] 7.786614 9.497741 8.969965 5.940088 NA 7.542609 9.144099 7.607686
## [1353] 8.771018 7.188651 NA 6.576988 6.954978 7.122098 6.730025 9.062626
## [1361] 8.894809 9.214496 6.569895 7.184828 4.882248 NA 9.059166 7.590547
## [1369] 8.793881 7.286042 8.828608 5.949624 6.868879 6.464595 8.838646 6.157210
## [1377] 6.782040 NA 8.644295 NA 5.837221 NA NA 8.580363
## [1385] 9.108486 8.237437 7.041446 7.273545 6.296281 6.297124 9.035816 5.810717
## [1393] 8.469659 7.187200 7.507970 7.560718 8.149117 6.362837 6.393893 6.050606
## [1401] 7.179960 7.397345 9.241021 9.421085 7.406305 6.954829 6.272717 9.554284
## [1409] 5.854085 4.833701 7.748312 7.058135 7.301651 7.494997 7.391257 8.953865
## [1417] 8.986276 NA 8.737953 6.531378 7.149149 NA 6.527743 7.915282
## [1425] NA 6.490277 7.439441 8.866865 8.641525 6.804749 9.124498 5.460755
## [1433] NA NA NA 9.504694 9.308873 5.295031 8.094654 NA
## [1441] 6.477212 7.748565 NA 5.856729 6.871200 5.818116 6.976818 5.780542
## [1449] 7.727395 5.889732 9.082842 8.726531 8.775693 6.295759 6.650413 NA
## [1457] 6.145449 5.321141plot(x=hfi$pf_score,y=hfi$pf_expression_control,
xlim =c(0,11),
ylim = c(0,11)
)
### Exercise 3 Looking at your plot from the previous exercise, describe the relationship between these two variables. Make sure to discuss the form, direction, and strength of the relationship as well as any unusual observations.
The relationship with the scatterplot indicates a positive correlation since both the x and y values are increasing and it is strongly correlated since the dots are close together especially at around 10.
Exercise 4
Using plot_ss, choose a line that does a good job of minimizing the sum of squares
I ran the plot_ss a bunch of times and I got the a variety of sum of squares but the smallest sum of squares is probably 952
## I had to filter out the NAs for the graph to work properly
hifi <- hfi %>%
select(pf_expression_control,pf_score) %>%
filter(pf_score != "NA" & pf_expression_control != "NA")
## I cant get the graph to print so I saved it as an imageExercise 5
write the equation of the regression line. What does the slope tell us in the context of the relationship between human freedom and the amount of political pressure on media content?
The equation is y_hat = 5.153687 + 0.349862 * pf_expression_control and The slope tells us that the greater the human freedom there is the greater the amount of political pressure on media content.
m1 <- lm(hf_score ~ pf_expression_control, data = hfi)
summary(m1)##
## Call:
## lm(formula = hf_score ~ pf_expression_control, data = hfi)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.6198 -0.4908 0.1031 0.4703 2.2933
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.153687 0.046070 111.87 <2e-16 ***
## pf_expression_control 0.349862 0.008067 43.37 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.667 on 1376 degrees of freedom
## (80 observations deleted due to missingness)
## Multiple R-squared: 0.5775, Adjusted R-squared: 0.5772
## F-statistic: 1881 on 1 and 1376 DF, p-value: < 2.2e-16Exercise 6
what is the residual for this prediction?
The residual may underestimate the pf score so the predicted value is below the expected value. We can expect the residual to be a positive value.
y_hat <- 5.153687 + 0.349862 * 6.7
y_hat## [1] 7.497762Exercise 7
Is there any apparent pattern in the residuals plot? What does this indicate about the linearity of the relationship between the two variables?
It is really difficult to see any linear trends here. This may indicate that there is no relationship between the two variables.
ggplot(data = m1, aes(x = .fitted, y = .resid)) +
geom_point() +
geom_hline(yintercept = 0, linetype = "dashed") +
xlab("Fitted values") +
ylab("Residuals")
Exercise 8
Based on the histogram and the normal probability plot, does the nearly normal residuals condition appear to be met? It is difficult to tell from the histogram but from the probablity plot it does seem to be linear and thus we can say that the conditions are met.
Exercise 9
Based on the residuals vs. fitted plot, does the constant variability condition appear to be met?
Yes there are no crazy points fanning out and the points all seem to stay together.
Exercise 10
Choose another freedom variable and a variable you think would strongly correlate with it.. Produce a scatterplot of the two variables and fit a linear model. At a glance, does there seem to be a linear relationship?
It seems like there is a linear relationship
ggplot(data = hfi, aes(x=pf_religion,y=hf_score)) +
geom_point() +
stat_smooth(method = "lm", se = FALSE)## `geom_smooth()` using formula 'y ~ x'## Warning: Removed 90 rows containing non-finite values (stat_smooth).## Warning: Removed 90 rows containing missing values (geom_point).
### Exercise 11 How does this relationship compare to the relationship between pf_expression_control and pf_score? Use the R2 values from the two model summaries to compare. Does your independent variable seem to predict your dependent one better? Why or why not? This relationship doesnt seem to compare well compared to the other reslationship. The R2 value is approximately 14.92% while the other relationship is 57.75% varability.
m2 <- lm(hf_score ~ pf_religion, data = hfi)
summary(m2)##
## Call:
## lm(formula = hf_score ~ pf_religion, data = hfi)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.10229 -0.58501 -0.04865 0.77466 2.00693
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.7081 0.1502 31.34 <2e-16 ***
## pf_religion 0.2917 0.0188 15.51 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.9377 on 1366 degrees of freedom
## (90 observations deleted due to missingness)
## Multiple R-squared: 0.1497, Adjusted R-squared: 0.1491
## F-statistic: 240.6 on 1 and 1366 DF, p-value: < 2.2e-16summary(m1)##
## Call:
## lm(formula = hf_score ~ pf_expression_control, data = hfi)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.6198 -0.4908 0.1031 0.4703 2.2933
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.153687 0.046070 111.87 <2e-16 ***
## pf_expression_control 0.349862 0.008067 43.37 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.667 on 1376 degrees of freedom
## (80 observations deleted due to missingness)
## Multiple R-squared: 0.5775, Adjusted R-squared: 0.5772
## F-statistic: 1881 on 1 and 1376 DF, p-value: < 2.2e-16Exercise 12
What’s one freedom relationship you were most surprised about and why? Display the model diagnostics for the regression model analyzing this relationship. I found this relationship between government and personal freedom score pretty interesting.
ggplot(data = hfi, aes(x=ef_government,y=pf_score)) +
geom_point() +
stat_smooth(method = "lm", se = FALSE)## `geom_smooth()` using formula 'y ~ x'## Warning: Removed 80 rows containing non-finite values (stat_smooth).## Warning: Removed 80 rows containing missing values (geom_point).
m3 <- lm(pf_score ~ ef_government, data = hfi)
summary(m3)##
## Call:
## lm(formula = pf_score ~ ef_government, data = hfi)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.7533 -0.9556 0.1408 1.1908 2.5672
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 8.74374 0.18774 46.573 <2e-16 ***
## ef_government -0.23892 0.02854 -8.372 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.342 on 1376 degrees of freedom
## (80 observations deleted due to missingness)
## Multiple R-squared: 0.04847, Adjusted R-squared: 0.04778
## F-statistic: 70.1 on 1 and 1376 DF, p-value: < 2.2e-16