Homework 7
Ben Jaffe
4/15/2021
Problem 2: NFL Data
Consider the first three variables, listed below, in the data concerning the performance of the 28 National Footbal Leagues in 1976.
Y = games won (per 14-game season)
x1 = rushing yards (per 14-game season)
x2 = passing yards (per 14-game season)
library(tidyverse)## -- Attaching packages --------------------------------------- tidyverse 1.3.0 --## v ggplot2 3.3.3 v purrr 0.3.4
## v tibble 3.0.6 v dplyr 1.0.3
## v tidyr 1.1.2 v stringr 1.4.0
## v readr 1.4.0 v forcats 0.5.1## -- Conflicts ------------------------------------------ tidyverse_conflicts() --
## x dplyr::filter() masks stats::filter()
## x dplyr::lag() masks stats::lag()library(dplyr)
library(tidyr)
# Load Data ( make sure URL is on one line)
nfl<-read.csv("https://raw.githubusercontent.com/kitadasmalley/sp21_MATH376LMT/main/data/nlf1976.csv", header = TRUE)b) Make a 3D scatterplot with the least squares fitted plane.
library(scatterplot3d)
nfl3d<-scatterplot3d(x=nfl$x1, y=nfl$x2, z=nfl$y,
pch=16, angle = 60)
nfl3d$plane3d(mlr)
c) Report the hat matrix H and I-H. Verify whether H(I-H) = 0. (It’s okay to round)
response_matrix<-nfl$y
design_matrix<-matrix(c(rep(1, dim(nfl)[1]), nfl$x1, nfl$x2), nrow=dim(nfl)[1])
hat_matrix<-design_matrix%*%solve(t(design_matrix)%*%design_matrix)%*%t(design_matrix)
I<-diag(dim(nfl)[1])
hat_matrix## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.03872158 0.020353402 0.043703757 0.0192077090 0.045201201
## [2,] 0.02035340 0.116329612 -0.024537978 0.1156249931 -0.032478890
## [3,] 0.04370376 -0.024537978 0.232453827 0.0316058863 0.239069414
## [4,] 0.01920771 0.115624993 0.031605886 0.1353163755 0.023983866
## [5,] 0.04520120 -0.032478890 0.239069414 0.0239838658 0.246470264
## [6,] 0.01873486 0.117474169 0.035459619 0.1390688923 0.027678934
## [7,] 0.03106173 0.049701759 0.111613292 0.0812375029 0.110622830
## [8,] 0.02280440 0.100527261 0.011609002 0.1088826723 0.005333847
## [9,] 0.05093199 -0.031932552 -0.014879599 -0.0684251661 -0.008669878
## [10,] 0.04168949 -0.005235669 0.145735253 0.0242365221 0.150125209
## [11,] 0.04932543 -0.039373068 0.122095665 -0.0276228687 0.129627791
## [12,] 0.03436545 0.042603165 0.032138186 0.0431195135 0.031466660
## [13,] 0.03350357 0.042678654 0.068907645 0.0566950850 0.068393874
## [14,] 0.03414575 0.047820574 -0.005413327 0.0359541548 -0.006749741
## [15,] 0.03541174 0.034872312 0.056461418 0.0422550755 0.056634598
## [16,] 0.04423755 0.001353802 -0.024467813 -0.0298242231 -0.021468335
## [17,] 0.03707341 0.042982217 -0.088951139 -0.0007934651 -0.090201904
## [18,] 0.01882873 0.127095972 -0.055476026 0.1179050949 -0.064580147
## [19,] 0.03273391 0.050693393 0.030001576 0.0525724981 0.028550672
## [20,] 0.03886945 0.027430513 -0.026607252 0.0023854029 -0.026098599
## [21,] 0.04286926 0.004097741 0.010217971 -0.0136362497 0.013111951
## [22,] 0.04679578 -0.016453301 0.025116485 -0.0341768298 0.030032749
## [23,] 0.04915999 -0.031976317 0.062513375 -0.0401074883 0.069074423
## [24,] 0.02068338 0.107748522 0.038586033 0.1279095400 0.031744779
## [25,] 0.02962772 0.068299495 0.006041361 0.0660648345 0.002807738
## [26,] 0.04595712 -0.024186192 0.131363766 -0.0050096556 0.137492419
## [27,] 0.02489412 0.106830569 -0.136094852 0.0627082956 -0.143632332
## [28,] 0.04910753 -0.018344158 -0.058265546 -0.0671379676 -0.053543393
## [,6] [,7] [,8] [,9] [,10]
## [1,] 0.0187348617 0.031061733 0.022804403 0.050931992 0.041689490
## [2,] 0.1174741692 0.049701759 0.100527261 -0.031932552 -0.005235669
## [3,] 0.0354596193 0.111613292 0.011609002 -0.014879599 0.145735253
## [4,] 0.1390688923 0.081237503 0.108882672 -0.068425166 0.024236522
## [5,] 0.0276789345 0.110622830 0.005333847 -0.008669878 0.150125209
## [6,] 0.1430445283 0.084538641 0.111209280 -0.073467951 0.026038023
## [7,] 0.0845386412 0.087294587 0.060812388 -0.033600806 0.073812964
## [8,] 0.1112092800 0.060812388 0.091726441 -0.034898966 0.015530835
## [9,] -0.0734679507 -0.033600806 -0.034898966 0.159924906 0.017125304
## [10,] 0.0260380233 0.073812964 0.015530835 0.017125304 0.098087236
## [11,] -0.0283008612 0.039910749 -0.019800002 0.078545482 0.089696977
## [12,] 0.0433318100 0.037804756 0.041505079 0.028884996 0.033038220
## [13,] 0.0581800753 0.058779592 0.047473242 0.004267272 0.052277327
## [14,] 0.0351247067 0.019554897 0.040301434 0.046945037 0.012771943
## [15,] 0.0429300839 0.047019513 0.038264586 0.023006848 0.046673180
## [16,] -0.0335757708 -0.019121984 -0.005684668 0.121798272 0.008220949
## [17,] -0.0047418214 -0.030893728 0.022430574 0.109116712 -0.030393342
## [18,] 0.1192111938 0.038539857 0.105511056 -0.025703770 -0.022687627
## [19,] 0.0531053175 0.041434138 0.048636685 0.019489121 0.030975471
## [20,] -0.0001691798 -0.004721377 0.018063732 0.088347759 0.004057351
## [21,] -0.0160569747 0.002265041 0.002414552 0.095000886 0.026057843
## [22,] -0.0370848853 -0.001563397 -0.014182062 0.112553574 0.036255555
## [23,] -0.0424813615 0.010426001 -0.022524305 0.108380154 0.057643753
## [24,] 0.1315191147 0.080493149 0.102725533 -0.062544710 0.028809795
## [25,] 0.0666287237 0.038340702 0.061058899 0.011917293 0.016377910
## [26,] -0.0046276745 0.054282302 -0.004282485 0.052053680 0.092776002
## [27,] 0.0602444183 -0.019487130 0.073855800 0.055154947 -0.062524261
## [28,] -0.0730159137 -0.050157972 -0.029304812 0.170679162 -0.007172212
## [,11] [,12] [,13] [,14] [,15] [,16]
## [1,] 0.049325430 0.03436545 0.033503566 0.034145749 0.03541174 0.044237546
## [2,] -0.039373068 0.04260317 0.042678654 0.047820574 0.03487231 0.001353802
## [3,] 0.122095665 0.03213819 0.068907645 -0.005413327 0.05646142 -0.024467813
## [4,] -0.027622869 0.04311951 0.056695085 0.035954155 0.04225508 -0.029824223
## [5,] 0.129627791 0.03146666 0.068393874 -0.006749741 0.05663460 -0.021468335
## [6,] -0.028300861 0.04333181 0.058180075 0.035124707 0.04293008 -0.033575771
## [7,] 0.039910749 0.03780476 0.058779592 0.019554897 0.04701951 -0.019121984
## [8,] -0.019800002 0.04150508 0.047473242 0.040301434 0.03826459 -0.005684668
## [9,] 0.078545482 0.02888500 0.004267272 0.046945037 0.02300685 0.121798272
## [10,] 0.089696977 0.03303822 0.052277327 0.012771943 0.04667318 0.008220949
## [11,] 0.111688805 0.02961149 0.036886792 0.018039332 0.04051140 0.050591555
## [12,] 0.029611493 0.03631927 0.036707479 0.036416260 0.03585089 0.031887946
## [13,] 0.036886792 0.03670748 0.046035016 0.028640538 0.04073358 0.011012984
## [14,] 0.018039332 0.03641626 0.028640538 0.044315698 0.03133380 0.048710568
## [15,] 0.040511402 0.03585089 0.040733583 0.031333804 0.03839091 0.024686824
## [16,] 0.050591555 0.03188795 0.011012984 0.048710568 0.02468682 0.098956750
## [17,] 0.005161411 0.03509932 0.006162956 0.062018358 0.01982684 0.100100422
## [18,] -0.053967237 0.04328589 0.037814753 0.054277886 0.03172426 0.009720901
## [19,] 0.022855965 0.03705114 0.038396073 0.036804391 0.03628491 0.026226641
## [20,] 0.029763847 0.03429616 0.017657325 0.048957748 0.02671531 0.078022111
## [21,] 0.055363580 0.03250326 0.020548455 0.041354452 0.02952981 0.076994942
## [22,] 0.074413351 0.03074231 0.018656255 0.038365801 0.02968326 0.086015526
## [23,] 0.094017199 0.02968322 0.023844184 0.030580162 0.03326605 0.078032406
## [24,] -0.020068438 0.04245777 0.056291345 0.034540146 0.04248231 -0.027085805
## [25,] 0.004302624 0.03844368 0.037183135 0.041732370 0.03466866 0.024835794
## [26,] 0.101656141 0.03112301 0.043417633 0.015960018 0.04308705 0.032448594
## [27,] -0.054193657 0.04056137 0.011824827 0.071484715 0.01923648 0.072560006
## [28,] 0.059260544 0.02970169 -0.002969666 0.056012324 0.01845831 0.134814059
## [,17] [,18] [,19] [,20] [,21]
## [1,] 0.0370734074 0.018828731 0.03273391 0.0388694512 0.042869258
## [2,] 0.0429822165 0.127095972 0.05069339 0.0274305131 0.004097741
## [3,] -0.0889511386 -0.055476026 0.03000158 -0.0266072517 0.010217971
## [4,] -0.0007934651 0.117905095 0.05257250 0.0023854029 -0.013636250
## [5,] -0.0902019043 -0.064580147 0.02855067 -0.0260985986 0.013111951
## [6,] -0.0047418214 0.119211194 0.05310532 -0.0001691798 -0.016056975
## [7,] -0.0308937277 0.038539857 0.04143414 -0.0047213775 0.002265041
## [8,] 0.0224305740 0.105511056 0.04863669 0.0180637318 0.002414552
## [9,] 0.1091167118 -0.025703770 0.01948912 0.0883477595 0.095000886
## [10,] -0.0303933419 -0.022687627 0.03097547 0.0040573508 0.026057843
## [11,] 0.0051614107 -0.053967237 0.02285596 0.0297638473 0.055363580
## [12,] 0.0350993153 0.043285887 0.03705114 0.0342961623 0.032503259
## [13,] 0.0061629563 0.037814753 0.03839607 0.0176573254 0.020548455
## [14,] 0.0620183578 0.054277886 0.03680439 0.0489577484 0.041354452
## [15,] 0.0198268372 0.031724259 0.03628491 0.0267153091 0.029529810
## [16,] 0.1001004216 0.009720901 0.02622664 0.0780221114 0.076994942
## [17,] 0.1300805346 0.061944136 0.03275560 0.0888148916 0.071474795
## [18,] 0.0619441358 0.142754289 0.05186659 0.0366621903 0.007288065
## [19,] 0.0327555954 0.051866589 0.03869570 0.0316989973 0.028064168
## [20,] 0.0888148916 0.036662190 0.03169900 0.0675068637 0.061149386
## [21,] 0.0714747952 0.007288065 0.02806417 0.0611493859 0.064022505
## [22,] 0.0699846601 -0.015937990 0.02422891 0.0634695329 0.072231846
## [23,] 0.0483117525 -0.037426368 0.02227682 0.0534152853 0.069966743
## [24,] -0.0023627988 0.108811743 0.05114848 0.0027010431 -0.010901193
## [25,] 0.0428321544 0.073453853 0.04157672 0.0347678834 0.024656985
## [26,] -0.0096747090 -0.039863322 0.02642301 0.0188915364 0.042730507
## [27,] 0.1353834635 0.134232541 0.04464008 0.0819935313 0.046247299
## [28,] 0.1364587151 -0.005286509 0.02081303 0.1019585532 0.100432380
## [,22] [,23] [,24] [,25] [,26]
## [1,] 0.046795785 0.049159990 0.020683375 0.029627722 0.045957125
## [2,] -0.016453301 -0.031976317 0.107748522 0.068299495 -0.024186192
## [3,] 0.025116485 0.062513375 0.038586033 0.006041361 0.131363766
## [4,] -0.034176830 -0.040107488 0.127909540 0.066064835 -0.005009656
## [5,] 0.030032749 0.069074423 0.031744779 0.002807738 0.137492419
## [6,] -0.037084885 -0.042481362 0.131519115 0.066628724 -0.004627675
## [7,] -0.001563397 0.010426001 0.080493149 0.038340702 0.054282302
## [8,] -0.014182062 -0.022524305 0.102725533 0.061058899 -0.004282485
## [9,] 0.112553574 0.108380154 -0.062544710 0.011917293 0.052053680
## [10,] 0.036255555 0.057643753 0.028809795 0.016377910 0.092776002
## [11,] 0.074413351 0.094017199 -0.020068438 0.004302624 0.101656141
## [12,] 0.030742312 0.029683218 0.042457773 0.038443677 0.031123015
## [13,] 0.018656255 0.023844184 0.056291345 0.037183135 0.043417633
## [14,] 0.038365801 0.030580162 0.034540146 0.041732370 0.015960018
## [15,] 0.029683262 0.033266054 0.042482314 0.034668656 0.043087051
## [16,] 0.086015526 0.078032406 -0.027085805 0.024835794 0.032448594
## [17,] 0.069984660 0.048311752 -0.002362799 0.042832154 -0.009674709
## [18,] -0.015937990 -0.037426368 0.108811743 0.073453853 -0.039863322
## [19,] 0.024228909 0.022276821 0.051148484 0.041576717 0.026423007
## [20,] 0.063469533 0.053415285 0.002701043 0.034767883 0.018891536
## [21,] 0.072231846 0.069966743 -0.010901193 0.024656985 0.042730507
## [22,] 0.085681611 0.087295290 -0.029437172 0.016366305 0.057846015
## [23,] 0.087295290 0.096283976 -0.033665279 0.009187479 0.078031381
## [24,] -0.029437172 -0.033665279 0.121280149 0.062842789 0.001209535
## [25,] 0.016366305 0.009187479 0.062842789 0.049071888 0.009718390
## [26,] 0.057846015 0.078031381 0.001209535 0.009718390 0.097286845
## [27,] 0.028478725 -0.005390526 0.054904854 0.068619827 -0.056923180
## [28,] 0.114622088 0.102182001 -0.062824620 0.018574793 0.030812257
## [,27] [,28]
## [1,] 0.024894119 0.049107526
## [2,] 0.106830569 -0.018344158
## [3,] -0.136094852 -0.058265546
## [4,] 0.062708296 -0.067137968
## [5,] -0.143632332 -0.053543393
## [6,] 0.060244418 -0.073015914
## [7,] -0.019487130 -0.050157972
## [8,] 0.073855800 -0.029304812
## [9,] 0.055154947 0.170679162
## [10,] -0.062524261 -0.007172212
## [11,] -0.054193657 0.059260544
## [12,] 0.040561374 0.029701695
## [13,] 0.011824827 -0.002969666
## [14,] 0.071484715 0.056012324
## [15,] 0.019236478 0.018458305
## [16,] 0.072560006 0.134814059
## [17,] 0.135383464 0.136458715
## [18,] 0.134232541 -0.005286509
## [19,] 0.044640084 0.020813034
## [20,] 0.081993531 0.101958553
## [21,] 0.046247299 0.100432380
## [22,] 0.028478725 0.114622088
## [23,] -0.005390526 0.102182001
## [24,] 0.054904854 -0.062824620
## [25,] 0.068619827 0.018574793
## [26,] -0.056923180 0.030812257
## [27,] 0.191257285 0.093132778
## [28,] 0.093132778 0.191002555I-hat_matrix## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.96127842 -0.020353402 -0.043703757 -0.0192077090 -0.045201201
## [2,] -0.02035340 0.883670388 0.024537978 -0.1156249931 0.032478890
## [3,] -0.04370376 0.024537978 0.767546173 -0.0316058863 -0.239069414
## [4,] -0.01920771 -0.115624993 -0.031605886 0.8646836245 -0.023983866
## [5,] -0.04520120 0.032478890 -0.239069414 -0.0239838658 0.753529736
## [6,] -0.01873486 -0.117474169 -0.035459619 -0.1390688923 -0.027678934
## [7,] -0.03106173 -0.049701759 -0.111613292 -0.0812375029 -0.110622830
## [8,] -0.02280440 -0.100527261 -0.011609002 -0.1088826723 -0.005333847
## [9,] -0.05093199 0.031932552 0.014879599 0.0684251661 0.008669878
## [10,] -0.04168949 0.005235669 -0.145735253 -0.0242365221 -0.150125209
## [11,] -0.04932543 0.039373068 -0.122095665 0.0276228687 -0.129627791
## [12,] -0.03436545 -0.042603165 -0.032138186 -0.0431195135 -0.031466660
## [13,] -0.03350357 -0.042678654 -0.068907645 -0.0566950850 -0.068393874
## [14,] -0.03414575 -0.047820574 0.005413327 -0.0359541548 0.006749741
## [15,] -0.03541174 -0.034872312 -0.056461418 -0.0422550755 -0.056634598
## [16,] -0.04423755 -0.001353802 0.024467813 0.0298242231 0.021468335
## [17,] -0.03707341 -0.042982217 0.088951139 0.0007934651 0.090201904
## [18,] -0.01882873 -0.127095972 0.055476026 -0.1179050949 0.064580147
## [19,] -0.03273391 -0.050693393 -0.030001576 -0.0525724981 -0.028550672
## [20,] -0.03886945 -0.027430513 0.026607252 -0.0023854029 0.026098599
## [21,] -0.04286926 -0.004097741 -0.010217971 0.0136362497 -0.013111951
## [22,] -0.04679578 0.016453301 -0.025116485 0.0341768298 -0.030032749
## [23,] -0.04915999 0.031976317 -0.062513375 0.0401074883 -0.069074423
## [24,] -0.02068338 -0.107748522 -0.038586033 -0.1279095400 -0.031744779
## [25,] -0.02962772 -0.068299495 -0.006041361 -0.0660648345 -0.002807738
## [26,] -0.04595712 0.024186192 -0.131363766 0.0050096556 -0.137492419
## [27,] -0.02489412 -0.106830569 0.136094852 -0.0627082956 0.143632332
## [28,] -0.04910753 0.018344158 0.058265546 0.0671379676 0.053543393
## [,6] [,7] [,8] [,9] [,10]
## [1,] -0.0187348617 -0.031061733 -0.022804403 -0.050931992 -0.041689490
## [2,] -0.1174741692 -0.049701759 -0.100527261 0.031932552 0.005235669
## [3,] -0.0354596193 -0.111613292 -0.011609002 0.014879599 -0.145735253
## [4,] -0.1390688923 -0.081237503 -0.108882672 0.068425166 -0.024236522
## [5,] -0.0276789345 -0.110622830 -0.005333847 0.008669878 -0.150125209
## [6,] 0.8569554717 -0.084538641 -0.111209280 0.073467951 -0.026038023
## [7,] -0.0845386412 0.912705413 -0.060812388 0.033600806 -0.073812964
## [8,] -0.1112092800 -0.060812388 0.908273559 0.034898966 -0.015530835
## [9,] 0.0734679507 0.033600806 0.034898966 0.840075094 -0.017125304
## [10,] -0.0260380233 -0.073812964 -0.015530835 -0.017125304 0.901912764
## [11,] 0.0283008612 -0.039910749 0.019800002 -0.078545482 -0.089696977
## [12,] -0.0433318100 -0.037804756 -0.041505079 -0.028884996 -0.033038220
## [13,] -0.0581800753 -0.058779592 -0.047473242 -0.004267272 -0.052277327
## [14,] -0.0351247067 -0.019554897 -0.040301434 -0.046945037 -0.012771943
## [15,] -0.0429300839 -0.047019513 -0.038264586 -0.023006848 -0.046673180
## [16,] 0.0335757708 0.019121984 0.005684668 -0.121798272 -0.008220949
## [17,] 0.0047418214 0.030893728 -0.022430574 -0.109116712 0.030393342
## [18,] -0.1192111938 -0.038539857 -0.105511056 0.025703770 0.022687627
## [19,] -0.0531053175 -0.041434138 -0.048636685 -0.019489121 -0.030975471
## [20,] 0.0001691798 0.004721377 -0.018063732 -0.088347759 -0.004057351
## [21,] 0.0160569747 -0.002265041 -0.002414552 -0.095000886 -0.026057843
## [22,] 0.0370848853 0.001563397 0.014182062 -0.112553574 -0.036255555
## [23,] 0.0424813615 -0.010426001 0.022524305 -0.108380154 -0.057643753
## [24,] -0.1315191147 -0.080493149 -0.102725533 0.062544710 -0.028809795
## [25,] -0.0666287237 -0.038340702 -0.061058899 -0.011917293 -0.016377910
## [26,] 0.0046276745 -0.054282302 0.004282485 -0.052053680 -0.092776002
## [27,] -0.0602444183 0.019487130 -0.073855800 -0.055154947 0.062524261
## [28,] 0.0730159137 0.050157972 0.029304812 -0.170679162 0.007172212
## [,11] [,12] [,13] [,14] [,15]
## [1,] -0.049325430 -0.03436545 -0.033503566 -0.034145749 -0.03541174
## [2,] 0.039373068 -0.04260317 -0.042678654 -0.047820574 -0.03487231
## [3,] -0.122095665 -0.03213819 -0.068907645 0.005413327 -0.05646142
## [4,] 0.027622869 -0.04311951 -0.056695085 -0.035954155 -0.04225508
## [5,] -0.129627791 -0.03146666 -0.068393874 0.006749741 -0.05663460
## [6,] 0.028300861 -0.04333181 -0.058180075 -0.035124707 -0.04293008
## [7,] -0.039910749 -0.03780476 -0.058779592 -0.019554897 -0.04701951
## [8,] 0.019800002 -0.04150508 -0.047473242 -0.040301434 -0.03826459
## [9,] -0.078545482 -0.02888500 -0.004267272 -0.046945037 -0.02300685
## [10,] -0.089696977 -0.03303822 -0.052277327 -0.012771943 -0.04667318
## [11,] 0.888311195 -0.02961149 -0.036886792 -0.018039332 -0.04051140
## [12,] -0.029611493 0.96368073 -0.036707479 -0.036416260 -0.03585089
## [13,] -0.036886792 -0.03670748 0.953964984 -0.028640538 -0.04073358
## [14,] -0.018039332 -0.03641626 -0.028640538 0.955684302 -0.03133380
## [15,] -0.040511402 -0.03585089 -0.040733583 -0.031333804 0.96160909
## [16,] -0.050591555 -0.03188795 -0.011012984 -0.048710568 -0.02468682
## [17,] -0.005161411 -0.03509932 -0.006162956 -0.062018358 -0.01982684
## [18,] 0.053967237 -0.04328589 -0.037814753 -0.054277886 -0.03172426
## [19,] -0.022855965 -0.03705114 -0.038396073 -0.036804391 -0.03628491
## [20,] -0.029763847 -0.03429616 -0.017657325 -0.048957748 -0.02671531
## [21,] -0.055363580 -0.03250326 -0.020548455 -0.041354452 -0.02952981
## [22,] -0.074413351 -0.03074231 -0.018656255 -0.038365801 -0.02968326
## [23,] -0.094017199 -0.02968322 -0.023844184 -0.030580162 -0.03326605
## [24,] 0.020068438 -0.04245777 -0.056291345 -0.034540146 -0.04248231
## [25,] -0.004302624 -0.03844368 -0.037183135 -0.041732370 -0.03466866
## [26,] -0.101656141 -0.03112301 -0.043417633 -0.015960018 -0.04308705
## [27,] 0.054193657 -0.04056137 -0.011824827 -0.071484715 -0.01923648
## [28,] -0.059260544 -0.02970169 0.002969666 -0.056012324 -0.01845831
## [,16] [,17] [,18] [,19] [,20]
## [1,] -0.044237546 -0.0370734074 -0.018828731 -0.03273391 -0.0388694512
## [2,] -0.001353802 -0.0429822165 -0.127095972 -0.05069339 -0.0274305131
## [3,] 0.024467813 0.0889511386 0.055476026 -0.03000158 0.0266072517
## [4,] 0.029824223 0.0007934651 -0.117905095 -0.05257250 -0.0023854029
## [5,] 0.021468335 0.0902019043 0.064580147 -0.02855067 0.0260985986
## [6,] 0.033575771 0.0047418214 -0.119211194 -0.05310532 0.0001691798
## [7,] 0.019121984 0.0308937277 -0.038539857 -0.04143414 0.0047213775
## [8,] 0.005684668 -0.0224305740 -0.105511056 -0.04863669 -0.0180637318
## [9,] -0.121798272 -0.1091167118 0.025703770 -0.01948912 -0.0883477595
## [10,] -0.008220949 0.0303933419 0.022687627 -0.03097547 -0.0040573508
## [11,] -0.050591555 -0.0051614107 0.053967237 -0.02285596 -0.0297638473
## [12,] -0.031887946 -0.0350993153 -0.043285887 -0.03705114 -0.0342961623
## [13,] -0.011012984 -0.0061629563 -0.037814753 -0.03839607 -0.0176573254
## [14,] -0.048710568 -0.0620183578 -0.054277886 -0.03680439 -0.0489577484
## [15,] -0.024686824 -0.0198268372 -0.031724259 -0.03628491 -0.0267153091
## [16,] 0.901043250 -0.1001004216 -0.009720901 -0.02622664 -0.0780221114
## [17,] -0.100100422 0.8699194654 -0.061944136 -0.03275560 -0.0888148916
## [18,] -0.009720901 -0.0619441358 0.857245711 -0.05186659 -0.0366621903
## [19,] -0.026226641 -0.0327555954 -0.051866589 0.96130430 -0.0316989973
## [20,] -0.078022111 -0.0888148916 -0.036662190 -0.03169900 0.9324931363
## [21,] -0.076994942 -0.0714747952 -0.007288065 -0.02806417 -0.0611493859
## [22,] -0.086015526 -0.0699846601 0.015937990 -0.02422891 -0.0634695329
## [23,] -0.078032406 -0.0483117525 0.037426368 -0.02227682 -0.0534152853
## [24,] 0.027085805 0.0023627988 -0.108811743 -0.05114848 -0.0027010431
## [25,] -0.024835794 -0.0428321544 -0.073453853 -0.04157672 -0.0347678834
## [26,] -0.032448594 0.0096747090 0.039863322 -0.02642301 -0.0188915364
## [27,] -0.072560006 -0.1353834635 -0.134232541 -0.04464008 -0.0819935313
## [28,] -0.134814059 -0.1364587151 0.005286509 -0.02081303 -0.1019585532
## [,21] [,22] [,23] [,24] [,25]
## [1,] -0.042869258 -0.046795785 -0.049159990 -0.020683375 -0.029627722
## [2,] -0.004097741 0.016453301 0.031976317 -0.107748522 -0.068299495
## [3,] -0.010217971 -0.025116485 -0.062513375 -0.038586033 -0.006041361
## [4,] 0.013636250 0.034176830 0.040107488 -0.127909540 -0.066064835
## [5,] -0.013111951 -0.030032749 -0.069074423 -0.031744779 -0.002807738
## [6,] 0.016056975 0.037084885 0.042481362 -0.131519115 -0.066628724
## [7,] -0.002265041 0.001563397 -0.010426001 -0.080493149 -0.038340702
## [8,] -0.002414552 0.014182062 0.022524305 -0.102725533 -0.061058899
## [9,] -0.095000886 -0.112553574 -0.108380154 0.062544710 -0.011917293
## [10,] -0.026057843 -0.036255555 -0.057643753 -0.028809795 -0.016377910
## [11,] -0.055363580 -0.074413351 -0.094017199 0.020068438 -0.004302624
## [12,] -0.032503259 -0.030742312 -0.029683218 -0.042457773 -0.038443677
## [13,] -0.020548455 -0.018656255 -0.023844184 -0.056291345 -0.037183135
## [14,] -0.041354452 -0.038365801 -0.030580162 -0.034540146 -0.041732370
## [15,] -0.029529810 -0.029683262 -0.033266054 -0.042482314 -0.034668656
## [16,] -0.076994942 -0.086015526 -0.078032406 0.027085805 -0.024835794
## [17,] -0.071474795 -0.069984660 -0.048311752 0.002362799 -0.042832154
## [18,] -0.007288065 0.015937990 0.037426368 -0.108811743 -0.073453853
## [19,] -0.028064168 -0.024228909 -0.022276821 -0.051148484 -0.041576717
## [20,] -0.061149386 -0.063469533 -0.053415285 -0.002701043 -0.034767883
## [21,] 0.935977495 -0.072231846 -0.069966743 0.010901193 -0.024656985
## [22,] -0.072231846 0.914318389 -0.087295290 0.029437172 -0.016366305
## [23,] -0.069966743 -0.087295290 0.903716024 0.033665279 -0.009187479
## [24,] 0.010901193 0.029437172 0.033665279 0.878719851 -0.062842789
## [25,] -0.024656985 -0.016366305 -0.009187479 -0.062842789 0.950928112
## [26,] -0.042730507 -0.057846015 -0.078031381 -0.001209535 -0.009718390
## [27,] -0.046247299 -0.028478725 0.005390526 -0.054904854 -0.068619827
## [28,] -0.100432380 -0.114622088 -0.102182001 0.062824620 -0.018574793
## [,26] [,27] [,28]
## [1,] -0.045957125 -0.024894119 -0.049107526
## [2,] 0.024186192 -0.106830569 0.018344158
## [3,] -0.131363766 0.136094852 0.058265546
## [4,] 0.005009656 -0.062708296 0.067137968
## [5,] -0.137492419 0.143632332 0.053543393
## [6,] 0.004627675 -0.060244418 0.073015914
## [7,] -0.054282302 0.019487130 0.050157972
## [8,] 0.004282485 -0.073855800 0.029304812
## [9,] -0.052053680 -0.055154947 -0.170679162
## [10,] -0.092776002 0.062524261 0.007172212
## [11,] -0.101656141 0.054193657 -0.059260544
## [12,] -0.031123015 -0.040561374 -0.029701695
## [13,] -0.043417633 -0.011824827 0.002969666
## [14,] -0.015960018 -0.071484715 -0.056012324
## [15,] -0.043087051 -0.019236478 -0.018458305
## [16,] -0.032448594 -0.072560006 -0.134814059
## [17,] 0.009674709 -0.135383464 -0.136458715
## [18,] 0.039863322 -0.134232541 0.005286509
## [19,] -0.026423007 -0.044640084 -0.020813034
## [20,] -0.018891536 -0.081993531 -0.101958553
## [21,] -0.042730507 -0.046247299 -0.100432380
## [22,] -0.057846015 -0.028478725 -0.114622088
## [23,] -0.078031381 0.005390526 -0.102182001
## [24,] -0.001209535 -0.054904854 0.062824620
## [25,] -0.009718390 -0.068619827 -0.018574793
## [26,] 0.902713155 0.056923180 -0.030812257
## [27,] 0.056923180 0.808742715 -0.093132778
## [28,] -0.030812257 -0.093132778 0.808997445round(hat_matrix%*%(I-hat_matrix), 2)## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
## [1,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [2,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [3,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [4,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [5,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [6,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [7,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [8,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [9,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [10,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [11,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [12,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [13,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [14,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [15,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [16,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [17,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [18,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [19,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [20,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [21,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [22,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [23,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [24,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [25,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [26,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [27,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [28,] 0 0 0 0 0 0 0 0 0 0 0 0 0
## [,14] [,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24] [,25]
## [1,] 0 0 0 0 0 0 0 0 0 0 0 0
## [2,] 0 0 0 0 0 0 0 0 0 0 0 0
## [3,] 0 0 0 0 0 0 0 0 0 0 0 0
## [4,] 0 0 0 0 0 0 0 0 0 0 0 0
## [5,] 0 0 0 0 0 0 0 0 0 0 0 0
## [6,] 0 0 0 0 0 0 0 0 0 0 0 0
## [7,] 0 0 0 0 0 0 0 0 0 0 0 0
## [8,] 0 0 0 0 0 0 0 0 0 0 0 0
## [9,] 0 0 0 0 0 0 0 0 0 0 0 0
## [10,] 0 0 0 0 0 0 0 0 0 0 0 0
## [11,] 0 0 0 0 0 0 0 0 0 0 0 0
## [12,] 0 0 0 0 0 0 0 0 0 0 0 0
## [13,] 0 0 0 0 0 0 0 0 0 0 0 0
## [14,] 0 0 0 0 0 0 0 0 0 0 0 0
## [15,] 0 0 0 0 0 0 0 0 0 0 0 0
## [16,] 0 0 0 0 0 0 0 0 0 0 0 0
## [17,] 0 0 0 0 0 0 0 0 0 0 0 0
## [18,] 0 0 0 0 0 0 0 0 0 0 0 0
## [19,] 0 0 0 0 0 0 0 0 0 0 0 0
## [20,] 0 0 0 0 0 0 0 0 0 0 0 0
## [21,] 0 0 0 0 0 0 0 0 0 0 0 0
## [22,] 0 0 0 0 0 0 0 0 0 0 0 0
## [23,] 0 0 0 0 0 0 0 0 0 0 0 0
## [24,] 0 0 0 0 0 0 0 0 0 0 0 0
## [25,] 0 0 0 0 0 0 0 0 0 0 0 0
## [26,] 0 0 0 0 0 0 0 0 0 0 0 0
## [27,] 0 0 0 0 0 0 0 0 0 0 0 0
## [28,] 0 0 0 0 0 0 0 0 0 0 0 0
## [,26] [,27] [,28]
## [1,] 0 0 0
## [2,] 0 0 0
## [3,] 0 0 0
## [4,] 0 0 0
## [5,] 0 0 0
## [6,] 0 0 0
## [7,] 0 0 0
## [8,] 0 0 0
## [9,] 0 0 0
## [10,] 0 0 0
## [11,] 0 0 0
## [12,] 0 0 0
## [13,] 0 0 0
## [14,] 0 0 0
## [15,] 0 0 0
## [16,] 0 0 0
## [17,] 0 0 0
## [18,] 0 0 0
## [19,] 0 0 0
## [20,] 0 0 0
## [21,] 0 0 0
## [22,] 0 0 0
## [23,] 0 0 0
## [24,] 0 0 0
## [25,] 0 0 0
## [26,] 0 0 0
## [27,] 0 0 0
## [28,] 0 0 0d) Construct the ANOVA table and test for significance of regression given the normality assumption.
betaMat<-solve(t(design_matrix)%*%design_matrix)%*%t(design_matrix)%*%response_matrix
hat_matrix<-design_matrix%*%solve(t(design_matrix)%*%design_matrix)%*%t(design_matrix)
y_hat_matrix<-design_matrix%*%betaMat
error_vector<-response_matrix-y_hat_matrix
n<-dim(nfl)[1]
p<-2
ms_res<-sum(error_vector^2)/(n-p-1)
ss_res<-t(response_matrix)%*%(I-hat_matrix)%*%response_matrix
ones<-matrix(c(rep(1, dim(nfl)[1])),
nrow=dim(nfl)[1])
J<-ones%*%t(ones)
ss_tot<-t(response_matrix)%*%(I-(J/n))%*%response_matrix
ss_reg<-t(response_matrix)%*%(hat_matrix-(J/n))%*%response_matrix
ms_res<-ss_res/(n-p-1)
ms_reg<-ss_reg/p
f_stat<-ms_reg/ms_res
p_value<-pf(f_stat, p, n-p-1, lower.tail = FALSE)
# SS(Res)
ss_res## [,1]
## [1,] 128.5531# SS(Reg)
ss_reg## [,1]
## [1,] 198.4112# MS(Res)
ms_res## [,1]
## [1,] 5.142124# MS(Reg)
ms_reg## [,1]
## [1,] 99.2056# F Statistic
f_stat## [,1]
## [1,] 19.29273# P Value
p_value## [,1]
## [1,] 8.556137e-06# Regression Degrees of Freedom
p## [1] 2# Residual Degrees of Freedom
n-p-1## [1] 25Verifying:
anova(mlr)## Analysis of Variance Table
##
## Response: nfl$y
## Df Sum Sq Mean Sq F value Pr(>F)
## nfl$x1 1 115.068 115.068 22.378 7.492e-05 ***
## nfl$x2 1 83.343 83.343 16.208 0.0004635 ***
## Residuals 25 128.553 5.142
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1summary(mlr)##
## Call:
## lm(formula = nfl$y ~ nfl$x1 + nfl$x2, data = nfl)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.4613 -1.4362 0.0489 1.6283 3.5213
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.218e+01 3.112e+00 -3.913 0.000619 ***
## nfl$x1 5.520e-03 1.132e-03 4.875 5.15e-05 ***
## nfl$x2 3.522e-03 8.749e-04 4.026 0.000464 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.268 on 25 degrees of freedom
## Multiple R-squared: 0.6068, Adjusted R-squared: 0.5754
## F-statistic: 19.29 on 2 and 25 DF, p-value: 8.556e-06e) Report a good estimate of the nuisance parameter σ.
# Old estimate is sqrt(ms_res)
sqrt(ms_res)## [,1]
## [1,] 2.267625# New estimate is sqrt(diag(covMat))f) Create the covariance matrix for beta hat. Print it.
ms_res<-sum(error_vector^2)/(n-p-1)
covMat<-ms_res*solve(t(design_matrix)%*%design_matrix)
covMat## [,1] [,2] [,3]
## [1,] 9.682627559 -2.782428e-03 -1.705263e-03
## [2,] -0.002782428 1.281857e-06 3.640215e-08
## [3,] -0.001705263 3.640215e-08 7.655303e-07g) Report the standard errors of beta_not_hat, beta_one_hat, and beta_two_hat.
se_bj<-sqrt(diag(covMat))
# beta 0 hat standard error
beta0_se<-se_bj[1]
beta0_se## [1] 3.111692# beta 1 hat standard error
beta1_se<-se_bj[2]
beta1_se## [1] 0.001132191# beta 2 hat standard error
beta2_se<-se_bj[3]
beta2_se## [1] 0.0008749459h) Based on the sample, give an estimate of the covariance between beta_one_hat and beta_two_hat.
covMat[2]-covMat[3]## [1] -0.001077165i) Perform t-tests for the slope parameters. Report their test statistics, reference distribution, degrees of freedom, and p-value.
# Test Statistics for beta_0, beta_1, and beta_2
t_test<-betaMat/se_bj
t_test## [,1]
## [1,] -3.913135
## [2,] 4.875239
## [3,] 4.025902# R-Squared
r_squared<-ss_reg/ss_tot
r_squared## [,1]
## [1,] 0.6068283# P-Value(s)
n<-dim(nfl)[1]
p<-2
pt(abs(t_test), df= n-p-1, lower.tail = FALSE)*2## [,1]
## [1,] 6.190716e-04
## [2,] 5.148197e-05
## [3,] 4.635451e-04These tests use an F distribution, with n-p-1 and p degrees of freedom (this would be 25), and a p-value of 6.190716e-04, 5.148197e-05, and 4.635451e-04 for the intercept, x1, and x2, respectively.
j) Interpret each of these tests on the context of the data.
# Rejection Region
qf(0.05/2, df1=n-p-1, df2=p, lower.tail = FALSE)## [1] 39.4579Based on these tests, the test statistics do not exceed the rejection region. However, their p-values do indicate there is some significance.