c:ms:2023:schedule:w10.lecture.note
Table of Contents
R code
set.seed(401)
sn <- 25
x <- rnorm(sn, 100, 10)
x
y <- 1.4 * x + 2 + rnorm(sn, 0, 10)
y
df <- data.frame(x, y)
# density graph
ggplot(data=df, aes(y)) +
geom_histogram() +
geom_vline(aes(xintercept=mean(y)),
color="red", linetype="dashed", size=1) +
coord_flip()
ggplot(data=df, aes(y)) +
geom_density(color="blue", size=1.5) +
geom_vline(aes(xintercept=mean(y)),
color="red", linetype="dashed", size=1) +
coord_flip()
lm.mod <- lm(y~x, data=df)
summary(lm.mod)
str(lm.mod)
inc.y <- lm.mod$coefficients[1]
slope.x <- lm.mod$coefficients[2]
inc.y
slope.x
ggplot(data=df, aes(x,y)) +
geom_point(color="blue", size=1.5, pch=1.5) +
geom_hline(aes(yintercept=mean(y))) +
geom_abline(intercept=inc.y, slope=slope.x)
ggplot(data=df, aes(x,y)) +
geom_point(color="blue", size=2.5, pch=2) +
geom_hline(aes(yintercept=mean(y)), size=1.5, color="red") +
geom_abline(intercept=inc.y, slope=slope.x, size=1.5, color="darkgreen")
################################
################################
################################
################################
set.seed(101)
sn <- 400
x <- rnorm(sn, 100, 10)
x
y <- 1.4*x + 2 + rnorm(sn, 0, 16)
y
df <- data.frame(x,y)
# density graph
ggplot(data=df, aes(y)) +
geom_histogram() +
geom_vline(aes(xintercept=mean(y)),
color="red", linetype="dashed", size=1) +
coord_flip()
ggplot(data=df, aes(y)) +
geom_density(color="blue", size=1.5) +
geom_vline(aes(xintercept=mean(y)),
color="red", linetype="dashed", size=1) +
coord_flip()
ggplot(data=df, aes(x,y)) +
geom_point(color="blue", size=1.5, pch=1.5) +
geom_hline(aes(yintercept=mean(y)), size=1, color="darkgreen") +
stat_smooth(method = "lm",
formula = y ~ x,
geom = "smooth", color="red", size=1)
ggplot(data=df, aes(x,y)) +
geom_point(color="blue", size=1.5, pch=2) +
geom_hline(aes(yintercept=mean(y)), size=1, color="darkgreen") +
geom_abline(intercept=10, slope=1.5, size=1.5, color="red")
lm.mod2 <- lm(y~x, data=df)
sum.lm.mod2 <- summary(lm.mod2)
sum.lm.mod2
lm.mod2$coefficients[2]
lm.mod2$coefficients[1]
b <- lm.mod2$coefficients[2]
a <- lm.mod2$coefficients[1]
ggplot(data=df, aes(x,y)) +
geom_point(color="blue", size=1.5, pch=2) +
geom_hline(aes(yintercept=mean(y)), size=1, color="darkgreen") +
geom_abline(intercept=a, slope=b, size=1.5, color="red")
lm.mod2$residuals
sum(lm.mod2$residuals)
ss.res <- sum(lm.mod2$residuals^2)
mean.y <- mean(df$y)
var.tot <- var(df$y)
df.tot <- length(df$y)-1
ss.tot <- var.tot*df.tot
ss.tot
y.hat <- lm.mod2$fitted.values
y.hat - mean(df$y)
explained <- y.hat - mean(df$y)
ss.exp <- sum(explained^2)
ss.exp
ss.res
ss.exp + ss.res
ss.tot
r.square <- ss.exp / ss.tot
r.square
sum.lm.mod2
r.coeff <- sqrt(r.square)
r.coeff
cor(x,y)
R. output
> set.seed(401)
> sn <- 25
> x <- rnorm(sn, 100, 10)
> x
[1] 99.04030 112.53423 111.25717 95.37048 106.51630 110.03586 99.37429 83.40702 91.38017 80.14344
[11] 95.16165 105.55799 100.47560 95.35164 103.18120 101.21572 115.59812 104.79399 89.67882 86.01922
[21] 114.26808 113.21215 110.42156 104.10994 107.89136
> y <- 1.4 * x + 2 + rnorm(sn, 0, 10)
> y
[1] 147.7866 178.1177 167.8750 124.8276 147.9924 133.5853 144.6882 102.0537 140.3838 112.9193 125.8841
[12] 135.8684 137.4363 129.0042 159.6048 137.0136 161.4669 147.8364 127.3562 122.0032 168.4221 138.2663
[23] 147.7574 135.0859 153.9057
> df <- data.frame(x, y)
> # density graph
> ggplot(data=df, aes(y)) +
+ geom_histogram() +
+ geom_vline(aes(xintercept=mean(y)),
+ color="red", linetype="dashed", size=1) +
+ coord_flip()
`stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
>
> ggplot(data=df, aes(y)) +
+ geom_density(color="blue", size=1.5) +
+ geom_vline(aes(xintercept=mean(y)),
+ color="red", linetype="dashed", size=1) +
+ coord_flip()
>
> lm.mod <- lm(y~x, data=df)
> summary(lm.mod)
Call:
lm(formula = y ~ x, data = df)
Residuals:
Min 1Q Median 3Q Max
-19.958 -6.345 -0.137 6.596 20.954
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -5.9221 22.5365 -0.263 0.795
x 1.4492 0.2212 6.553 1.1e-06 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 10.75 on 23 degrees of freedom
Multiple R-squared: 0.6512, Adjusted R-squared: 0.636
F-statistic: 42.94 on 1 and 23 DF, p-value: 1.097e-06
> str(lm.mod)
List of 12
$ coefficients : Named num [1:2] -5.92 1.45
..- attr(*, "names")= chr [1:2] "(Intercept)" "x"
$ residuals : Named num [1:25] 10.18 20.95 12.56 -7.46 -0.45 ...
..- attr(*, "names")= chr [1:25] "1" "2" "3" "4" ...
$ effects : Named num [1:25] -705.43 -70.46 7.59 -7.34 -3.9 ...
..- attr(*, "names")= chr [1:25] "(Intercept)" "x" "" "" ...
$ rank : int 2
$ fitted.values: Named num [1:25] 138 157 155 132 148 ...
..- attr(*, "names")= chr [1:25] "1" "2" "3" "4" ...
$ assign : int [1:2] 0 1
$ qr :List of 5
..$ qr : num [1:25, 1:2] -5 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 ...
.. ..- attr(*, "dimnames")=List of 2
.. .. ..$ : chr [1:25] "1" "2" "3" "4" ...
.. .. ..$ : chr [1:2] "(Intercept)" "x"
.. ..- attr(*, "assign")= int [1:2] 0 1
..$ qraux: num [1:2] 1.2 1.24
..$ pivot: int [1:2] 1 2
..$ tol : num 1e-07
..$ rank : int 2
..- attr(*, "class")= chr "qr"
$ df.residual : int 23
$ xlevels : Named list()
$ call : language lm(formula = y ~ x, data = df)
$ terms :Classes 'terms', 'formula' language y ~ x
.. ..- attr(*, "variables")= language list(y, x)
.. ..- attr(*, "factors")= int [1:2, 1] 0 1
.. .. ..- attr(*, "dimnames")=List of 2
.. .. .. ..$ : chr [1:2] "y" "x"
.. .. .. ..$ : chr "x"
.. ..- attr(*, "term.labels")= chr "x"
.. ..- attr(*, "order")= int 1
.. ..- attr(*, "intercept")= int 1
.. ..- attr(*, "response")= int 1
.. ..- attr(*, ".Environment")=<environment: R_GlobalEnv>
.. ..- attr(*, "predvars")= language list(y, x)
.. ..- attr(*, "dataClasses")= Named chr [1:2] "numeric" "numeric"
.. .. ..- attr(*, "names")= chr [1:2] "y" "x"
$ model :'data.frame': 25 obs. of 2 variables:
..$ y: num [1:25] 148 178 168 125 148 ...
..$ x: num [1:25] 99 112.5 111.3 95.4 106.5 ...
..- attr(*, "terms")=Classes 'terms', 'formula' language y ~ x
.. .. ..- attr(*, "variables")= language list(y, x)
.. .. ..- attr(*, "factors")= int [1:2, 1] 0 1
.. .. .. ..- attr(*, "dimnames")=List of 2
.. .. .. .. ..$ : chr [1:2] "y" "x"
.. .. .. .. ..$ : chr "x"
.. .. ..- attr(*, "term.labels")= chr "x"
.. .. ..- attr(*, "order")= int 1
.. .. ..- attr(*, "intercept")= int 1
.. .. ..- attr(*, "response")= int 1
.. .. ..- attr(*, ".Environment")=<environment: R_GlobalEnv>
.. .. ..- attr(*, "predvars")= language list(y, x)
.. .. ..- attr(*, "dataClasses")= Named chr [1:2] "numeric" "numeric"
.. .. .. ..- attr(*, "names")= chr [1:2] "y" "x"
- attr(*, "class")= chr "lm"
> inc.y <- lm.mod$coefficients[1]
> slope.x <- lm.mod$coefficients[2]
> inc.y
(Intercept)
-5.92206
> slope.x
x
1.449211
>
> ggplot(data=df, aes(x,y)) +
+ geom_point(color="blue", size=1.5, pch=1.5) +
+ geom_hline(aes(yintercept=mean(y))) +
+ geom_abline(intercept=inc.y, slope=slope.x)
>
>
> ggplot(data=df, aes(x,y)) +
+ geom_point(color="blue", size=2.5, pch=2) +
+ geom_hline(aes(yintercept=mean(y)), size=1.5, color="red") +
+ geom_abline(intercept=inc.y, slope=slope.x, size=1.5, color="darkgreen")
>
> ################################
> ################################
> ################################
> ################################
>
> set.seed(101)
> sn <- 400
> x <- rnorm(sn, 100, 10)
> x
[1] 96.73964 105.52462 93.25056 102.14359 103.10769 111.73966 106.18790 98.87266 109.17028 97.76741
[11] 105.26448 92.05156 114.27756 85.33180 97.63317 98.06662 91.50245 100.58465 91.82330 79.49692
[21] 98.36244 107.08522 97.32019 85.36078 107.44436 85.89610 104.67068 98.80680 104.67239 104.98136
[31] 108.94937 102.79152 110.07866 79.26894 111.89853 92.75626 101.67984 109.20335 83.28395 104.48469
[41] 104.82459 107.58214 76.80673 95.40495 88.94616 104.02928 105.68935 92.93917 97.09909 85.16122
[51] 88.49745 97.25529 105.77901 86.03097 107.49058 89.48813 101.65381 111.29809 111.73722 95.72137
[61] 97.40198 85.88827 93.58642 101.12458 104.22604 103.86835 93.12202 101.48902 99.42350 99.25177
[71] 115.09897 116.19937 111.53158 99.22396 81.81065 89.62555 103.02492 87.22054 101.38339 99.49016
[81] 118.52148 111.11675 94.88625 94.56119 82.71073 104.70750 100.05387 113.48046 107.24097 115.52549
[91] 113.25470 99.65735 96.38987 92.79835 102.82015 92.09474 95.55095 113.64993 104.97454 91.85604
[101] 102.68066 94.07792 121.33486 111.72749 107.46761 97.69491 100.87772 78.16260 95.33368 116.85960
[111] 94.32079 99.53257 98.43019 116.02242 107.68654 92.28371 93.69318 91.69719 94.08887 109.81085
[121] 93.38395 92.27582 79.81527 94.66415 104.34728 92.28833 92.46059 97.00642 116.63966 87.55670
[131] 92.16866 102.44831 98.56113 83.91369 109.51580 81.80868 117.83672 118.87139 114.90719 96.19400
[141] 90.90625 96.61906 85.88116 102.17543 106.70126 97.12141 104.69303 95.29929 97.60734 95.52538
[151] 93.81170 102.52963 92.46632 107.32277 95.97413 71.77000 104.62974 121.32870 97.29513 102.48525
[161] 100.38116 103.94069 84.95915 84.13109 90.72882 107.76197 92.19316 87.21433 99.98572 81.49022
[171] 104.51505 95.67053 107.13603 109.60695 103.81535 112.18073 99.82863 99.61791 112.43734 90.44141
[181] 109.15425 90.60662 101.12125 105.53013 105.31742 91.26238 98.13151 97.86290 97.95989 117.19709
[191] 102.02033 105.12656 114.52400 103.63865 91.24151 99.85439 92.75507 119.69370 94.63598 99.73768
[201] 98.35968 86.16725 104.23511 92.09511 112.09925 108.94517 98.98801 102.97123 101.97298 98.43016
[211] 115.36571 78.32330 105.98448 100.43112 112.95027 107.06303 103.45545 99.20103 104.54808 112.76252
[221] 112.64838 102.69254 98.79456 107.95271 94.85972 95.93407 112.19719 100.83711 105.89902 94.82581
[231] 107.69463 108.01970 93.03140 111.77853 105.85845 95.33106 103.85650 94.65394 110.56668 97.93907
[241] 106.07012 94.51936 79.00024 102.50813 99.45055 93.40272 85.44143 100.23729 105.47908 91.91099
[251] 97.60977 96.46881 108.19599 96.54720 88.17116 89.68679 99.24911 108.28585 89.64015 98.52844
[261] 97.18546 86.26314 115.58450 94.25055 121.87335 107.94292 102.02560 99.77950 103.04531 88.90751
[271] 107.65473 99.77974 90.96002 104.00002 88.50712 101.88168 102.17290 89.50914 99.24877 84.32011
[281] 88.26203 78.51694 103.41715 109.04971 110.96490 85.28739 97.19403 108.46140 87.14086 96.87564
[291] 96.37157 114.12454 97.44798 103.87018 105.24941 106.15145 96.67184 107.45329 96.88735 97.65749
[301] 88.34143 97.60201 111.84383 102.07508 99.56367 90.28922 105.74223 87.40135 110.61679 95.66006
[311] 102.39177 94.97671 82.96123 105.23772 115.64172 75.33747 98.51179 121.10837 96.53805 109.82183
[321] 103.91431 94.90006 89.50354 100.46007 89.91172 93.43750 100.94018 100.17924 83.96476 104.83076
[331] 97.75577 95.62602 113.27768 95.31522 86.60042 116.65433 105.38950 109.99385 103.41489 89.41373
[341] 99.58401 87.07172 91.06136 105.41909 109.69636 97.55011 108.60180 93.24156 94.99956 118.32033
[351] 96.70728 102.61308 88.87839 110.62210 102.05000 79.75672 112.27375 111.95344 107.72516 85.90462
[361] 95.67570 95.57712 113.48992 89.25368 96.64918 107.50832 119.97207 117.91908 87.67024 68.22790
[371] 89.15158 100.60763 98.18234 103.41311 111.06792 106.88816 104.70992 83.59933 93.20977 104.86308
[381] 102.26175 93.53319 98.86714 105.20524 125.86743 105.42192 110.65204 92.49852 107.93791 92.50237
[391] 103.06302 113.18370 97.33460 102.65806 111.39838 90.13455 93.26333 80.95256 92.82078 96.16458
> y <- 1.4*x + 2 + rnorm(sn, 0, 16)
> y
[1] 132.30690 116.26373 102.92148 152.03568 148.62125 145.23458 128.72418 140.81982 180.13428 126.71891
[11] 140.75356 126.70752 151.67496 140.90857 167.19847 138.26867 140.09513 127.80099 101.22968 98.87192
[21] 124.49299 138.55446 146.98584 147.30567 141.97227 121.66819 146.76882 160.20161 137.46257 130.32682
[31] 137.63163 157.48389 161.87376 117.07605 170.70415 115.97677 168.13299 143.25823 120.14394 151.71639
[41] 157.05003 147.87415 103.89740 137.70645 122.89482 161.45022 175.63248 124.98146 118.90559 125.17165
[51] 113.71257 131.85370 122.43841 126.65040 126.63737 125.40707 149.53018 127.40238 150.91523 137.99144
[61] 120.33013 115.81479 133.10345 154.33507 124.77469 129.11777 127.55762 121.86201 108.75323 149.03593
[71] 175.48256 138.87299 176.42295 174.48624 113.85711 123.20494 158.95204 116.79922 128.38950 133.72567
[81] 183.09255 173.82009 162.87062 151.14890 111.74930 156.72017 124.95247 158.56504 135.95007 183.62406
[91] 159.29435 128.60357 121.00659 118.18438 130.73640 112.98394 133.61012 147.12472 153.10322 102.67619
[101] 125.39519 123.09994 167.12412 149.23502 152.03587 142.92851 151.76348 102.75455 140.86985 161.79683
[111] 152.06027 144.62060 139.54836 167.39453 154.49285 141.98107 117.93898 122.40050 123.26812 153.71793
[121] 122.53166 136.58723 113.27211 154.18945 173.45069 126.17369 147.05687 149.68243 200.90285 100.62195
[131] 148.73329 136.97472 162.00172 105.04047 152.31765 102.60661 170.85247 187.91323 147.18637 145.62508
[141] 140.79521 108.84048 125.81853 126.80585 173.41216 166.45171 153.27515 120.51269 148.82013 122.36233
[151] 117.01803 148.12499 141.53806 151.77987 136.56074 97.81333 175.77054 148.36930 141.97085 137.33414
[161] 108.67386 156.53995 149.56735 120.63470 143.62663 150.52260 128.25124 107.21147 161.80511 134.13153
[171] 137.35287 164.00363 156.56514 153.90992 155.91806 147.69572 146.18249 146.38569 146.19223 135.74346
[181] 146.02366 105.76156 139.97315 154.56684 152.23720 120.44186 135.72005 154.93419 150.38464 148.97005
[191] 142.97852 160.25597 135.58243 149.65804 117.34308 143.64464 115.23345 172.41222 161.51218 160.49027
[201] 133.47454 105.47177 141.60362 117.41335 148.44390 114.41137 143.26866 165.60948 148.84450 165.70723
[211] 163.44605 116.68974 153.01589 131.20094 139.70546 168.97821 133.99155 127.38999 171.32937 144.22951
[221] 164.24872 150.77626 164.65702 140.13966 135.62057 128.49762 137.64412 128.19980 136.26003 121.43766
[231] 149.36358 150.85180 131.03067 148.53701 157.45806 153.61989 126.77119 136.71470 154.16814 138.81539
[241] 147.06123 151.12100 84.09791 141.80539 150.59611 126.72361 117.92938 127.57624 139.99515 131.70418
[251] 128.11498 120.18725 128.94675 143.67165 131.06991 99.86916 137.92815 131.20339 130.32497 147.18538
[261] 124.40672 137.60417 164.26852 139.64674 192.99892 146.51951 139.86154 152.50096 159.88686 130.82308
[271] 153.38218 138.98802 154.75733 169.94912 109.17631 146.50361 146.73210 126.22433 135.90030 127.78037
[281] 133.76225 96.75362 159.10296 159.61588 162.24551 117.37702 139.97459 129.37033 132.40218 151.45761
[291] 135.77740 164.45266 144.74040 138.94407 135.33135 131.65770 127.91329 155.32920 139.97307 150.81138
[301] 126.12238 140.95811 154.78033 133.01955 145.56363 135.27575 175.07892 148.32427 145.09870 139.97886
[311] 132.28771 154.59219 128.29751 159.62249 150.59405 111.25710 139.08130 186.21618 131.23893 159.99042
[321] 140.45829 116.31435 123.08005 134.65362 141.07042 124.29293 137.22687 123.44081 75.29330 144.73415
[331] 165.51711 134.46748 140.63536 142.15189 155.84088 142.25283 180.66935 133.44466 144.29671 119.22711
[341] 157.47883 115.72553 131.52360 153.41154 158.16958 144.40760 145.22891 123.02795 120.76962 149.04446
[351] 130.01780 147.60471 152.36399 156.37299 140.40351 93.06820 178.60071 154.76698 145.23716 103.35464
[361] 116.57163 158.37971 149.82117 142.89029 149.58717 160.62458 168.02319 182.00714 101.06283 75.58043
[371] 133.84178 150.09934 134.95391 126.05753 142.03125 143.39621 167.25819 135.76302 119.84304 145.65798
[381] 155.74441 82.92507 118.81283 124.15648 208.13760 126.77023 165.50725 132.27584 148.85673 107.16469
[391] 136.34497 153.84423 137.19628 142.71654 155.03963 123.66223 128.68534 94.58241 140.02013 143.52958
> df <- data.frame(x,y)
> # density graph
> ggplot(data=df, aes(y)) +
+ geom_histogram() +
+ geom_vline(aes(xintercept=mean(y)),
+ color="red", linetype="dashed", size=1) +
+ coord_flip()
`stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
>
> ggplot(data=df, aes(y)) +
+ geom_density(color="blue", size=1.5) +
+ geom_vline(aes(xintercept=mean(y)),
+ color="red", linetype="dashed", size=1) +
+ coord_flip()
>
> ggplot(data=df, aes(x,y)) +
+ geom_point(color="blue", size=1.5, pch=1.5) +
+ geom_hline(aes(yintercept=mean(y)), size=1, color="darkgreen") +
+ stat_smooth(method = "lm",
+ formula = y ~ x,
+ geom = "smooth", color="red", size=1)
>
>
>
> ggplot(data=df, aes(x,y)) +
+ geom_point(color="blue", size=1.5, pch=2) +
+ geom_hline(aes(yintercept=mean(y)), size=1, color="darkgreen") +
+ geom_abline(intercept=10, slope=1.5, size=1.5, color="red")
>
> lm.mod2 <- lm(y~x, data=df)
> sum.lm.mod2 <- summary(lm.mod2)
> sum.lm.mod2
Call:
lm(formula = y ~ x, data = df)
Residuals:
Min 1Q Median 3Q Max
-48.386 -10.834 -0.276 8.934 37.376
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.90168 7.62463 0.118 0.906
x 1.39426 0.07609 18.324 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 14.66 on 398 degrees of freedom
Multiple R-squared: 0.4576, Adjusted R-squared: 0.4562
F-statistic: 335.8 on 1 and 398 DF, p-value: < 2.2e-16
>
> lm.mod2$coefficients[2]
x
1.394256
> lm.mod2$coefficients[1]
(Intercept)
0.9016802
>
> b <- lm.mod2$coefficients[2]
> a <- lm.mod2$coefficients[1]
>
> ggplot(data=df, aes(x,y)) +
+ geom_point(color="blue", size=1.5, pch=2) +
+ geom_hline(aes(yintercept=mean(y)), size=1, color="darkgreen") +
+ geom_abline(intercept=a, slope=b, size=1.5, color="red")
>
> lm.mod2$residuals
1 2 3 4 5 6 7 8 9
-3.4746174 -31.7662994 -27.9953774 8.7196574 3.9610280 -11.4608149 -20.2306343 2.0643286 27.0212529
10 11 12 13 14 15 16 17 18
-10.4955793 -6.9137777 -2.5376141 -8.5589127 21.0324963 30.1711479 0.6369969 11.6155907 -13.3414636
19 20 21 22 23 24 25 26 27
-27.6971994 -12.8688367 -13.5511384 -11.6514560 10.3948753 27.3891900 -8.7343750 1.0053441 -0.0705950
28 29 30 31 32 33 34 35 36
21.5379341 -9.3792404 -16.9457691 -15.1733880 13.2644964 7.4942308 5.6531643 13.7872501 -14.2508958
37 38 39 40 41 42 43 44 45
25.4635703 -9.9008955 3.1230920 5.1362849 9.9960200 -3.0245953 -4.0925309 3.7858216 -2.0205986
46 47 48 49 50 51 52 53 54
15.5050718 27.3727744 -5.5012269 -17.3771050 5.5334144 -10.5772242 -4.6467711 -25.9463125 5.7994998
55 56 57 58 59 60 61 62 63
-24.1337116 -0.2639915 6.8970486 -28.6773575 -5.7767647 3.6296544 -16.3748583 -4.8371426 1.7183225
64 65 66 67 68 69 70 71 72
12.4398255 -21.4447985 -16.6030004 -3.1800045 -20.5413749 -30.7702866 9.7518575 14.1034204 -24.0403803
73 74 75 76 77 78 79 80 81
20.0176731 35.2409359 -1.1095846 -2.6577263 14.4072229 -5.7102332 -13.8665975 -5.8907808 16.9415652
82 83 84 85 86 87 88 89 90
17.9931867 29.6732031 18.4046991 -4.4723224 9.8294162 -15.4499433 -0.5574663 -14.4729952 21.6502481
91 92 93 94 95 96 97 98 99
0.4866024 -11.2459893 -14.2872615 -12.1019636 -13.5229133 -16.3214061 -0.5140732 -12.2340766 5.8401374
100 101 102 103 104 105 106 107 108
-26.2963375 -18.6696323 -8.9704600 -2.9494460 -7.4434017 1.2968079 5.8150963 10.2124233 -7.1258268
109 110 111 112 113 114 115 116 117
7.0485934 -2.0370668 19.6512446 4.9450222 1.4097695 4.7278671 3.4485490 12.4122630 -13.5949983
118 119 120 121 122 123 124 125 126
-6.3505617 -8.8175514 -0.2882130 -8.5711641 7.0294129 1.0874987 21.3017029 27.0621633 -3.4015577
127 128 129 130 131 132 133 134 135
17.2414383 13.5289414 37.3755979 -22.3562059 19.3248889 -6.7661424 23.6805735 -12.8583893 -1.2771132
136 137 138 139 140 141 142 143 144
-12.3573364 5.6562192 21.2743769 -13.9253690 10.6043118 13.1469240 -26.7729251 5.1765012 -16.5545580
145 146 147 148 149 150 151 152 153
23.7415811 30.1379044 6.4045552 -13.2606073 11.8288094 -11.7261952 -14.6812000 4.2707380 11.7146435
154 155 156 157 158 159 160 161 162
1.2427601 1.8465318 -3.1541202 28.9882028 -21.6956671 5.4148290 -6.4582415 -32.1848782 10.7183154
163 164 165 166 167 168 169 170 171
30.2108509 2.4327253 16.2257323 -0.6268704 -1.1913196 -15.2893256 21.4977219 19.6116057 -9.2695699
172 173 174 175 176 177 178 179 180
29.7127197 6.2883910 0.1880614 10.2711793 -9.6146350 6.0941300 6.5911259 -11.4759098 8.7432830
181 182 183 184 185 186 187 188 189
-7.0670145 -21.4689644 -1.9174567 6.5291240 4.4960517 -7.7029489 -2.0020893 17.5865667 12.9017805
190 191 192 193 194 195 196 197 198
-15.3344050 -0.1656470 12.7809341 -24.9950438 4.2575262 -10.7726382 3.5203550 -14.9925564 4.6268558
199 200 201 202 203 204 205 206 207
28.6637087 20.5287197 -4.5657238 -15.5691273 -4.6285154 -11.8925051 -8.7528513 -38.3877830 4.3523274
208 209 210 211 212 213 214 215 216
21.1395345 5.7663682 27.5686900 1.6950174 6.5853138 4.3446907 -9.7274554 -18.6778369 18.8032432
217 218 219 220 221 222 223 224 225
-11.1535346 -11.8233452 24.6608838 -13.8920218 6.2863442 6.6948766 26.0104146 -11.2757601 2.4601369
226 227 228 229 230 231 232 233 234
-6.1607243 -19.6891885 -13.2946477 -12.2920126 -11.6754864 -1.6920089 -0.6570117 0.4193871 -8.2125793
235 236 237 238 239 240 241 242 243
8.9625719 19.8022864 -18.9330501 3.8411704 -0.8918261 1.3615594 -1.7293776 18.4351146 -26.9503418
244 245 246 247 248 249 250 251 252
-2.0188815 11.0348899 -4.4053943 -2.0995328 -13.0819055 -7.9713881 2.6550375 -8.8797268 -15.2166647
253 254 255 256 257 258 259 260 261
-22.8078616 8.1584308 7.2350489 -26.0788823 -1.3522121 -20.6765143 4.4419549 8.9098171 -11.9963947
262 263 264 265 266 267 268 269 270
16.4295674 2.2124356 7.3356495 22.1745595 -4.8822450 -3.2899706 12.4810919 15.3136167 5.9615531
271 272 273 274 275 276 277 278 279
2.3822204 -1.0321771 27.0340866 24.0447723 -15.1269755 3.5527724 3.3752225 0.5239755 -3.3795901
280 281 282 283 284 285 286 287 288
9.3148527 9.8006871 -13.6207847 14.0112681 6.6709718 6.6303285 -2.4371321 3.5595267 -22.7543261
289 290 291 292 293 294 295 296 297
10.0038149 15.4864625 0.5090688 4.4321302 7.9712729 -6.7792422 -12.3149711 -17.2463007 -7.7736970
298 299 300 301 302 303 304 305 306
4.6101104 3.9856124 13.7501494 2.0501165 3.9742149 -2.0603007 -10.2009494 5.8446841 8.4877582
307 308 309 310 311 312 313 314 315
26.7454828 25.5627249 -10.0311236 5.7025416 -11.3743320 21.2686430 11.7266151 11.9924647 -11.5418205
316 317 318 319 320 321 322 323 324
5.3156858 0.8289472 16.4584012 -4.2615220 5.9689829 -5.3265662 -16.9023231 -2.6124888 -6.3151418
325 326 327 328 329 330 331 332 333
14.8087721 -6.8845717 -4.4112764 -17.1363971 -42.6767643 -2.3284644 28.3188400 0.2386305 -18.2044305
334 335 336 337 338 339 340 341 342
8.3563734 34.1960270 -21.2948682 32.8277064 -20.8166195 -0.7918255 -6.3402167 17.7315363 -6.5764400
343 344 345 346 347 348 349 350 351
3.6590539 5.5286408 4.3230679 7.4960704 -7.0915009 -7.8763515 -12.5857940 -16.8260691 -5.7185980
352 353 354 355 356 357 358 359 360
3.6341103 27.5430616 1.2357534 -2.7820224 -19.0347891 21.1606580 -2.2264760 -5.8609865 -17.3200922
361 362 363 364 365 366 367 368 369
-17.7264836 24.2190429 -9.3145369 17.5461131 13.9317778 9.8287519 -0.1502889 16.6960559 -22.0736253
370 371 372 373 374 375 376 377 378
-20.4484231 8.6399497 8.9248462 -2.8391012 -19.0285217 -13.7275614 -6.5349466 20.3640549 18.3024546
379 380 381 382 383 384 385 386 387
-11.0169399 -1.4497000 12.2636443 -48.3858395 -19.9349754 -23.4282517 31.7444754 -21.1166158 10.3282794
388 389 390 391 392 393 394 395 396
2.4075270 -2.5380510 -22.7089945 -8.2529627 -4.8645207 0.5852366 -1.3167768 -1.1799373 -2.9101056
397 398 399 400
-2.2493160 -19.1878786 9.7025061 8.5498421
> sum(lm.mod2$residuals)
[1] -1.481315e-13
> ss.res <- sum(lm.mod2$residuals^2)
>
> mean.y <- mean(df$y)
> var.tot <- var(df$y)
> df.tot <- length(df$y)-1
> ss.tot <- var.tot*df.tot
> ss.tot
[1] 157720.1
>
> y.hat <- lm.mod2$fitted.values
> y.hat - mean(df$y)
1 2 3 4 5 6 7
-4.188193540 8.060324036 -9.052855916 3.346310443 4.690509466 16.725688058 8.985106285
8 9 10 11 12 13 14
-1.214214749 13.143314113 -2.755217099 7.697625625 -10.724577336 20.264160488 -20.093633982
15 16 17 18 19 20 21
-2.942382245 -2.338036133 -11.490167661 1.172749237 -11.042829145 -28.228953240 -1.925583097
22 23 24 25 26 27 28
10.236203716 -3.378744942 -20.053229354 10.736932962 -19.306862002 6.869709427 -1.306037575
29 30 31 32 33 34 35
6.872098565 7.302876361 12.835307738 4.249684406 14.409821034 -28.546825176 16.947194881
36 37 38 39 40 41 42
-9.742041972 2.699714560 13.189420340 -22.948863166 6.610398182 7.084302239 10.929033016
43 44 45 46 47 48 49
-31.979775017 -6.049083432 -15.054289797 5.975442811 8.290000729 -9.487019465 -3.687016096
50 51 52 53 54 55 56
-20.331471466 -15.679915057 -3.469240752 8.415010924 -19.118811218 10.801373996 -14.298645187
57 58 59 60 61 62 63
2.663423441 16.110024023 16.722288536 -5.607919185 -3.264716559 -19.317777113 -8.584576990
64 65 66 67 68 69 70
1.925536197 6.249777464 5.751065411 -9.232080329 2.433672581 -0.446194773 -0.685638984
71 72 73 74 75 76 77
21.409428922 22.943662437 16.435569549 -0.724402520 -25.003016473 -14.107044909 4.575107276
78 79 80 81 82 83 84
-17.460253140 2.286391149 -0.353258888 26.181272643 15.857191683 -6.772291669 -7.225505548
85 86 87 88 89 90 91
-23.748085276 6.921045005 0.432700698 19.152802236 10.453353668 22.004103260 18.838035602
92 93 94 95 96 97 98
-0.120152760 -4.675861233 -9.683360572 4.289601082 -10.664362592 -5.845518829 19.389092184
99 100 101 102 103 104 105
7.293378323 -10.997182870 4.095114946 -7.899310626 30.103856034 16.708711308 10.769351738
106 107 108 109 110 111 112
-2.856291313 1.581352807 -30.089335260 -6.148449365 23.864189832 -7.560682256 -0.294126975
113 114 115 116 117 118 119
-1.831121148 22.696954827 11.074591845 -10.400899658 -8.435732181 -11.218648193 -7.884035535
120 121 122 123 124 125 126
14.036434436 -8.866881966 -10.411890997 -27.785100861 -7.081957406 6.418817120 -10.394457998
127 128 129 130 131 132 133
-10.154276177 -3.816224498 23.557545329 -16.991555363 -10.561309080 3.771155620 -1.648565341
134 135 136 137 138 139 140
-22.070852707 13.625053083 -25.005765812 25.226543589 26.669147778 21.142029261 -4.948941984
141 142 143 144 145 146 147
-12.321426981 -4.356307611 -19.327682961 3.390695591 9.700866033 -3.655906788 6.900883453
148 149 150 151 152 153 154
-6.196409564 -2.978389475 -5.881183034 -8.270480396 3.884543420 -10.146292007 10.567405765
155 156 157 158 159 160 161
-5.255499753 -39.002263512 6.812631665 30.095258615 -3.413686972 3.822670480 0.889031722
162 163 164 165 166 167 168
5.851921144 -20.613210572 -21.767732731 -12.568810769 11.179765694 -10.527150723 -17.468909475
169 170 171 172 173 174 175
0.337678648 -25.449786671 6.652731499 -5.678800699 10.307043017 13.752146484 5.677168696
176 177 178 179 180 181 182
17.340645806 0.118653098 -0.175147804 17.698434213 -12.969529318 13.120963429 -12.739187464
183 184 185 186 187 188 189
1.920897652 8.068003086 7.771435656 -11.824895795 -2.247567141 -2.622081296 -2.486849390
190 191 192 193 194 195 196
24.334742967 3.174454748 7.505325330 20.607767499 5.430807189 -11.853987746 0.154576491
197 198 199 200 201 202 203
-9.743700390 27.815654880 -7.121233632 -0.008155611 -1.929440828 -18.928807803 6.262422434
204 205 206 207 208 209 210
-10.663850175 17.227042523 12.829445847 -1.053376512 4.500240233 3.108426135 -1.831165164
211 212 213 214 215 216 217
21.781326872 -29.865278310 8.701490814 0.958687155 18.413587198 10.205262822 5.175374020
218 219 220 221 222 223 224
-0.756373622 6.698772801 18.151818084 17.992667603 4.111674430 -1.323103072 11.445710456
225 226 227 228 229 230 231
-6.809271836 -5.311364177 17.363597791 1.524741376 8.582337678 -6.856559994 11.085882713
232 233 234 235 236 237 238
11.539103178 -9.358425291 16.779881273 8.525774195 -6.152106461 5.734534353 -7.096181047
239 240 241 242 243 244 245
15.090255018 -2.515877757 8.820896097 -7.283823851 -28.921459521 3.854562893 -0.408487563
246 247 248 249 250 251 252
-8.840705400 -19.940791225 0.688439411 7.996832897 -10.920567370 -2.975005986 -4.565789371
253 254 255 256 257 258 259
11.784903101 -4.456491985 -16.134849344 -14.021667702 -0.689347013 11.910194696 -14.086689781
260 261 262 263 264 265 266
-1.694142756 -3.566597677 -18.795107076 22.086377996 -7.658616347 30.854647422 11.432050544
267 268 269 270 271 272 273
3.181799384 0.050155739 4.603531162 -15.108185808 11.030246473 0.050485562 -12.246460775
274 275 276 277 278 279 280
5.934639390 -15.666422989 2.981129235 3.387171035 -14.269352250 -0.689822699 -21.504192760
281 282 283 284 285 286 287
-16.008148268 -29.595299776 5.121979266 12.975202346 15.645476443 -20.155556179 -3.554648742
288 289 290 291 292 293 294
12.154944408 -17.571341210 -3.998566035 -4.701373931 20.050816630 -3.200583582 5.753606266
295 296 297 298 299 300 301
7.676609253 8.934290166 -4.282721357 10.749382792 -3.982246646 -2.908474925 -15.897442211
302 303 304 305 306 307 308
-2.985817392 16.870919093 3.250788947 -0.250759772 -13.181717777 8.363724287 -17.208161234
309 310 311 312 313 314 315
15.160112909 -5.693393330 3.692329779 -6.646157354 -23.398814345 7.660317317 22.166157510
316 317 318 319 320 321 322
-34.028292908 -1.717352317 29.788066013 -4.469258871 14.051731373 5.815147000 -6.753033299
323 324 325 326 327 328 329
-14.277169928 0.999047979 -13.708058852 -8.792211285 1.668441234 0.607500727 -21.999649132
330 331 332 333 334 335 336
7.092908553 -2.771438717 -5.740856271 18.870084347 -6.174191334 -18.324853141 23.577989327
337 338 339 340 341 342 343
7.871933488 14.291574349 5.118826524 -14.402386692 -0.222410421 -17.667739509 -12.105163697
344 345 346 347 348 349 350
7.913192957 13.876800158 -3.058182742 12.350704495 -9.065405731 -6.614299669 25.900822413
351 352 353 354 355 356 357
-4.233308503 4.000891801 -15.148776649 15.167525530 3.215819010 -27.866724081 17.470338913
358 359 360 361 362 363 364
17.023749956 11.128436252 -19.294980783 -5.671591247 -5.809038961 19.165998537 -14.625527113
365 366 367 368 369 370 371
-4.314316955 10.826117811 28.203766718 25.341376354 -16.833258109 -43.940853091 -14.767882592
372 373 374 375 376 377 378
1.204788347 -2.176695408 5.116340351 15.789100676 9.961452371 6.924422052 -22.509145717
379 380 381 382 383 384 385
-9.109732108 7.137970613 3.511054869 -8.658795795 -1.221903897 7.615023765 36.423415403
386 387 388 389 390 391 392
7.917140730 15.209261430 -10.101392648 11.425071990 -10.096024661 4.628228753 18.739043938
393 394 395 396 397 398 399
-3.358662973 4.063609447 16.249858053 -13.397371823 -9.035050482 -26.199424528 -9.652082743
400
-4.989967840
> explained <- y.hat - mean(df$y)
> ss.exp <- sum(explained^2)
> ss.exp
[1] 72172.76
> ss.res
[1] 85547.32
>
> ss.exp + ss.res
[1] 157720.1
> ss.tot
[1] 157720.1
>
> r.square <- ss.exp / ss.tot
> r.square
[1] 0.4576003
> sum.lm.mod2
Call:
lm(formula = y ~ x, data = df)
Residuals:
Min 1Q Median 3Q Max
-48.386 -10.834 -0.276 8.934 37.376
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.90168 7.62463 0.118 0.906
x 1.39426 0.07609 18.324 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 14.66 on 398 degrees of freedom
Multiple R-squared: 0.4576, Adjusted R-squared: 0.4562
F-statistic: 335.8 on 1 and 398 DF, p-value: < 2.2e-16
>
> r.coeff <- sqrt(r.square)
> r.coeff
[1] 0.6764616
> cor(x,y)
[1] 0.6764616
>
R. Graph output
##############################
c/ms/2023/schedule/w10.lecture.note.txt · Last modified: 2023/05/08 01:56 by hkimscil