c:ms:2025:schedule:w13.lecture.note
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Table of Contents
MR
# multiple regression: a simple e.g.
#
#
rm(list=ls())
d <- read.csv("http://commres.net/wiki/_media/regression01-bankaccount.csv")
d
colnames(d) <- c("y", "x1", "x2")
d
# y = 통장갯수
# x1 = 인컴
# x2 = 부양가족수
lm.y.x1 <- lm(y ~ x1, data=d)
summary(lm.y.x1)
anova(lm.y.x1)
cor(d$x1, d$y)^2
summary(lm.y.x1)$r.squared
lm.y.x2 <- lm(y ~ x2, data=d)
summary(lm.y.x2)
anova(lm.y.x2)
cor(d$x2, d$y)^2
summary(lm.y.x2)$r.squared
lm.y.x1x2 <- lm(y ~ x1+x2, data=d)
summary(lm.y.x1x2)
anova(lm.y.x1x2)
bcd <- summary(lm.y.x1x2)$r.squared
bcd
bc.cd <- summary(lm.y.x1)$r.squared + summary(lm.y.x2)$r.squared
bc.cd
bc.cd - bcd # note that this is the amount that shared by the two IVs
lm.y.x1x2$coefficient
# y.hat = 6.399103 + (0.01184145)*x1 + (−0.54472725)*x2
a <- lm.y.x1x2$coefficient[1]
b1 <- lm.y.x1x2$coefficient[2]
b2 <- lm.y.x1x2$coefficient[3]
a
b1
b2
y.pred <- a + (b1 * d$x1) + (b2 * d$x2)
y.pred
# or
y.pred2 <- predict(lm.y.x1x2)
head(y.pred == y.pred2)
y.real <- d$y
y.real
y.mean <- mean(d$y)
y.mean
deviation.score <- d$y - y.mean
ds <- deviation.score
res <- y.real - y.pred
reg <- y.pred - y.mean
y.mean
# remember y is sum of res + reg + y.mean
y2 <- res + reg + y.mean
d$y==y2
ss.tot <- sum(ds^2)
ss.res <- sum(res^2)
ss.reg <- sum(reg^2)
ss.tot2 <- var(d$y) * (length(d$y)-1)
ss.tot
ss.tot2
ss.res
ss.reg
ss.res+ss.reg
k <- 3 # # of parameters a, b1, b2
df.tot <- length(y)-1
df.reg <- k - 1
df.res <- df.tot - df.reg
ms.reg <- ss.reg/df.reg
ms.res <- ss.res/df.res
ms.reg
ms.res
f.val <- ms.reg/ms.res
f.val
p.val <- pf(f.val, df.reg, df.res, lower.tail = F)
p.val
# double check
summary(lm.y.x1x2)
anova(lm.y.x1x2)
summary(lm(y~x2+x1, data=d))
anova(lm(y~x2+x1, data=d))
# note on 2 t-tests in summary
# anova에서의 x1, x2에 대한 테스트와
# lm에서의 x1, x2에 대한 테스트 (t-test) 간에
# 차이가 있음에 주의 (x1, x2에 대한 Pr 값이
# 다름). 그 이유는
# t-tests는 __pr__ 테스트로 테스트를
# (spr, zero_order_r 테스트가 아님) 하고
# anova test는 x1 전체에 대한 테스트 하고
# x2는 x1에 대한 테스트 외에 나머지를 가지고
# 테스트하기 때문에 그러함
# 또한 anova test에서 두번째 IV의 F값은
# summary(lm)에서 두번 째 IV의 t값의 제곱값
# 임을 이해. 이는 두번 째 IV의 설명력을 나
# 타내는 부분이 lm과 anova 모두 같기 때문
# 반면에 첫번째 IV의 경우에는 lm 분석 때에는
# 고유의 설명력만을 가지고 (semi-partial cor^2)
# 판단을 하는 반면에, anova는 x2와 공유하는
# 설명력도 포함해서 분석하기 때문에 t값의
# 제곱이 F값이 되지 못함
# beta에 대한 설명
# beta coefficient (standardized b)
# beta <- b * (sd(x)/sd(y))
beta1 <- b1 * (sd(x1)/sd(y))
beta2 <- b2 * (sd(x2)/sd(y))
beta1
beta2
# install.packages("lm.beta")
library(lm.beta)
lm.beta(lm.y.x1x2)
#######################################################
# partial correlation coefficient and pr2
# x2's explanation?
# understand with diagrams first
# then calculate with r
lm.tmp.1 <- lm(x2~x1, data=d)
res.x2.x1 <- lm.tmp.1$residuals
lm.tmp.2 <- lm(y~x1, data=d)
res.y.x1 <- lm.tmp.2$residuals
lm.tmp.3 <- lm(res.y.x1 ~ res.x2.x1, data=d)
summary(lm.tmp.3)
summary(lm.tmp.3)$r.squared
sqrt(summary(lm.tmp.3)$r.squared)
# install.packages("ppcor")
library(ppcor)
partial.r <- pcor.test(d$y, d$x2, d$x1)
str(partial.r)
partial.r$estimate
summary(lm.tmp.3)
summary(lm.tmp.3)$r.square
partial.r$estimate^2
# x1's own explanation?
lm.tmp.4 <- lm(x1~x2, data=d)
res.x1.x2 <- lm.tmp.4$residuals
lm.tmp.5 <- lm(y~x2, data=d)
res.y.x2 <- lm.tmp.5$residuals
lm.tmp.6 <- lm(res.y.x2 ~ res.x1.x2, data=d)
summary(lm.tmp.6)
partial.r <- pcor.test(d$y, d$x1, d$x2)
str(partial.r)
partial.r$estimate # this is partial correlation, not pr^2
# in order to get pr2, you should ^2
partial.r$estimate^2
summary(lm.tmp.6)$r.squared
#######################################################
# semipartial correlation coefficient and spr2
#
spr.y.x2.x1 <- spcor.test(d$y,d$x2,d$x1)
spr.y.x1.x2 <- spcor.test(d$y,d$x1,d$x2)
spr.y.x2.x1
spr.y.x1.x2
spr2.y.x2.x1 <- spr.y.x2.x1$estimate^2
spr2.y.x1.x2 <- spr.y.x1.x2$estimate^2
spr2.y.x2.x1
spr2.y.x1.x2
lm.tmp.7 <- lm(y ~ res.x2.x1, data = d)
summary(lm.tmp.7)
spr2.y.x2.x1
lm.tmp.8 <- lm(y~res.x1.x2, data = d)
summary(lm.tmp.8)
spr2.y.x1.x2
bcd # remember bcd in the above?
bd <- spr2.y.x2.x1 + spr2.y.x1.x2
bd
bcd - bd
#######################################################
# get the common area that explain the y variable
# 1.
summary(lm.y.x2)
all.x2 <- summary(lm.y.x2)$r.squared
all.x2
spr2.y.x2.x1
cma.1 <- all.x2 - spr2.y.x2.x1
cma.1
# 2.
summary(lm.y.x1)
all.x1 <- summary(lm.y.x1)$r.squared
all.x1
spr2.y.x1.x2
cma.2 <- all.x1 - spr2.y.x1.x2
cma.2
# OR 3.
summary(lm.y.x1x2)
r2.y.x1x2 <- summary(lm.y.x1x2)$r.square
r2.y.x1x2
spr2.y.x1.x2
spr2.y.x2.x1
cma.3 <- r2.y.x1x2 - (spr2.y.x1.x2 + spr2.y.x2.x1)
bcd - bd
cma.3
cma.1
cma.2
cma.3
# OR 애초에 simple regression과 multiple
# regression에서 얻은 R2을 가지고
# 공통설명력을 알아볼 수도 있었다.
r2.y.x1 <- summary(lm.y.x1)$r.square
r2.y.x2 <- summary(lm.y.x2)$r.square
r2.y.x1
r2.y.x2
r2.y.x1x2 <- summary(lm.y.x1x2)$r.square
r2.y.x1x2
cma.4 <- r2.y.x1 + r2.y.x2 - r2.y.x1x2
cma.4
# Note that sorting out unique and common
# explanation area is only possible with
# semi-partial correlation determinant
# NOT partial correlation determinant
# because only semi-partial correlation
# shares the same denominator (as total
# y).
#############################################
output
> # multiple regression: a simple e.g.
> #
> #
> rm(list=ls())
> d <- read.csv("http://commres.net/wiki/_media/regression01-bankaccount.csv")
> d
bankaccount income famnum
1 6 220 5
2 5 190 6
3 7 260 3
4 7 200 4
5 8 330 2
6 10 490 4
7 8 210 3
8 11 380 2
9 9 320 1
10 9 270 3
>
> colnames(d) <- c("y", "x1", "x2")
> d
y x1 x2
1 6 220 5
2 5 190 6
3 7 260 3
4 7 200 4
5 8 330 2
6 10 490 4
7 8 210 3
8 11 380 2
9 9 320 1
10 9 270 3
> # y = 통장갯수
> # x1 = 인컴
> # x2 = 부양가족수
> lm.y.x1 <- lm(y ~ x1, data=d)
> summary(lm.y.x1)
Call:
lm(formula = y ~ x1, data = d)
Residuals:
Min 1Q Median 3Q Max
-1.519 -0.897 -0.130 1.006 1.580
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.61778 1.24152 2.91 0.019 *
x1 0.01527 0.00413 3.70 0.006 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 1.18 on 8 degrees of freedom
Multiple R-squared: 0.631, Adjusted R-squared: 0.585
F-statistic: 13.7 on 1 and 8 DF, p-value: 0.00605
> anova(lm.y.x1)
Analysis of Variance Table
Response: y
Df Sum Sq Mean Sq F value Pr(>F)
x1 1 18.9 18.93 13.7 0.006 **
Residuals 8 11.1 1.38
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> cor(d$x1, d$y)^2
[1] 0.6311
> summary(lm.y.x1)$r.squared
[1] 0.6311
>
>
> lm.y.x2 <- lm(y ~ x2, data=d)
> summary(lm.y.x2)
Call:
lm(formula = y ~ x2, data = d)
Residuals:
Min 1Q Median 3Q Max
-1.254 -0.888 -0.485 0.496 2.592
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 10.791 1.119 9.64 1.1e-05 ***
x2 -0.846 0.312 -2.71 0.027 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 1.4 on 8 degrees of freedom
Multiple R-squared: 0.479, Adjusted R-squared: 0.414
F-statistic: 7.36 on 1 and 8 DF, p-value: 0.0265
> anova(lm.y.x2)
Analysis of Variance Table
Response: y
Df Sum Sq Mean Sq F value Pr(>F)
x2 1 14.4 14.38 7.36 0.027 *
Residuals 8 15.6 1.95
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> cor(d$x2, d$y)^2
[1] 0.4793
> summary(lm.y.x2)$r.squared
[1] 0.4793
>
>
> lm.y.x1x2 <- lm(y ~ x1+x2, data=d)
> summary(lm.y.x1x2)
Call:
lm(formula = y ~ x1 + x2, data = d)
Residuals:
Min 1Q Median 3Q Max
-1.217 -0.578 -0.151 0.664 1.191
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 6.39910 1.51654 4.22 0.0039 **
x1 0.01184 0.00356 3.33 0.0127 *
x2 -0.54473 0.22636 -2.41 0.0470 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.93 on 7 degrees of freedom
Multiple R-squared: 0.798, Adjusted R-squared: 0.74
F-statistic: 13.8 on 2 and 7 DF, p-value: 0.0037
> anova(lm.y.x1x2)
Analysis of Variance Table
Response: y
Df Sum Sq Mean Sq F value Pr(>F)
x1 1 18.93 18.93 21.88 0.0023 **
x2 1 5.01 5.01 5.79 0.0470 *
Residuals 7 6.06 0.87
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> bcd <- summary(lm.y.x1x2)$r.squared
> bcd
[1] 0.7981
> bc.cd <- summary(lm.y.x1)$r.squared + summary(lm.y.x2)$r.squared
> bc.cd
[1] 1.11
> bc.cd - bcd # note that this is the amount that shared by the two IVs
[1] 0.3123
>
>
> lm.y.x1x2$coefficient
(Intercept) x1 x2
6.39910 0.01184 -0.54473
> # y.hat = 6.399103 + (0.01184145)*x1 + (−0.54472725)*x2
> a <- lm.y.x1x2$coefficient[1]
> b1 <- lm.y.x1x2$coefficient[2]
> b2 <- lm.y.x1x2$coefficient[3]
> a
(Intercept)
6.399
> b1
x1
0.01184
> b2
x2
-0.5447
>
> y.pred <- a + (b1 * d$x1) + (b2 * d$x2)
> y.pred
[1] 6.281 5.381 7.844 6.588 9.217 10.023 7.252 9.809 9.644 7.962
> # or
> y.pred2 <- predict(lm.y.x1x2)
> head(y.pred == y.pred2)
1 2 3 4 5 6
TRUE TRUE TRUE TRUE TRUE TRUE
>
> y.real <- d$y
> y.real
[1] 6 5 7 7 8 10 8 11 9 9
> y.mean <- mean(d$y)
> y.mean
[1] 8
>
> deviation.score <- d$y - y.mean
> ds <- deviation.score
> res <- y.real - y.pred
> reg <- y.pred - y.mean
> y.mean
[1] 8
> # remember y is sum of res + reg + y.mean
> y2 <- res + reg + y.mean
> d$y==y2
[1] TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
>
> ss.tot <- sum(ds^2)
> ss.res <- sum(res^2)
> ss.reg <- sum(reg^2)
>
> ss.tot2 <- var(d$y) * (length(d$y)-1)
> ss.tot
[1] 30
> ss.tot2
[1] 30
> ss.res
[1] 6.056
> ss.reg
[1] 23.94
> ss.res+ss.reg
[1] 30
>
> k <- 3 # # of parameters a, b1, b2
> df.tot <- length(d$y)-1
> df.reg <- k - 1
> df.res <- df.tot - df.reg
>
> ms.reg <- ss.reg/df.reg
> ms.res <- ss.res/df.res
> ms.reg
[1] 11.97
> ms.res
[1] 0.8652
> f.val <- ms.reg/ms.res
> f.val
[1] 13.84
> p.val <- pf(f.val, df.reg, df.res, lower.tail = F)
> p.val
[1] 0.003696
>
> # double check
> summary(lm.y.x1x2)
Call:
lm(formula = y ~ x1 + x2, data = d)
Residuals:
Min 1Q Median 3Q Max
-1.217 -0.578 -0.151 0.664 1.191
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 6.39910 1.51654 4.22 0.0039 **
x1 0.01184 0.00356 3.33 0.0127 *
x2 -0.54473 0.22636 -2.41 0.0470 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.93 on 7 degrees of freedom
Multiple R-squared: 0.798, Adjusted R-squared: 0.74
F-statistic: 13.8 on 2 and 7 DF, p-value: 0.0037
> anova(lm.y.x1x2)
Analysis of Variance Table
Response: y
Df Sum Sq Mean Sq F value Pr(>F)
x1 1 18.93 18.93 21.88 0.0023 **
x2 1 5.01 5.01 5.79 0.0470 *
Residuals 7 6.06 0.87
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
>
> summary(lm(y~x2+x1, data=d))
Call:
lm(formula = y ~ x2 + x1, data = d)
Residuals:
Min 1Q Median 3Q Max
-1.217 -0.578 -0.151 0.664 1.191
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 6.39910 1.51654 4.22 0.0039 **
x2 -0.54473 0.22636 -2.41 0.0470 *
x1 0.01184 0.00356 3.33 0.0127 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.93 on 7 degrees of freedom
Multiple R-squared: 0.798, Adjusted R-squared: 0.74
F-statistic: 13.8 on 2 and 7 DF, p-value: 0.0037
> anova(lm(y~x2+x1, data=d))
Analysis of Variance Table
Response: y
Df Sum Sq Mean Sq F value Pr(>F)
x2 1 14.38 14.38 16.6 0.0047 **
x1 1 9.57 9.57 11.1 0.0127 *
Residuals 7 6.06 0.87
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
>
> # note on 2 t-tests in summary
> # anova에서의 x1, x2에 대한 테스트와
> # lm에서의 x1, x2에 대한 테스트 (t-test) 간에
> # 차이가 있음에 주의 (x1, x2에 대한 Pr 값이
> # 다름). 그 이유는
> # t-tests는 __pr__ 테스트로 테스트를
> # (spr, zero_order_r 테스트가 아님) 하고
> # anova test는 x1 전체에 대한 테스트 하고
> # x2는 x1에 대한 테스트 외에 나머지를 가지고
> # 테스트하기 때문에 그러함
>
> # 또한 anova test에서 두번째 IV의 F값은
> # summary(lm)에서 두번 째 IV의 t값의 제곱값
> # 임을 이해. 이는 두번 째 IV의 설명력을 나
> # 타내는 부분이 lm과 anova 모두 같기 때문
> # 반면에 첫번째 IV의 경우에는 lm 분석 때에는
> # 고유의 설명력만을 가지고 (semi-partial cor^2)
> # 판단을 하는 반면에, anova는 x2와 공유하는
> # 설명력도 포함해서 분석하기 때문에 t값의
> # 제곱이 F값이 되지 못함
>
> # beta에 대한 설명
> # beta coefficient (standardized b)
> # beta <- b * (sd(x)/sd(y))
> beta1 <- b1 * (sd(d$x1)/sd(d$y))
> beta2 <- b2 * (sd(d$x2)/sd(d$y))
> beta1
x1
0.6161
> beta2
x2
-0.4459
>
> # install.packages("lm.beta")
> library(lm.beta)
경고메시지(들):
패키지 ‘lm.beta’는 R 버전 4.3.3에서 작성되었습니다
> lm.beta(lm.y.x1x2)
Call:
lm(formula = y ~ x1 + x2, data = d)
Standardized Coefficients::
(Intercept) x1 x2
NA 0.6161 -0.4459
>
> #######################################################
> # partial correlation coefficient and pr2
> # x2's explanation?
> # understand with diagrams first
> # then calculate with r
> lm.tmp.1 <- lm(x2~x1, data=d)
> res.x2.x1 <- lm.tmp.1$residuals
>
> lm.tmp.2 <- lm(y~x1, data=d)
> res.y.x1 <- lm.tmp.2$residuals
>
> lm.tmp.3 <- lm(res.y.x1 ~ res.x2.x1, data=d)
> summary(lm.tmp.3)
Call:
lm(formula = res.y.x1 ~ res.x2.x1, data = d)
Residuals:
Min 1Q Median 3Q Max
-1.217 -0.578 -0.151 0.664 1.191
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 6.28e-18 2.75e-01 0.00 1.000
res.x2.x1 -5.45e-01 2.12e-01 -2.57 0.033 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.87 on 8 degrees of freedom
Multiple R-squared: 0.453, Adjusted R-squared: 0.384
F-statistic: 6.62 on 1 and 8 DF, p-value: 0.033
> summary(lm.tmp.3)$r.squared
[1] 0.4527
> sqrt(summary(lm.tmp.3)$r.squared)
[1] 0.6729
> # install.packages("ppcor")
> library(ppcor)
> partial.r <- pcor.test(d$y, d$x2, d$x1)
> str(partial.r)
'data.frame': 1 obs. of 6 variables:
$ estimate : num -0.673
$ p.value : num 0.047
$ statistic: num -2.41
$ n : int 10
$ gp : num 1
$ Method : chr "pearson"
> partial.r$estimate
[1] -0.6729
> summary(lm.tmp.3)
Call:
lm(formula = res.y.x1 ~ res.x2.x1, data = d)
Residuals:
Min 1Q Median 3Q Max
-1.217 -0.578 -0.151 0.664 1.191
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 6.28e-18 2.75e-01 0.00 1.000
res.x2.x1 -5.45e-01 2.12e-01 -2.57 0.033 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.87 on 8 degrees of freedom
Multiple R-squared: 0.453, Adjusted R-squared: 0.384
F-statistic: 6.62 on 1 and 8 DF, p-value: 0.033
> summary(lm.tmp.3)$r.square
[1] 0.4527
> partial.r$estimate^2
[1] 0.4527
>
>
> # x1's own explanation?
> lm.tmp.4 <- lm(x1~x2, data=d)
> res.x1.x2 <- lm.tmp.4$residuals
>
> lm.tmp.5 <- lm(y~x2, data=d)
> res.y.x2 <- lm.tmp.5$residuals
>
> lm.tmp.6 <- lm(res.y.x2 ~ res.x1.x2, data=d)
> summary(lm.tmp.6)
Call:
lm(formula = res.y.x2 ~ res.x1.x2, data = d)
Residuals:
Min 1Q Median 3Q Max
-1.217 -0.578 -0.151 0.664 1.191
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.33e-17 2.75e-01 0.00 1.0000
res.x1.x2 1.18e-02 3.33e-03 3.55 0.0075 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.87 on 8 degrees of freedom
Multiple R-squared: 0.612, Adjusted R-squared: 0.564
F-statistic: 12.6 on 1 and 8 DF, p-value: 0.00746
>
> partial.r <- pcor.test(d$y, d$x1, d$x2)
> str(partial.r)
'data.frame': 1 obs. of 6 variables:
$ estimate : num 0.783
$ p.value : num 0.0127
$ statistic: num 3.33
$ n : int 10
$ gp : num 1
$ Method : chr "pearson"
> partial.r$estimate # this is partial correlation, not pr^2
[1] 0.7825
> # in order to get pr2, you should ^2
> partial.r$estimate^2
[1] 0.6123
> summary(lm.tmp.6)$r.squared
[1] 0.6123
>
> #######################################################
> # semipartial correlation coefficient and spr2
> #
> spr.y.x2.x1 <- spcor.test(d$y,d$x2,d$x1)
> spr.y.x1.x2 <- spcor.test(d$y,d$x1,d$x2)
> spr.y.x2.x1
estimate p.value statistic n gp Method
1 -0.4087 0.2748 -1.185 10 1 pearson
> spr.y.x1.x2
estimate p.value statistic n gp Method
1 0.5647 0.1132 1.81 10 1 pearson
> spr2.y.x2.x1 <- spr.y.x2.x1$estimate^2
> spr2.y.x1.x2 <- spr.y.x1.x2$estimate^2
> spr2.y.x2.x1
[1] 0.167
> spr2.y.x1.x2
[1] 0.3189
>
> lm.tmp.7 <- lm(y ~ res.x2.x1, data = d)
> summary(lm.tmp.7)
Call:
lm(formula = y ~ res.x2.x1, data = d)
Residuals:
Min 1Q Median 3Q Max
-1.862 -1.171 -0.494 0.549 3.077
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 8.000 0.559 14.31 5.5e-07 ***
res.x2.x1 -0.545 0.430 -1.27 0.24
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 1.77 on 8 degrees of freedom
Multiple R-squared: 0.167, Adjusted R-squared: 0.0629
F-statistic: 1.6 on 1 and 8 DF, p-value: 0.241
> spr2.y.x2.x1
[1] 0.167
>
> lm.tmp.8 <- lm(y~res.x1.x2, data = d)
> summary(lm.tmp.8)
Call:
lm(formula = y ~ res.x1.x2, data = d)
Residuals:
Min 1Q Median 3Q Max
-2.664 -0.608 -0.149 1.219 2.290
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 8.00000 0.50540 15.83 2.5e-07 ***
res.x1.x2 0.01184 0.00612 1.94 0.089 .
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 1.6 on 8 degrees of freedom
Multiple R-squared: 0.319, Adjusted R-squared: 0.234
F-statistic: 3.74 on 1 and 8 DF, p-value: 0.089
> spr2.y.x1.x2
[1] 0.3189
>
> bcd # remember bcd in the above?
[1] 0.7981
> bd <- spr2.y.x2.x1 + spr2.y.x1.x2
> bd
[1] 0.4859
> bcd - bd
[1] 0.3123
>
> #######################################################
> # get the common area that explain the y variable
> # 1.
> summary(lm.y.x2)
Call:
lm(formula = y ~ x2, data = d)
Residuals:
Min 1Q Median 3Q Max
-1.254 -0.888 -0.485 0.496 2.592
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 10.791 1.119 9.64 1.1e-05 ***
x2 -0.846 0.312 -2.71 0.027 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 1.4 on 8 degrees of freedom
Multiple R-squared: 0.479, Adjusted R-squared: 0.414
F-statistic: 7.36 on 1 and 8 DF, p-value: 0.0265
> all.x2 <- summary(lm.y.x2)$r.squared
> all.x2
[1] 0.4793
> spr2.y.x2.x1
[1] 0.167
> cma.1 <- all.x2 - spr2.y.x2.x1
> cma.1
[1] 0.3123
>
> # 2.
> summary(lm.y.x1)
Call:
lm(formula = y ~ x1, data = d)
Residuals:
Min 1Q Median 3Q Max
-1.519 -0.897 -0.130 1.006 1.580
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.61778 1.24152 2.91 0.019 *
x1 0.01527 0.00413 3.70 0.006 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 1.18 on 8 degrees of freedom
Multiple R-squared: 0.631, Adjusted R-squared: 0.585
F-statistic: 13.7 on 1 and 8 DF, p-value: 0.00605
> all.x1 <- summary(lm.y.x1)$r.squared
> all.x1
[1] 0.6311
> spr2.y.x1.x2
[1] 0.3189
> cma.2 <- all.x1 - spr2.y.x1.x2
> cma.2
[1] 0.3123
>
> # OR 3.
> summary(lm.y.x1x2)
Call:
lm(formula = y ~ x1 + x2, data = d)
Residuals:
Min 1Q Median 3Q Max
-1.217 -0.578 -0.151 0.664 1.191
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 6.39910 1.51654 4.22 0.0039 **
x1 0.01184 0.00356 3.33 0.0127 *
x2 -0.54473 0.22636 -2.41 0.0470 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.93 on 7 degrees of freedom
Multiple R-squared: 0.798, Adjusted R-squared: 0.74
F-statistic: 13.8 on 2 and 7 DF, p-value: 0.0037
> r2.y.x1x2 <- summary(lm.y.x1x2)$r.square
> r2.y.x1x2
[1] 0.7981
> spr2.y.x1.x2
[1] 0.3189
> spr2.y.x2.x1
[1] 0.167
> cma.3 <- r2.y.x1x2 - (spr2.y.x1.x2 + spr2.y.x2.x1)
> bcd - bd
[1] 0.3123
> cma.3
[1] 0.3123
>
> cma.1
[1] 0.3123
> cma.2
[1] 0.3123
> cma.3
[1] 0.3123
>
> # OR 애초에 simple regression과 multiple
> # regression에서 얻은 R2을 가지고
> # 공통설명력을 알아볼 수도 있었다.
> r2.y.x1 <- summary(lm.y.x1)$r.square
> r2.y.x2 <- summary(lm.y.x2)$r.square
> r2.y.x1
[1] 0.6311
> r2.y.x2
[1] 0.4793
> r2.y.x1x2 <- summary(lm.y.x1x2)$r.square
> r2.y.x1x2
[1] 0.7981
> cma.4 <- r2.y.x1 + r2.y.x2 - r2.y.x1x2
> cma.4
[1] 0.3123
>
> # Note that sorting out unique and common
> # explanation area is only possible with
> # semi-partial correlation determinant
> # NOT partial correlation determinant
> # because only semi-partial correlation
> # shares the same denominator (as total
> # y).
> #############################################
>
explanation. added
# ex. # resid(lm(y~x1, data=df)) = bc / delta.y # resid(lm(y~x2, data=df)) = cd / delta.y # resid(lm(y~x1+x2, data=df)) = bcd / delta.y # b / delta.y = ? # ce / delta.x2 = ?
# exp.added
spcor.test(df$y, df$x1, df$x2)
spcor.test(df$y, df$x1, df$x2)$estimate
spcor.test(df$y, df$x1, df$x2)$estimate^2
spcor.test(df$y, df$x2, df$x1)$estimate^2
summary(lm(y~x1+x2, data=df))$r.square
b <- spcor.test(df$y, df$x1, df$x2)$estimate^2
d <- spcor.test(df$y, df$x2, df$x1)$estimate^2
bcd <- summary(lm(y~x1+x2, data=df))$r.square
summary(lm(df$y~df$x1+df$x2, data=df))$r.square -
(spcor.test(df$y, df$x1, df$x2)$estimate^2 +
spcor.test(df$y, df$x2, df$x1)$estimate^2)
bcd - (b + d)
Another one
c/ms/2025/schedule/w13.lecture.note.1749424134.txt.gz · Last modified: 2025/06/09 08:08 by hkimscil