c:ms:multiple_regression_lecture_note_for_r
Table of Contents
Multiple regression with pr, spr, zero-order r
# 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
# attach(d)
lm.y.x1 <- lm(y ~ x1, data=d)
summary(lm.y.x1)
lm.y.x2 <- lm(y ~ x2, data=d)
summary(lm.y.x2)
lm.y.x1x2 <- lm(y ~ x1+x2, data=d)
summary(lm.y.x1x2)
lm.y.x1x2$coefficient
# y.hat = 6.399103 + (0.011841)*x1 + (−0.544727)*x2
a <- lm.y.x1x2$coefficient[1]
b1 <- lm.y.x1x2$coefficient[2]
b2 <- lm.y.x1x2$coefficient[3]
y.pred <- a + (b1 * x1) + (b2 * x2)
y.pred
y.real <- y
y.real
y.mean <- mean(y)
y.mean
res <- y.real - y.pred
reg <- y.pred - y.mean
ss.res <- sum(res^2)
ss.reg <- sum(reg^2)
ss.tot <- var(y) * (length(y)-1)
ss.tot
ss.res
ss.reg
ss.res+ss.reg
# slope test
summary(lm.y.x1x2)
# note on 2 t-tests
# 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?
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)
# install.packages("ppcor")
library(ppcor)
pcor(d)
spcor(d)
partial.r <- pcor.test(y, x2, x1)
partial.r
str(partial.r)
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$residuals
lm.tmp.6 <- lm(res.y.x2 ~ res.x1.x2, data=d)
summary(lm.tmp.6)
partial.r2 <- pcor.test(y, x1, x2)
str(partial.r2)
partial.r2$estimate^2
#######################################################
# semipartial correlation coefficient and spr2
#
spr.1 <- spcor.test(y,x2,x1)
spr.2 <- spcor.test(y,x1,x2)
spr.1
spr.2
spr.1$estimate^2
spr.2$estimate^2
lm.tmp.7 <- lm(y ~ res.x2.x1, data = d)
summary(lm.tmp.7)
#######################################################
# get the common area that explain the y variable
# 1.
summary(lm.y.x2)
all.x2 <- summary(lm.y.x2)$r.squared
sp.x2 <- spr.1$estimate^2
all.x2
sp.x2
cma.1 <- all.x2 - sp.x2
cma.1
# 2.
summary(lm.y.x1)
all.x1 <- summary(lm.y.x1)$r.squared
sp.x1 <- spr.2$estimate^2
all.x1
sp.x1
cma.2 <- all.x1 - sp.x1
cma.2
# OR 3.
summary(lm.y.x1x2)
r2.y.x1x2 <- summary(lm.y.x1x2)$r.square
r2.y.x1x2
sp.x1
sp.x2
cma.3 <- r2.y.x1x2 - (sp.x1 + sp.x2)
cma.3
cma.1
cma.2
cma.3
# Note that sorting out unique and common
# explanation area is only possible with
# semi-partial correlation determinant
# NOT partial correlation determinant
# because only semi-partial correlation
# shares the same denominator (as total
# y).
#############################################
Output
Multiple regression
>
> lm.y.x1x2 <- lm(y ~ x1+x2, data=d)
> summary(lm.y.x1x2)
Call:
lm(formula = y ~ x1 + x2, data = d)
Residuals:
Min 1Q Median 3Q Max
-1.2173 -0.5779 -0.1515 0.6642 1.1906
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 6.399103 1.516539 4.220 0.00394 **
x1 0.011841 0.003561 3.325 0.01268 *
x2 -0.544727 0.226364 -2.406 0.04702 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.9301 on 7 degrees of freedom
Multiple R-squared: 0.7981, Adjusted R-squared: 0.7404
F-statistic: 13.84 on 2 and 7 DF, p-value: 0.003696
>
>
> lm.y.x1x2$coefficient
(Intercept) x1 x2
6.39910298 0.01184145 -0.54472725
> # y.hat = 6.399103 + (0.011841)*x1 + (−0.544727)*x2
> a <- lm.y.x1x2$coefficient[1]
> b1 <- lm.y.x1x2$coefficient[2]
> b2 <- lm.y.x1x2$coefficient[3]
>
> y.pred <- a + (b1 * x1) + (b2 * x2)
> y.pred
[1] 6.280586 5.380616 7.843699 6.588485 9.217328 10.022506
[7] 7.251626 9.809401 9.643641 7.962113
> y.real <- y
> y.real
[1] 6 5 7 7 8 10 8 11 9 9
> y.mean <- mean(y)
> y.mean
[1] 8
>
> res <- y.real - y.pred
> reg <- y.pred - y.mean
> ss.res <- sum(res^2)
> ss.reg <- sum(reg^2)
>
> ss.tot <- var(y) * (length(y)-1)
> ss.tot
[1] 30
> ss.res
[1] 6.056235
> ss.reg
[1] 23.94376
> ss.res+ss.reg
[1] 30
>
> # slope test
> summary(lm.y.x1x2)
Call:
lm(formula = y ~ x1 + x2, data = d)
Residuals:
Min 1Q Median 3Q Max
-1.2173 -0.5779 -0.1515 0.6642 1.1906
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 6.399103 1.516539 4.220 0.00394 **
x1 0.011841 0.003561 3.325 0.01268 *
x2 -0.544727 0.226364 -2.406 0.04702 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.9301 on 7 degrees of freedom
Multiple R-squared: 0.7981, Adjusted R-squared: 0.7404
F-statistic: 13.84 on 2 and 7 DF, p-value: 0.003696
> # note on 2 t-tests
>
> # beta coefficient (standardized b)
> # beta <- b * (sd(x)/sd(y))
> beta1 <- b1 * (sd(x1)/sd(y))
> beta2 <- b2 * (sd(x2)/sd(y))
> beta1
x1
0.616097
> beta2
x2
-0.4458785
>
> # install.packages("lm.beta")
> library(lm.beta)
> lm.beta(lm.y.x1x2)
Call:
lm(formula = y ~ x1 + x2, data = d)
Standardized Coefficients::
(Intercept) x1 x2
NA 0.6160970 -0.4458785
>
> #######################################################
> # partial correlation coefficient and pr2
> # x2's explanation?
> lm.tmp.1 <- lm(x2~x1, data=d)
> res.x2.x1 <- lm.tmp.1$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
-1.2173 -0.5779 -0.1515 0.6642 1.1906
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 6.281e-18 2.751e-01 0.000 1.000
res.x2.x1 -5.447e-01 2.117e-01 -2.573 0.033 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.8701 on 8 degrees of freedom
Multiple R-squared: 0.4527, Adjusted R-squared: 0.3843
F-statistic: 6.618 on 1 and 8 DF, p-value: 0.033
>
> # install.packages("ppcor")
> library(ppcor)
> pcor(d)
$estimate
y x1 x2
y 1.0000000 0.7825112 -0.6728560
x1 0.7825112 1.0000000 0.3422911
x2 -0.6728560 0.3422911 1.0000000
$p.value
y x1 x2
y 0.00000000 0.01267595 0.04702022
x1 0.01267595 0.00000000 0.36723388
x2 0.04702022 0.36723388 0.00000000
$statistic
y x1 x2
y 0.000000 3.3251023 -2.4064253
x1 3.325102 0.0000000 0.9638389
x2 -2.406425 0.9638389 0.0000000
$n
[1] 10
$gp
[1] 1
$method
[1] "pearson"
> spcor(d)
$estimate
y x1 x2
y 1.0000000 0.5646726 -0.4086619
x1 0.7171965 1.0000000 0.2078919
x2 -0.6166940 0.2470028 1.0000000
$p.value
y x1 x2
y 0.00000000 0.113182 0.2748117
x1 0.02964029 0.000000 0.5914441
x2 0.07691195 0.521696 0.0000000
$statistic
y x1 x2
y 0.000000 1.8101977 -1.1846548
x1 2.722920 0.0000000 0.5623159
x2 -2.072679 0.6744045 0.0000000
$n
[1] 10
$gp
[1] 1
$method
[1] "pearson"
> partial.r <- pcor.test(y, x2, x1)
> partial.r
estimate p.value statistic n gp Method
1 -0.672856 0.04702022 -2.406425 10 1 pearson
> 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$estimate^2
[1] 0.4527352
>
> # x1's own explanation?
> lm.tmp.4 <- lm(x1~x2, data=d)
> res.x1.x2 <- lm.tmp.4$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
-1.2173 -0.5779 -0.1515 0.6642 1.1906
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.330e-17 2.751e-01 0.000 1.00000
res.x1.x2 1.184e-02 3.331e-03 3.555 0.00746 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.8701 on 8 degrees of freedom
Multiple R-squared: 0.6123, Adjusted R-squared: 0.5639
F-statistic: 12.64 on 1 and 8 DF, p-value: 0.007458
>
> partial.r2 <- pcor.test(y, x1, x2)
> str(partial.r2)
'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.r2$estimate^2
[1] 0.6123238
> #######################################################
>
> # semipartial correlation coefficient and spr2
> #
> spr.1 <- spcor.test(y,x2,x1)
> spr.2 <- spcor.test(y,x1,x2)
> spr.1
estimate p.value statistic n gp Method
1 -0.4086619 0.2748117 -1.184655 10 1 pearson
> spr.2
estimate p.value statistic n gp Method
1 0.5646726 0.113182 1.810198 10 1 pearson
> spr.1$estimate^2
[1] 0.1670045
> spr.2$estimate^2
[1] 0.3188552
>
> lm.tmp.7 <- lm(y ~ res.x2.x1, data = d)
> summary(lm.tmp.7)
Call:
lm(formula = y ~ res.x2.x1, data = d)
Residuals:
Min 1Q Median 3Q Max
-1.8617 -1.1712 -0.4940 0.5488 3.0771
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 8.0000 0.5589 14.314 5.54e-07 ***
res.x2.x1 -0.5447 0.4301 -1.266 0.241
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 1.767 on 8 degrees of freedom
Multiple R-squared: 0.167, Adjusted R-squared: 0.06288
F-statistic: 1.604 on 1 and 8 DF, p-value: 0.241
> #######################################################
>
> # get the common area that explain the y variable
> # 1.
> summary(lm.y.x2)
Call:
lm(formula = y ~ x2, data = d)
Residuals:
Min 1Q Median 3Q Max
-1.2537 -0.8881 -0.4851 0.4963 2.5920
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 10.7910 1.1195 9.639 1.12e-05 ***
x2 -0.8458 0.3117 -2.713 0.0265 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 1.397 on 8 degrees of freedom
Multiple R-squared: 0.4793, Adjusted R-squared: 0.4142
F-statistic: 7.363 on 1 and 8 DF, p-value: 0.02651
> all.x2 <- summary(lm.y.x2)$r.squared
> sp.x2 <- spr.1$estimate^2
> all.x2
[1] 0.4792703
> sp.x2
[1] 0.1670045
> cma.1 <- all.x2 - sp.x2
> cma.1
[1] 0.3122658
>
> # 2.
> summary(lm.y.x1)
Call:
lm(formula = y ~ x1, data = d)
Residuals:
Min 1Q Median 3Q Max
-1.5189 -0.8969 -0.1297 1.0058 1.5800
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.617781 1.241518 2.914 0.01947 *
x1 0.015269 0.004127 3.700 0.00605 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 1.176 on 8 degrees of freedom
Multiple R-squared: 0.6311, Adjusted R-squared: 0.585
F-statistic: 13.69 on 1 and 8 DF, p-value: 0.006046
> all.x1 <- summary(lm.y.x1)$r.squared
> sp.x1 <- spr.2$estimate^2
> all.x1
[1] 0.631121
> sp.x1
[1] 0.3188552
> cma.2 <- all.x1 - sp.x1
> cma.2
[1] 0.3122658
>
> # OR 3.
> summary(lm.y.x1x2)
Call:
lm(formula = y ~ x1 + x2, data = d)
Residuals:
Min 1Q Median 3Q Max
-1.2173 -0.5779 -0.1515 0.6642 1.1906
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 6.399103 1.516539 4.220 0.00394 **
x1 0.011841 0.003561 3.325 0.01268 *
x2 -0.544727 0.226364 -2.406 0.04702 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.9301 on 7 degrees of freedom
Multiple R-squared: 0.7981, Adjusted R-squared: 0.7404
F-statistic: 13.84 on 2 and 7 DF, p-value: 0.003696
> r2.y.x1x2 <- summary(lm.y.x1x2)$r.square
> r2.y.x1x2
[1] 0.7981255
> sp.x1
[1] 0.3188552
> sp.x2
[1] 0.1670045
> cma.3 <- r2.y.x1x2 - (sp.x1 + sp.x2)
> cma.3
[1] 0.3122658
>
> cma.1
[1] 0.3122658
> cma.2
[1] 0.3122658
> cma.3
[1] 0.3122658
> # 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).
> #############################################
>
>
>
Simple regression
> # 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
> # attach(d)
> 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
-1.5189 -0.8969 -0.1297 1.0058 1.5800
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.617781 1.241518 2.914 0.01947 *
x1 0.015269 0.004127 3.700 0.00605 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 1.176 on 8 degrees of freedom
Multiple R-squared: 0.6311, Adjusted R-squared: 0.585
F-statistic: 13.69 on 1 and 8 DF, p-value: 0.006046
단순회귀분석에서 (simple regression) F-test와 t-test는 (slope test) 기본적으로 똑 같은 테스트를 말한다. 왜냐하면 F-test에 기여하는 독립변인이 오직하나이고 그 하나가 slope test에 (t-test) 사용되기 때문이다. 이것은 t-test의 t값과 F-test의 F값의 관계에서도 나타난다.
$$ t^2 = F $$
> t.cal <- 3.7 > t.cal^2 [1] 13.69 > F.cal <- 13.69 > F.cal [1] 13.69
Simple regression에서 설명한 것처럼 기울기에 (slope) 대한 t-test는 기울기가 y 변인의 variability를 (평균을 중심으로 흔들림을) 설명하는 데 기여했는가를 테스트 하기 위한 것이다. 기울기가 0 이라면 이는 평균을 (평균선이 기울기가 0이다) 사용하는 것과 같으므로 기울기의 효과가 없음을 의미한다. 따라서 b와 b zero의 차이가 통계학적으로 의미있었는가를 t-test한다.
$$ \text{t calculated value} = \frac {b - 0}{se} $$
위에서 $se$는 아래처럼 구한다고 언급하였다.
\begin{eqnarray*} se & = & \sqrt{\frac{1}{n-2} * \frac{\text{SSE}}{\text{SSx}}} \\ & = & \sqrt{\frac {\text{MSE}} {\text{SSx}}} \\ \text{note that MSE } & = & \text{mean square error } \\ & = & \text{ms.res } \end{eqnarray*}
위에서 구한 t값의 p value는 R에서
summary(lm.y.x1) n <- length(y) k <- 1 # num of predictor variables sse <- sum(lm.y.x1$residuals^2) # ss.res ssx1 <- sum((x1-mean(x1))^2) b <- lm.y.x1$coefficient[2] se <- sqrt((1/(n-2))*(sse/ssx1)) t.b.cal <- (b - 0) / se t.b.cal p.value <- 2 * pt(t.b.cal, n-k-1, lower.tail=F) p.value # checck t.b.cal f.cal <- t.b.cal^2 f.cal p.value
> summary(lm.y.x1)
Call:
lm(formula = y ~ x1, data = d)
Residuals:
Min 1Q Median 3Q Max
-1.5189 -0.8969 -0.1297 1.0058 1.5800
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.617781 1.241518 2.914 0.01947 *
x1 0.015269 0.004127 3.700 0.00605 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 1.176 on 8 degrees of freedom
Multiple R-squared: 0.6311, Adjusted R-squared: 0.585
F-statistic: 13.69 on 1 and 8 DF, p-value: 0.006046
> n <- length(y)
> k <- 1 # num of predictor variables
> sse <- sum(lm.y.x1$residuals^2)
> ssx1 <- sum((x1-mean(x1))^2)
> b <- lm.y.x1$coefficient[2]
> se <-sqrt((1/(n-2))*(sse/ssx1))
> se <-sqrt(mse/ssx1)
> t.b.cal <- (b - 0) / se
> t.b.cal
x1
3.699639
> p.value <- 2 * pt(t.b.cal, n-k-1, lower.tail=F)
>
> # checck
> t.b.cal
x1
3.699639
> t.b.cal^2
x1
13.68733
> p.value
x1
0.006045749
>
>
>
> 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
-1.2537 -0.8881 -0.4851 0.4963 2.5920
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 10.7910 1.1195 9.639 1.12e-05 ***
x2 -0.8458 0.3117 -2.713 0.0265 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 1.397 on 8 degrees of freedom
Multiple R-squared: 0.4793, Adjusted R-squared: 0.4142
F-statistic: 7.363 on 1 and 8 DF, p-value: 0.02651
>
>
c/ms/multiple_regression_lecture_note_for_r.txt · Last modified: 2024/09/30 08:56 by hkimscil