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r:multiple_regression

# Multiple Regression

data file
University of New Mexico enrollment data (for 30 years)
ROLL: # of enrollment
UNEM: enemployment level
INC: income level

# data import
> str(datavar)
'data.frame':	29 obs. of  5 variables:
$YEAR : int 1 2 3 4 5 6 7 8 9 10 ...$ ROLL : int  5501 5945 6629 7556 8716 9369 9920 10167 11084 12504 ...
$UNEM : num 8.1 7 7.3 7.5 7 6.4 6.5 6.4 6.3 7.7 ...$ HGRAD: int  9552 9680 9731 11666 14675 15265 15484 15723 16501 16890 ...
$INC : int 1923 1961 1979 2030 2112 2192 2235 2351 2411 2475 ... >  two.predictor.model <- lm(ROLL ~ UNEM + HGRAD, datavar) summary(two.predictor.model) two.predictor.model three.predictor.model <- lm(ROLL ~ UNEM + HGRAD + INC, datavar) summary(three.predictor.model) three.predictor.model > two.predictor.model <- lm(ROLL ~ UNEM + HGRAD, datavar) > summary(two.predictor.model) Call: lm(formula = ROLL ~ UNEM + HGRAD, data = datavar) Residuals: Min 1Q Median 3Q Max -2102.2 -861.6 -349.4 374.5 3603.5 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -8.256e+03 2.052e+03 -4.023 0.00044 *** UNEM 6.983e+02 2.244e+02 3.111 0.00449 ** HGRAD 9.423e-01 8.613e-02 10.941 3.16e-11 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 1313 on 26 degrees of freedom Multiple R-squared: 0.8489, Adjusted R-squared: 0.8373 F-statistic: 73.03 on 2 and 26 DF, p-value: 2.144e-11 > two.predictor.model Call: lm(formula = ROLL ~ UNEM + HGRAD, data = datavar) Coefficients: (Intercept) UNEM HGRAD -8255.7511 698.2681 0.9423 >  > three.predictor.model <- lm(ROLL ~ UNEM + HGRAD + INC, datavar) > summary(three.predictor.model) Call: lm(formula = ROLL ~ UNEM + HGRAD + INC, data = datavar) Residuals: Min 1Q Median 3Q Max -1148.84 -489.71 -1.88 387.40 1425.75 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -9.153e+03 1.053e+03 -8.691 5.02e-09 *** UNEM 4.501e+02 1.182e+02 3.809 0.000807 *** HGRAD 4.065e-01 7.602e-02 5.347 1.52e-05 *** INC 4.275e+00 4.947e-01 8.642 5.59e-09 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 670.4 on 25 degrees of freedom Multiple R-squared: 0.9621, Adjusted R-squared: 0.9576 F-statistic: 211.5 on 3 and 25 DF, p-value: < 2.2e-16 > three.predictor.model Call: lm(formula = ROLL ~ UNEM + HGRAD + INC, data = datavar) Coefficients: (Intercept) UNEM HGRAD INC -9153.2545 450.1245 0.4065 4.2749 >  만약에 • unemployment rate (UNEM) = 9%, 12%, 3% • spring high school graduating class (HGRAD) = 100000, 98000, 78000 • a per capita income (INC) of \$30000, \$28000, \$36000
• 일 때, enrollment는 어떻게 predict할 수 있을까?

위에서 얻은 prediction model은 아래와 같다.
$$\hat{Y} = -9153.2545 + 450.1245 \cdot UNEM + 0.4065 \cdot HGRAD + 4.2749 \cdot INC$$
여기에 위의 정보를 대입해 보면 된다.

new.data <- data.frame(UNEM=c(9, 12, 3), HGRAD=c(100000, 98000, 78000), INC=c(30000, 28000, 36000))
predict(three.predictor.model, newdata=new.data)
> new.data <- data.frame(UNEM=c(9, 10, 15), HGRAD=c(100000, 98000, 78000), INC=c(30000, 28000, 19000))
> predict(three.predictor.model, newdata=new.data)
1        2        3
163792.0 154879.4 110526.6
> 

\begin{align*} \hat{Y} & = -9153.2545 + 450.1245 \cdot \text{UNEM} + 0.4065 \cdot \text{HGRAD} + 4.2749 \cdot \text{INC} \\ 163792.0 & = -9153.2545 + 450.1245 \cdot (9) + 0.4065 \cdot (100000) + 4.2749 \cdot (30000) \\ 154879.4 & = -9153.2545 + 450.1245 \cdot (10) + 0.4065 \cdot (98000) + 4.2749 \cdot (28000) \\ 110526.6 & = -9153.2545 + 450.1245 \cdot (15) + 0.4065 \cdot (78000) + 4.2749 \cdot (19000) \\ \end{align*}

beta coefficient 살펴보기
see beta coefficients

# install.packages('lm.beta')
# library(lm.beta)
lm.beta(three.predictor.model)
> # install.packages('lm.beta')
> # library(lm.beta)
> lm.beta(three.predictor.model)

Call:
lm(formula = ROLL ~ UNEM + HGRAD + INC, data = datavar)

Standardized Coefficients::
0.0000000   0.1553619   0.3656177   0.6061762

> 

by hand

# coefficient * (sd(x)/sd(y)) 이므로
#
attach(datavar)
sd.roll <- sd(ROLL)
sd.unem <- sd(UNEM)
sd.inc <- sd(INC)

b.unem <- three.predictor.model$coefficients[2] b.hgrad <- three.predictor.model$coefficients[3]
b.inc <- three.predictor.model$coefficients[4] ## or b.unem <- 4.501e+02 b.hgrad <- 4.065e-01 b.inc <- 4.275e+00 b.unem * (sd.unem / sd.roll) b.hgrad * (sd.hgrad / sd.roll) b.inc * (sd.inc / sd.roll) lm.beta(three.predictor.model)  output of the above > sd.roll <- sd(ROLL) > sd.unem <- sd(UNEM) > sd.hgrad <- sd(HGRAD) > sd.inc <- sd(INC) > > b.unem <- three.predictor.model$coefficients[2]
> b.hgrad <- three.predictor.model$coefficients[3] > b.inc <- three.predictor.model$coefficients[4]
>
> ## or
> b.unem <- 4.501e+02
> b.inc <- 4.275e+00
>
>
> b.unem * (sd.unem / sd.roll)
[1] 0.1554
[1] 0.3656
> b.inc * (sd.inc / sd.roll)
[1] 0.6062
>
> lm.beta(three.predictor.model)

Call:
lm(formula = ROLL ~ UNEM + HGRAD + INC, data = datavar)

Standardized Coefficients::
0.0000      0.1554      0.3656      0.6062

> 

> fit <- three.predictor.model
> step <- stepAIC(fit, direction="both")
Start:  AIC=381.2
ROLL ~ UNEM + HGRAD + INC

Df Sum of Sq      RSS AIC
<none>               11237313 381
- UNEM   1   6522098 17759411 392
- HGRAD  1  12852039 24089352 401
- INC    1  33568255 44805568 419
>


# etc

marketing <- read.csv("http://commres.net/wiki/_media/marketing_from_datarium.csv")
# install.packages("tidyverse", dep=TRUE)
library(tidyverse)
data("marketing", package = "datarium")
head(marketing, 4)
• Data, marketing is the impact of the amount of money spent on three advertising medias (youtube, facebook and newspaper) on sales.
• Note that to list all the independent (explanatory) variables, you could use lm (sales ~ ., data=“marketing”).
• You could also use - sign to subtract ivs. lm(sales ~ . - newspapers, data = “marketing”)
mod <- lm(sales ~ . , data=marketing)
summary(mod)
summary(mod)$coefficients mod2 <- lm(sales ~ . - newspaper, data = marketing) summary(mod2) > mod <- lm(sales ~ . , data=marketing) > summary(mod) Call: lm(formula = sales ~ ., data = marketing) Residuals: Min 1Q Median 3Q Max -10.5932 -1.0690 0.2902 1.4272 3.3951 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 3.526667 0.374290 9.422 <2e-16 *** youtube 0.045765 0.001395 32.809 <2e-16 *** facebook 0.188530 0.008611 21.893 <2e-16 *** newspaper -0.001037 0.005871 -0.177 0.86 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 2.023 on 196 degrees of freedom Multiple R-squared: 0.8972, Adjusted R-squared: 0.8956 F-statistic: 570.3 on 3 and 196 DF, p-value: < 2.2e-16 > summary(mod)$coefficients
Estimate  Std. Error    t value     Pr(>|t|)
(Intercept)  3.526667243 0.374289884  9.4222884 1.267295e-17
newspaper   -0.001037493 0.005871010 -0.1767146 8.599151e-01
> mod2  <- lm(sales ~ . - newspaper, data = marketing)
> summary(mod2)

Call:
lm(formula = sales ~ . - newspaper, data = marketing)

Residuals:
Min       1Q   Median       3Q      Max
-10.5572  -1.0502   0.2906   1.4049   3.3994

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  3.50532    0.35339   9.919   <2e-16 ***
youtube      0.04575    0.00139  32.909   <2e-16 ***
facebook     0.18799    0.00804  23.382   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.018 on 197 degrees of freedom
Multiple R-squared:  0.8972,	Adjusted R-squared:  0.8962
F-statistic: 859.6 on 2 and 197 DF,  p-value: < 2.2e-16

sales = 3.5 + 0.045*youtube + 0.187*facebook.

# Prediction

Predict(lm.fit, newdata=data.frame(x1=c(2,3,4), x2=c(3,3,3))

Year <- c(2017, 2017, 2017, 2017, 2017, 2017, 2017,
2017, 2017, 2017, 2017, 2017, 2016, 2016,
2016, 2016, 2016, 2016, 2016, 2016, 2016,
2016, 2016, 2016)
Month <- c(12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1,
12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1)
Interest_Rate <- c(2.75, 2.5, 2.5, 2.5, 2.5, 2.5,
2.5, 2.25, 2.25, 2.25, 2, 2, 2, 1.75,
1.75, 1.75, 1.75, 1.75, 1.75, 1.75,
1.75, 1.75, 1.75, 1.75)
Unemployment_Rate <- c(5.3, 5.3, 5.3, 5.3, 5.4, 5.6,
5.5, 5.5, 5.5, 5.6, 5.7, 5.9, 6, 5.9,
5.8, 6.1, 6.2, 6.1, 6.1, 6.1, 5.9, 6.2,
6.2, 6.1)
Stock_Index_Price <- c(1464, 1394, 1357, 1293, 1256,
1254, 1234, 1195, 1159, 1167, 1130, 1075,
1047, 965, 943, 958, 971, 949, 884, 866,
876, 822, 704, 719)

dat.stock <- data.frame(Year, Month, Interest_Rate, Unemployment_Rate, Stock_Index_Price)
dat.stock
Year Month Interest_Rate Unemployment_Rate Stock_Index_Price
1  2017    12          2.75               5.3              1464
2  2017    11          2.50               5.3              1394
3  2017    10          2.50               5.3              1357
4  2017     9          2.50               5.3              1293
5  2017     8          2.50               5.4              1256
6  2017     7          2.50               5.6              1254
7  2017     6          2.50               5.5              1234
8  2017     5          2.25               5.5              1195
9  2017     4          2.25               5.5              1159
10 2017     3          2.25               5.6              1167
11 2017     2          2.00               5.7              1130
12 2017     1          2.00               5.9              1075
13 2016    12          2.00               6.0              1047
14 2016    11          1.75               5.9               965
15 2016    10          1.75               5.8               943
16 2016     9          1.75               6.1               958
17 2016     8          1.75               6.2               971
18 2016     7          1.75               6.1               949
19 2016     6          1.75               6.1               884
20 2016     5          1.75               6.1               866
21 2016     4          1.75               5.9               876
22 2016     3          1.75               6.2               822
23 2016     2          1.75               6.2               704
24 2016     1          1.75               6.1               719
> 
lm.stock <- lm(Stock_Index_Price~Interest_Rate+Unemployment_Rate, data=dat.stock)
summary(lm.stock)
lm.stock$coefficient coefficients(lm.stock) > names(lm.stock) [1] "coefficients" "residuals" "effects" "rank" "fitted.values" [6] "assign" "qr" "df.residual" "xlevels" "call" [11] "terms" "model"  > lm.stock <- lm(Stock_Index_Price~Interest_Rate+Unemployment_Rate, data=dat.stock) > summary(lm.stock) Call: lm(formula = Stock_Index_Price ~ Interest_Rate + Unemployment_Rate, data = dat.stock) Residuals: Min 1Q Median 3Q Max -158.205 -41.667 -6.248 57.741 118.810 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 1798.4 899.2 2.000 0.05861 . Interest_Rate 345.5 111.4 3.103 0.00539 ** Unemployment_Rate -250.1 117.9 -2.121 0.04601 * --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 70.56 on 21 degrees of freedom Multiple R-squared: 0.8976, Adjusted R-squared: 0.8879 F-statistic: 92.07 on 2 and 21 DF, p-value: 4.043e-11 > lm.stock$coefficient
(Intercept)     Interest_Rate Unemployment_Rate
1798.4040          345.5401         -250.1466
> coefficients(lm.stock)
(Intercept)     Interest_Rate Unemployment_Rate
1798.4040          345.5401         -250.1466
>
> 
dat.stock.new.iv <- data.frame(Interest_Rate=c(5,6,2,1), Unemployment_Rate=c(8,8.5,4,3))
predict(lm.stock, newdata=dat.stock.new.iv)
> dat.stock.new.iv <- data.frame(Interest_Rate=c(5,6,2,1), Unemployment_Rate=c(8,8.5,4,3))
> predict(lm.stock, newdata=dat.stock.new.iv)
1        2        3        4
1524.932 1745.399 1488.898 1393.504
> 

# Partial, Semi-partial Correlation and R squared value

아래에서, summary(lm.stock)은

> lm.stock <- lm(Stock_Index_Price~Interest_Rate+Unemployment_Rate, data=dat.stock)
> summary(lm.stock)

Call:
lm(formula = Stock_Index_Price ~ Interest_Rate + Unemployment_Rate,
data = dat.stock)

Residuals:
Min       1Q   Median       3Q      Max
-158.205  -41.667   -6.248   57.741  118.810

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)         1798.4      899.2   2.000  0.05861 .
Interest_Rate        345.5      111.4   3.103  0.00539 **
Unemployment_Rate   -250.1      117.9  -2.121  0.04601 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 70.56 on 21 degrees of freedom
Multiple R-squared:  0.8976,	Adjusted R-squared:  0.8879
F-statistic: 92.07 on 2 and 21 DF,  p-value: 4.043e-11
> 
> anova(lm.stock)
Analysis of Variance Table

Response: Stock_Index_Price
Df Sum Sq Mean Sq  F value    Pr(>F)
Interest_Rate      1 894463  894463 179.6477 9.231e-12 ***
Unemployment_Rate  1  22394   22394   4.4977   0.04601 *
Residuals         21 104559    4979
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> 
 df SS ms F interest 1 (a) 894463 (1) 894463 179.6471179 unemp 1 (b) 22394 (2) 22394 4.497690299 res 21 (c) 104559 (3) 4979 total 23 1021416 (4) interest + enemp 916857 (5)

(4) = (1) + (2) + (3)
(5) = (1) + (2)

from the above summary(lm.stock)
Multiple R-squared: 0.8976, . . . .
F-statistic: 92.07 on 2 and 21 DF, p-value: 4.043e-11

\begin{eqnarray*} R^2 & = & \dfrac {SS_{reg}}{SS_{total}} \\ & = & \dfrac{(5)}{(4)}, \qquad \text{from the table} \\ & = & \dfrac {916857}{1021416} \\ & = & 0.897633286 \\ & = & 0.8976 \end{eqnarray*}

\begin{eqnarray*} F & = & \dfrac {MS_{reg}} {MS_{res}} \\ & = & \dfrac {(5)/(a)+(b)}{(3)/(c)} \\ & = & \dfrac {916857/2}{104559/21} \\ & = & 92.0724041 \\ & = & 92.07 \qquad (df = 2, 21) \end{eqnarray*}

그런데, 각각의 IV와 DV 간의 correlation 값은:

> cor.test(Stock_Index_Price, Interest_Rate)

Pearson's product-moment correlation

data:  Stock_Index_Price and Interest_Rate
t = 12.45, df = 22, p-value = 1.954e-11
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
0.8552498 0.9721916
sample estimates:
cor
0.9357932

> cor.test(Stock_Index_Price, Unemployment_Rate)

Pearson's product-moment correlation

data:  Stock_Index_Price and Unemployment_Rate
t = -11.196, df = 22, p-value = 1.487e-10
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
-0.9662308 -0.8264284
sample estimates:
cor
-0.9223376
> 

Stock_Index_Price ~ Interest_Rate = 0.9357932
Stock_Index_Price ~ Unemployment_Rate = -0.9223376

각각의 r square 값은
Stock_Index_Price ~ Interest_Rate -> 0.875708913
Stock_Index_Price ~ Unemployment_Rate -> 0.850706648

이 둘의 합은 1.726415562 로 1을 넘는다. 즉, 각 독립변인의 영향력은 중복 (독립변인 간의 상관관계) 되어 있다는 뜻.

install.packages("ppcor")
library(ppcor)
> pcor.test(Stock_Index_Price, Interest_Rate, Unemployment_Rate)
estimate     p.value statistic  n gp  Method
1 0.560649 0.005389174  3.102717 24  1 pearson
> spcor.test(Stock_Index_Price, Interest_Rate, Unemployment_Rate)
estimate   p.value statistic  n gp  Method
1 0.2166264 0.3207966  1.016852 24  1 pearson

0.2166264^2 = 0.046926997

b + c / a + b + c + d = 0.8757
b / a + b + c + d = 0.0467
c / a + b + c + d = 0.8288

> spcor.test(Stock_Index_Price, Unemployment_Rate, Interest_Rate)
estimate   p.value  statistic  n gp  Method
1 -0.1480697 0.5001535 -0.6861036 24  1 pearson
> 

(-0.1480697)^2 = 0.021924636 = 0.0219

d + c / a + b + c + d = 0.8507
d / a + b + c + d = 0.0219
c / a + b + c + d = 0.8288

위와 아래의 c / a + b + c + d = 0.8288 가 일치하므로, 독립변인 둘의 겹치는 영향력이 크다는 것을 알수 있다.

> lm.stock.int <- lm(Stock_Index_Price~Interest_Rate, data=dat.stock)
> summary(lm.stock.int)

Call:
lm(formula = Stock_Index_Price ~ Interest_Rate, data = dat.stock)

Residuals:
Min       1Q   Median       3Q      Max
-183.892  -30.181    4.455   56.608  101.057

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)     -99.46      95.21  -1.045    0.308
Interest_Rate   564.20      45.32  12.450 1.95e-11 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 75.96 on 22 degrees of freedom
Multiple R-squared:  0.8757,	Adjusted R-squared:  0.8701
F-statistic:   155 on 1 and 22 DF,  p-value: 1.954e-11

> lm.stock.unemp <- lm(Stock_Index_Price ~ Unemployment_Rate, data=dat.stock)
> summary(lm.stock.unemp)

Call:
lm(formula = Stock_Index_Price ~ Unemployment_Rate, data = dat.stock)

Residuals:
Min       1Q   Median       3Q      Max
-159.671  -41.996    2.089   72.381  151.226

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)         4471.3      304.2    14.7 7.41e-13 ***
Unemployment_Rate   -589.0       52.6   -11.2 1.49e-10 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 83.25 on 22 degrees of freedom
Multiple R-squared:  0.8507,	Adjusted R-squared:  0.8439
F-statistic: 125.4 on 1 and 22 DF,  p-value: 1.487e-10

>
> 
> anova(lm.stock.int, lm.stock)
Analysis of Variance Table

Model 1: Stock_Index_Price ~ Interest_Rate
Model 2: Stock_Index_Price ~ Interest_Rate + Unemployment_Rate
Res.Df    RSS Df Sum of Sq      F  Pr(>F)
1     22 126953
2     21 104559  1     22394 4.4977 0.04601 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

> anova(lm.stock.unemp, lm.stock)
Analysis of Variance Table

Model 1: Stock_Index_Price ~ Unemployment_Rate
Model 2: Stock_Index_Price ~ Interest_Rate + Unemployment_Rate
Res.Df    RSS Df Sum of Sq      F   Pr(>F)
1     22 152491
2     21 104559  1     47932 9.6269 0.005389 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> 

Alternatively,

> install.packages("lmtest")
> library(lmtest)
> lrtest(lm.stock.int, lm.stock)
Likelihood ratio test

Model 1: Stock_Index_Price ~ Interest_Rate
Model 2: Stock_Index_Price ~ Interest_Rate + Unemployment_Rate
#Df  LogLik Df  Chisq Pr(>Chisq)
1   3 -136.94
2   4 -134.61  1 4.6576    0.03092 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> 

# e.g. 5

#packages we will need to conduct to create and graph our data
library(MASS) #create data
library(car) #graph data
py1 =.6 #Cor between X1 (Practice Time) and Memory Errors
py2 =.4 #Cor between X2 (Performance Anxiety) and Memory Errors
p12= .3 #Cor between X1 (Practice Time) and X2 (Performance Anxiety)
Means.X1X2Y<- c(10,10,10) #set the means of X and Y variables
CovMatrix.X1X2Y <- matrix(c(1,p12,py1,
p12,1,py2,
py1,py2,1),3,3) # creates the covariate matrix
#build the correlated variables. Note: empirical=TRUE means make the correlation EXACTLY r.
# if we say empirical=FALSE, the correlation would be normally distributed around r
set.seed(42)
CorrDataT<-mvrnorm(n=100, mu=Means.X1X2Y,Sigma=CovMatrix.X1X2Y, empirical=TRUE)
#Convert them to a "Data.Frame" & add our labels to the vectors we created
CorrDataT<-as.data.frame(CorrDataT)
colnames(CorrDataT) <- c("Practice","Anxiety","Memory")
#make the scatter plots
scatterplot(Memory~Practice,CorrDataT)
scatterplot(Memory~Anxiety,CorrDataT)
scatterplot(Anxiety~Practice,CorrDataT)
# Pearson Correlations
ry1<-cor(CorrDataT$Memory,CorrDataT$Practice)
ry2<-cor(CorrDataT$Memory,CorrDataT$Anxiety)
r12<-cor(CorrDataT$Anxiety,CorrDataT$Practice)
ry1
ry2
r12
ry1^2
ry2^2
r12^2
> ry1
[1] 0.6
> ry2
[1] 0.4
> r12
[1] 0.3
>
> ry1^2
[1] 0.36
> ry2^2
[1] 0.16
> r12^2
[1] 0.09
> 
> lm.m.pa <- lm(Memory~Practice+Anxiety, data=CorrDataT)
> summary(lm.m.pa)

Call:
lm(formula = Memory ~ Practice + Anxiety, data = CorrDataT)

Residuals:
Min       1Q   Median       3Q      Max
-1.99998 -0.54360  0.04838  0.43878  2.74186

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  2.30769    0.96783   2.384  0.01905 *
Practice     0.52747    0.08153   6.469 4.01e-09 ***
Anxiety      0.24176    0.08153   2.965  0.00381 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.7739 on 97 degrees of freedom
Multiple R-squared:  0.4132,	Adjusted R-squared:  0.4011
F-statistic: 34.15 on 2 and 97 DF,  p-value: 5.919e-12


a+b+c+e = variance of y = total variance = 100% = 1 이라고 보면
r2 = a + b + c / a + b + c + e = a + b + c = 0.4132 (from summary(lm.m.pa))

이 중에서 우리는 이미
a + c = 0.62 = 0.36
b + c = 0.42 = 0.16

따라서 Practice를 제어한 Anxiety의 영향력은
r2 - (a+c) = 0.4132 - 0.36 = 0.0532
반대로, Anxiety를 제어한 Practice의 영향력은
r2 - (a+c) = 0.4132 - 0.16 = 0.2532