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====== Partial and semi-partial correlation ======
references
{{https://web.stanford.edu/~hastie/Papers/ESLII.pdf|The Elements of Statistical Learning}} or local copy 
or [[https://www.r-bloggers.com/2016/06/introduction-to-r-for-data-science-session-7-multiple-linear-regression-model-in-r-categorical-predictors-partial-and-part-correlation/| Introduction to R for Data Science :: Session 7 (Multiple Linear Regression Model in R  + Categorical Predictors, Partial and Part Correlation)]] 
====== Partial and semipartial ======
<code>
options(digits = 4)
HSGPA <- c(3.0, 3.2, 2.8, 2.5, 3.2, 3.8, 3.9, 3.8, 3.5, 3.1)
FGPA <-  c(2.8, 3.0, 2.8, 2.2, 3.3, 3.3, 3.5, 3.7, 3.4, 2.9)
SATV <-  c(500, 550, 450, 400, 600, 650, 700, 550, 650, 550)

scholar <- data.frame(FGPA, HSGPA, SATV) # collect into a data frame
describe(scholar) # provides descrptive information about each variable

corrs <- cor(scholar) # find the correlations and set them into an object called 'corrs'
corrs                 # print corrs

pairs(scholar)        # pairwise scatterplots
</code>

<code>
attach(scholar)
# freshman's gpa ~ hischool gpa + sat
mod.all <- lm(FGPA ~ HSGPA+ SATV, data = scholar)
summary(mod.all)
</code>
<code>> mod.all <- lm(FGPA ~ HSGPA+ SATV, data = scholar)
> summary(mod.all)

Call:
lm(formula = FGPA ~ HSGPA + SATV, data = scholar)

Residuals:
    Min      1Q  Median      3Q     Max 
-0.2431 -0.1125 -0.0286  0.1269  0.2716 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)  
(Intercept) 0.233102   0.456379    0.51    0.625  
HSGPA       0.845192   0.283816    2.98    0.021 *
SATV        0.000151   0.001405    0.11    0.917  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.192 on 7 degrees of freedom
Multiple R-squared:  0.851,	Adjusted R-squared:  0.809 
F-statistic: 20.1 on 2 and 7 DF,  p-value: 0.00126

> 
> </code>

[{{:pasted:20201203-233528.png?600}}]

SATV 는 significant한 역할을 하지 못한다는 t-test 결과이다. 이것이 사실일까?
우선 FGPA에 대해 SATV와 HSGPA를 가지고 regression을 각각 해보자
아래의 결과는 두개의 IV는 각각 종속변인인 FGPA를 significant하게 설명하고 있다.
즉, 둘이 같이 설명하려고 했을 때에만 그 설명력이 사라진다.
<WRAP clear/>
<code>
attach(scholar)
ma1 <- lm(FGPA ~ SATV)
ma2 <- lm(FGPA ~ HSGPA)
summary(ma1)
summary(ma2)
</code>

<code>> ma1 <- lm(FGPA ~ SATV)
> ma2 <- lm(FGPA ~ HSGPA)
> summary(ma1)

Call:
lm(formula = FGPA ~ SATV)

Residuals:
    Min      1Q  Median      3Q     Max 
-0.2804 -0.1305 -0.0566  0.0350  0.6481 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)   
(Intercept)  0.95633    0.54419    1.76   0.1169   
SATV         0.00381    0.00096    3.97   0.0041 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.27 on 8 degrees of freedom
Multiple R-squared:  0.663,	Adjusted R-squared:  0.621 
F-statistic: 15.8 on 1 and 8 DF,  p-value: 0.00412

> summary(ma2)

Call:
lm(formula = FGPA ~ HSGPA)

Residuals:
    Min      1Q  Median      3Q     Max 
-0.2434 -0.1094 -0.0266  0.1259  0.2797 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)    0.230      0.426    0.54  0.60412    
HSGPA          0.872      0.129    6.77  0.00014 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.179 on 8 degrees of freedom
Multiple R-squared:  0.851,	Adjusted R-squared:  0.833 
F-statistic: 45.8 on 1 and 8 DF,  p-value: 0.000143

> 
</code>

아래는 HSGPA의 영향력을 IV, DV 모두에게서 제거한 후, DV에 대한 IV의 영향력을 보는 작업이다. 
<code>
m1 <- lm(FGPA ~ HSGPA)
m2 <- lm(SATV ~ HSGPA)
res.m1 <- resid(m1)
# res.m1 <- m1$residuals
res.m2 <- resid(m2)
m.12 <- lm(res.m1 ~ res.m2)
summary(m.12)

</code>

결과는 아래에서 보는 것처럼 not significant하다. 

<code>

> m1 <- lm(FGPA ~ HSGPA)
> m2 <- lm(SATV ~ HSGPA)
> res.m1 <- resid(m1)
> res.m2 <- resid(m2)
> m.12 <- lm(res.m1 ~ res.m2)
> summary(m.12)

Call:
lm(formula = res.m1 ~ res.m2)

Residuals:
    Min      1Q  Median      3Q     Max 
-0.2431 -0.1125 -0.0286  0.1269  0.2716 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)
(Intercept) -6.50e-18   5.67e-02    0.00     1.00
res.m2       1.51e-04   1.31e-03    0.12     0.91

Residual standard error: 0.179 on 8 degrees of freedom
Multiple R-squared:  0.00165,	Adjusted R-squared:  -0.123 
F-statistic: 0.0132 on 1 and 8 DF,  p-value: 0.911

</code>
특히 위에서 R-squared value는 0.00165이고 이를 square root 한 값은 $\sqrt{0.00165} = 0.04064$ 이다. 이 값은 HSGPA의 영향력을 IV와 (SATV) DV에서 (FGPA) 모두 제거한 후의 correlation값이라고도 할 수 있다. 사실 이 숫자는 ''lm()''말고도 

<code>cor(res.m1, res.m2)
## 혹은 
cor.test(res.m1, res.m2)</code> 으로 확인해 볼 수 있다. 

<code>
> cor(res.m1, res.m2)
[1] 0.04064
> 
> cor.test(res.m1, res.m2)

	Pearson's product-moment correlation

data:  res.m1 and res.m2
t = 0.12, df = 8, p-value = 0.9
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.6045  0.6535
sample estimates:
    cor 
0.04064 
>
</code>

우리는 이것을 [[:multiple_regression#determining_ivs_role|partial correlation이라고 부른다는 것을 알고 있다]]. 이를 ppcor 패키지를 이용해서 테스트해보면

<code>
# install.packages("ppcor")
pcor.test(FGPA, SATV, HSGPA)
</code>
<code>> pcor.test(FGPA, SATV, HSGPA)
  estimate p.value statistic  n gp  Method
1  0.04064  0.9173    0.1076 10  1 pearson
> </code>
위에서 estimate 값인 0.04064가 위의 R-square 값에 square root값을 씌운 값이 된다. 이를 그림으로 나타내 보면 아래와 같다. 

[{{:pasted:20201203-235526.png?600}}]
반대의 경우도 실행을 해보면 즉, SATV의 영향력을 제어한 후, HSGPA의 영향력만을 볼 때
<code>
n1 <- lm(FGPA ~ SATV)
n2 <- lm(HSGPA ~ SATV)
res.n1 <- resid(n1)
res.n2 <- resid(n2)
n.12 <- lm(res.n1 ~ res.n2)
summary(n.12)
</code>

<code>> n1 <- lm(FGPA ~ SATV)
> n2 <- lm(HSGPA ~ SATV)
> res.n1 <- resid(n1)
> res.n2 <- resid(n2)
> n.12 <- lm(res.n1 ~ res.n2)
> summary(n.12)

Call:
lm(formula = res.n1 ~ res.n2)

Residuals:
    Min      1Q  Median      3Q     Max 
-0.2431 -0.1125 -0.0286  0.1269  0.2716 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)  
(Intercept) -2.04e-18   5.67e-02    0.00    1.000  
res.n2       8.45e-01   2.65e-01    3.18    0.013 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.179 on 8 degrees of freedom
Multiple R-squared:  0.559,	Adjusted R-squared:  0.504 
F-statistic: 10.1 on 1 and 8 DF,  p-value: 0.0129

> 
</code>
이제는 그 R-squared 값이 0.559임을 알게 되었다. 이의 제곱근은 $\sqrt{0.559} = 0.7477$ 이고, 이것이 SATV의 영향력을 IV, DV 모두에게서 제거한 후에 IV의 (HSGPA만의) 영향력을 보는 방법이라는 것을 안다. 

[{{:pasted:20201203-235353.png?600}}]

<code>
pcor.test(FGPA, HSGPA, SATV) 
</code>
<code>> pcor.test(FGPA, HSGPA, SATV) 
  estimate p.value statistic  n gp  Method
1   0.7476 0.02057     2.978 10  1 pearson
> </code>

<fc #ff0000>**위의 도식화된 분석으로 R의 multiple regression에서의 한 변인에 대한 t-test는 그 변인을 제외한 다른 IV들을 콘트롤하여 해당 IV와 DV에서 제거한 후에 본다는 사실을 알 수 있다.**</fc>  즉, ''lm(FGPA~SATV+HSGPA)'' 에서 독립변인 SATV의 t값은 HSGPA의 영향력을 제거하여 제어한 후에 살펴보고 이를 반영한다는 것을 말한다. 



또한 위의 설명은 [[:multiple_regression#in_r|다른 곳에서 언급했던]] Multiple regression에서의 summary(lm())과 anova(lm())이 차이를 보이는 이유를 설명하기도 한다 (여기서는 summary(mod)와 anova(mod)). anova는 변인을 순서대로 받고 다른 IV들에 대한 제어를 하지 않으므로 IV 순서에 따라서 그 분석 결과가 달라지기도 한다. 

아래의 결과를 살펴보면 anova() 결과 독립변인들의 p value 들과 summary() 에서의 독립변인들의 p value가 다른 이유가 다르다.

<code>
# anova()에서의 결과 
acs_k3      1  110211  110211   32.059 2.985e-08 ***
# summary(lm())에서의 결과
acs_k3        3.3884     2.3333   1.452    0.147    
</code>
아래는 [[:multiple_regression#in_r|Multiple Regression 설명에서 가져옴]]
<code>
dvar <- read.csv("http://commres.net/wiki/_media/elemapi2_.csv", fileEncoding="UTF-8-BOM")
mod <- lm(api00 ~ ell + acs_k3 + avg_ed + meals, data=dvar)
summary(mod)
anova(mod)
</code>

<code>
> dvar <- read.csv("http://commres.net/wiki/_media/elemapi2_.csv", fileEncoding="UTF-8-BOM")
> mod <- lm(api00 ~ ell + acs_k3 + avg_ed + meals, data=dvar)
> summary(mod)

Call:
lm(formula = api00 ~ ell + acs_k3 + avg_ed + meals, data = dvar)

Residuals:
     Min       1Q   Median       3Q      Max 
-187.020  -40.358   -0.313   36.155  173.697 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 709.6388    56.2401  12.618  < 2e-16 ***
ell          -0.8434     0.1958  -4.307 2.12e-05 ***
acs_k3        3.3884     2.3333   1.452    0.147    
avg_ed       29.0724     6.9243   4.199 3.36e-05 ***
meals        -2.9374     0.1948 -15.081  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 58.63 on 374 degrees of freedom
  (21 observations deleted due to missingness)
Multiple R-squared:  0.8326,	Adjusted R-squared:  0.8308 
F-statistic:   465 on 4 and 374 DF,  p-value: < 2.2e-16

> anova(mod)
Analysis of Variance Table

Response: api00
           Df  Sum Sq Mean Sq  F value    Pr(>F)    
ell         1 4502711 4502711 1309.762 < 2.2e-16 ***
acs_k3      1  110211  110211   32.059 2.985e-08 ***
avg_ed      1  998892  998892  290.561 < 2.2e-16 ***
meals       1  781905  781905  227.443 < 2.2e-16 ***
Residuals 374 1285740    3438                       
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> 
</code>


아래에서 mod2를 보면 
  * api00이 종속변인이고, 
  * ell, avg_ed, meals + acs_k3 가 독립변인인데, 
  * 그 순서가 이전 문서의 
  * ell, acs_k3, avg_ed, meals에서 바뀐것을 알 수 있다 (acs.k3가 맨 뒤로 감).
  * 즉, 
  * lm(api00 ~ ell + acs_k3 + avg_ed + meals)
  * lm(api00 ~ ell + avg_ed + meals + acs_k3)

anova는 독립변인에 대한 영향력을 다른 IV들을 고려하지 않고, 그냥 입력 순서대로 처리하므로, acs_k3를 마지막으로 보냄으로써, 다른 IV들이 DV에 대한 설명력을 모두 차지하고 그 나머지를 보여주게 된다. 이것이 t-test가 significant하지 않은 이유이다. 

<code>> mod2 <- lm(api00 ~ ell + avg_ed + meals + acs_k3, data=dvar)
> summary(mod2)

Call:
lm(formula = api00 ~ ell + avg_ed + meals + acs_k3, data = dvar)

Residuals:
    Min      1Q  Median      3Q     Max 
-186.90  -40.13   -0.07   35.96  174.12 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  711.681     56.468   12.60  < 2e-16 ***
ell           -0.845      0.197   -4.28  2.4e-05 ***
avg_ed        28.966      6.947    4.17  3.8e-05 ***
meals         -2.948      0.196  -15.02  < 2e-16 ***
acs_k3         3.336      2.340    1.43     0.15    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 58.8 on 371 degrees of freedom
Multiple R-squared:  0.832,	Adjusted R-squared:  0.831 
F-statistic:  461 on 4 and 371 DF,  p-value: <2e-16

> 
> anova(mod2)
Analysis of Variance Table

Response: api00
           Df  Sum Sq Mean Sq  F value Pr(>F)    
ell         1 4502711 4502711 1309.762 <2e-16 ***
avg_ed      1 1017041 1017041  295.840 <2e-16 ***
meals       1  866716  866716  252.113 <2e-16 ***
acs_k3      1    7250    7250    2.109 0.1473    
Residuals 374 1285740    3438                    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
</code>

이는 다른 독립변인들의 순서를 바꾸어도 마찬가지이다. mod3은 mod2에서 meals변인을 맨 앞으로 옮긴 예이다. 즉

  * mod  <- lm(api00 ~ ell + acs_k3 + avg_ed + meals)
  * mod2 <- lm(api00 ~ ell + avg_ed + meals + acs_k3)
  * mod3 <- lm(api00 ~ meals + ell + avg_ed + acs_k3)

summary(mod), summary(mod2), summary(mod3)의 결과는 서로 다르지 않지만, anova의 결과는 어떤 독립변인이 앞으로 오는가에 따라서 그 f값과  p-value가 달라진다. 물론, 만약에 독립변인들 간의 상관관계가 0이라면 순서가 영향을 주지는 않겠다.
<code>
> mod3 <- lm(api00 ~ meals + ell + avg_ed + acs_k3, data=dvar)
> summary(mod3)

Call:
lm(formula = api00 ~ meals + ell + avg_ed + acs_k3, data = dvar)

Residuals:
    Min      1Q  Median      3Q     Max 
-186.90  -40.13   -0.07   35.96  174.12 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  711.681     56.468   12.60  < 2e-16 ***
meals         -2.948      0.196  -15.02  < 2e-16 ***
ell           -0.845      0.197   -4.28  2.4e-05 ***
avg_ed        28.966      6.947    4.17  3.8e-05 ***
acs_k3         3.336      2.340    1.43     0.15    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 58.8 on 371 degrees of freedom
Multiple R-squared:  0.832,	Adjusted R-squared:  0.831 
F-statistic:  461 on 4 and 371 DF,  p-value: <2e-16

> anova(mod2)
Analysis of Variance Table

Response: api00
           Df  Sum Sq Mean Sq F value Pr(>F)    
ell         1 4480281 4480281 1297.34 <2e-16 ***
avg_ed      1 1014175 1014175  293.67 <2e-16 ***
meals       1  864080  864080  250.21 <2e-16 ***
acs_k3      1    7021    7021    2.03   0.15    
Residuals 371 1281224    3453                   
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
>
>
> anova(mod3)
Analysis of Variance Table

Response: api00
           Df  Sum Sq Mean Sq F value  Pr(>F)    
meals       1 6219897 6219897 1801.08 < 2e-16 ***
ell         1   82758   82758   23.96 1.5e-06 ***
avg_ed      1   55880   55880   16.18 7.0e-05 ***
acs_k3      1    7021    7021    2.03    0.15    
Residuals 371 1281224    3453                    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> </code>


====== e.g. Using ppcor.test with 4 var ======

<code>
options(digits = 4)

HSGPA <- c(3.0, 3.2, 2.8, 2.5, 3.2, 3.8, 3.9, 3.8, 3.5, 3.1)
FGPA <-  c(2.8, 3.0, 2.8, 2.2, 3.3, 3.3, 3.5, 3.7, 3.4, 2.9)
SATV <-  c(500, 550, 450, 400, 600, 650, 700, 550, 650, 550)
GREV <-  c(600, 670, 540, 800, 750, 820, 830, 670, 690, 600)
##GREV <- c(510, 670, 440, 800, 750, 420, 830, 470, 690, 600)

scholar <- data.frame(HSGPA, FGPA, SATV, GREV) # collect into a data frame
describe(scholar) # provides descrptive information about each variable

corrs <- cor(scholar) # find the correlations and set them into an object called 'corrs'
corrs                 # print corrs

pairs(scholar)        # pairwise scatterplots

pcor.test(scholar$GREV, scholar$FGPA, scholar[,c("SATV", "HSGPA")]) # working

reg3 <- lm(GREV ~ SATV + HSGPA)   # run linear regression
resid3 <- resid(reg3)     # find the residuals - GREV free of SATV and HSGPA

reg4 <- lm(FGPA ~ SATV + HSGPA)   # second regression
resid4 <- resid(reg4)     # second set of residuals - FGPA free of SATV and HSGPA

cor(resid3, resid4)       # correlation of residuals - partial correlation
</code>

<code>
> pcor.test(scholar$GREV, scholar$FGPA, scholar[,c("SATV", "HSGPA")]) # working
  estimate p.value statistic  n gp  Method
1   -0.535  0.1719    -1.551 10  2 pearson
> 
> reg3 <- lm(GREV ~ SATV + HSGPA)   # run linear regression
> resid3 <- resid(reg3)     # find the residuals - GREV free of SATV and HSGPA
> 
> reg4 <- lm(FGPA ~ SATV + HSGPA)   # second regression
> resid4 <- resid(reg4)     # second set of residuals - FGPA free of SATV and HSGPA
> 
> cor(resid3, resid4)       # correlation of residuals - partial correlation
[1] -0.535


</code>
----
----
학자인 A는 GRE점수는 (GREV) 학점에 신경을 쓰는 활동보다는 지능지수와 관련된다고 믿는 SATV의 영향력이 더 클것으로 생각된다. 그래서 SATV만의 영향력을 다른 변인을 콘트롤하여 살펴보고 싶다. 
<code>
pcor.test(scholar$GREV, scholar$SATV, scholar[, c("HSGPA", "FGPA")])

reg7 <- lm(GREV ~ HSGPA + FGPA)   # run linear regression
resid7 <- resid(reg7)     # find the residuals - HSGPA free of SATV

reg8 <- lm(SATV ~ HSGPA+ FGPA)   # second regression
resid8 <- resid(reg8)     # second set of residuals - FGPA free of SATV

cor(resid7, resid8)       # correlation of residuals - partial correlation

</code>

<code>
> pcor.test(scholar$GREV, scholar$SATV, scholar[, c("HSGPA", "FGPA")])
  estimate p.value statistic  n gp  Method
1   0.3179  0.4429    0.8213 10  2 pearson
> 
> reg7 <- lm(GREV ~ HSGPA + FGPA)   # run linear regression
> resid7 <- resid(reg7)     # find the residuals - HSGPA free of SATV
> 
> reg8 <- lm(SATV ~ HSGPA+ FGPA)   # second regression
> resid8 <- resid(reg8)     # second set of residuals - FGPA free of SATV
> 
> cor(resid7, resid8)       # correlation of residuals - partial correlation
[1] 0.3179
> 
> 
</code>

====== e.g., 독립변인 들이 서로 독립적일 때의 각각의 설명력 ======
In this example, the two IVs are orthogonal to each other (not correlated with each other). Hence, regress res.y.x2 against x1 would not result in any problem. 
<code>
n <- 32
set.seed(182)
u <-matrix(rnorm(2*n), ncol=2)
u0 <- cbind(u[,1] - mean(u[,1]), u[,2] - mean(u[,2]))
x <- svd(u0)$u
eps <- rnorm(n)
y <-  x %*% c(0.05, 1) + eps * 0.01
x1 <- x[,1]
x2 <- x[,2]
dset <- data.frame(y,x1,x2)
head(dset)
</code><code>
        y       x1      x2
1  0.2311 -0.42320  0.2536
2 -0.1708 -0.13428 -0.1573
3  0.1617  0.12404  0.1580
4  0.1111  0.10377  0.1214
5  0.2176  0.08796  0.1962
6  0.2054  0.02187  0.1950
>
</code><code>
round(cor(dset), 3)
</code><code>
       y    x1    x2
y  1.000 0.068 0.996
x1 0.068 1.000 0.000
x2 0.996 0.000 1.000
> </code>

<code>
lm.y.x1 <- lm(y ~ x1)
summary(lm.y.x1)
</code><code>
Call:
lm(formula = y ~ x1)

Residuals:
    Min      1Q  Median      3Q     Max 
-0.3750 -0.1013 -0.0229  0.1402  0.2985 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.00258    0.03242   -0.08     0.94
x1           0.06895    0.18341    0.38     0.71

Residual standard error: 0.183 on 30 degrees of freedom
Multiple R-squared:  0.00469,	Adjusted R-squared:  -0.0285 
F-statistic: 0.141 on 1 and 30 DF,  p-value: 0.71
</code>

<code>
lm.y.x1x2 <- lm(y ~ x1 + x2)
summary(lm.y.x1x2)
</code><code>
Call:
lm(formula = y ~ x1 + x2)

Residuals:
      Min        1Q    Median        3Q       Max 
-0.026697 -0.004072  0.000732  0.006664  0.017220 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) -0.00258    0.00168   -1.54     0.14    
x1           0.06895    0.00949    7.27  5.3e-08 ***
x2           1.00328    0.00949  105.72  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.00949 on 29 degrees of freedom
Multiple R-squared:  0.997,	Adjusted R-squared:  0.997 
F-statistic: 5.61e+03 on 2 and 29 DF,  p-value: <2e-16
> </code>
.... 
Std. Error for x1
SSres/n-2 =
Std. Error for x2

.... 
<code>
lm.y.x2 <- lm(y ~ x2)
summary(lm.y.x2)
</code>
<code>Call:
lm(formula = y ~ x2)

Residuals:
      Min        1Q    Median        3Q       Max 
-0.027366 -0.010654  0.002941  0.009922  0.027470 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) -0.002576   0.002770   -0.93     0.36    
x2           1.003276   0.015669   64.03   <2e-16 ***
---
Signif. codes:  
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.01567 on 30 degrees of freedom
Multiple R-squared:  0.9927,	Adjusted R-squared:  0.9925 
F-statistic:  4100 on 1 and 30 DF,  p-value: < 2.2e-16
</code>
<code>
res.lm.y.x2 <- lm.y.x2$resdiduals
</code>
<code>
d <- data.frame(X1=x1, X2=x2, Y=y, RY=res.lm.y.x2)
plot(d)
</code>

{{:pasted:20230524-000614.png}}

<code>> 1-0.9927
[1] 0.0073
</code>

X2이 설명하는 Y분산의 나머지를 (1-R<sup>2</sup> = 0.0073) 종속변인으로 하고 x1을 독립변인으로 하여 regression을 하면 figure의 RY축에 해당하는 관계가 나타난다. 특히 RY와 X1과의 관계가 선형적으로 바뀐것은 X1 자체로는 아무런 역할을 하지 못하는 것으로 나타나다가, X2가 개입되고 X2의 영향력으로 설명된 Y부분을 제외한 (제어한, controlling) 나머지에 대한 X1의 설명력이 significant하게 바뀐 결과이다.

<code>> lm.resyx2.x1 <- lm(lm.y.x2$residuals ~ x1)
> summary(lm.resyx2.x1)

Call:
lm(formula = lm.y.x2$residuals ~ x1)

Residuals:
       Min         1Q     Median         3Q        Max 
-0.0266967 -0.0040718  0.0007323  0.0066643  0.0172201 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) 2.220e-18  1.649e-03    0.00        1    
x1          6.895e-02  9.331e-03    7.39 3.11e-08 ***
---
Signif. codes:  
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.009331 on 30 degrees of freedom
Multiple R-squared:  0.6454,	Adjusted R-squared:  0.6336 
F-statistic: 54.61 on 1 and 30 DF,  p-value: 3.115e-08

> </code>

Actual correlation would look like the below. 
{{lm.y.x1.png?400}}
x1의 영향력은 y 총분산에 비해 크지 않다 (b / a + b = .469%)

{{pcor.y.x1.x2.v2.png?400}}
x2의 영향력을 control한 후에 x1영향력을 보면 64.54%에 달하게 된다. 

====== IV 각각의 설명력의 크기와 합쳐서 regression했을 때의 크기? ======
see https://www.researchgate.net/post/Why_is_the_Multiple_regression_model_not_significant_while_simple_regression_for_the_same_variables_is_significant

<code>
RSS = 3:10 #Right shoe size
LSS = rnorm(RSS, RSS, 0.1) #Left shoe size - similar to RSS
cor(LSS, RSS) #correlation ~ 0.99
 
weights = 120 + rnorm(RSS, 10*RSS, 10)
 
##Fit a joint model
m = lm(weights ~ LSS + RSS)

##F-value is very small, but neither LSS or RSS are significant
summary(m)
</code>


<code>> RSS = 3:10 #Right shoe size
> LSS = rnorm(RSS, RSS, 0.1) #Left shoe size - similar to RSS
> cor(LSS, RSS) #correlation ~ 0.99
[1] 0.9994836
> 
> weights = 120 + rnorm(RSS, 10*RSS, 10)
> 
> ##Fit a joint model
> m = lm(weights ~ LSS + RSS)
> 
> ##F-value is very small, but neither LSS or RSS are significant
> summary(m)

Call:
lm(formula = weights ~ LSS + RSS)

Residuals:
      1       2       3       4       5       6       7       8 
 4.8544  4.5254 -3.6333 -7.6402 -0.2467 -3.1997 -5.2665 10.6066 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  104.842      8.169  12.834 5.11e-05 ***
LSS          -14.162     35.447  -0.400    0.706    
RSS           26.305     35.034   0.751    0.487    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 7.296 on 5 degrees of freedom
Multiple R-squared:  0.9599,	Adjusted R-squared:  0.9439 
F-statistic: 59.92 on 2 and 5 DF,  p-value: 0.000321

> 
> ##Fitting RSS or LSS separately gives a significant result. 
> summary(lm(weights ~ LSS))

Call:
lm(formula = weights ~ LSS)

Residuals:
   Min     1Q Median     3Q    Max 
-6.055 -4.930 -2.925  4.886 11.854 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  103.099      7.543   13.67 9.53e-06 ***
LSS           12.440      1.097   11.34 2.81e-05 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 7.026 on 6 degrees of freedom
Multiple R-squared:  0.9554,	Adjusted R-squared:  0.948 
F-statistic: 128.6 on 1 and 6 DF,  p-value: 2.814e-05

> 
</code>





====== e. g. API00 data  using ppcor package ======
This part is an extension of the [[:multiple regression#in_r|R example in Multiple Regression]]. 

We are going to use spcor for identifying the effect of each IV. In order to do that, we need to reconstruct the data with the involved variable. 

<code>
dvar <- read.csv("http://commres.net/wiki/_media/elemapi2_.csv", fileEncoding="UTF-8-BOM")
mod <- lm(api00 ~ ell + acs_k3 + avg_ed + meals, data=dvar)
summary(mod)
anova(mod)

attach(dvar)
da1 <- data.frame(api00, ell, acs_k3, avg_ed, meals)
da2 <- data.frame(api00, ell, avg_ed, meals)

da1 <- na.omit(da1)
da2 <- na.omit(da2)

spcor(da1)
spcor(da2)

</code>


<code>
> spcor(da1)
$estimate
             api00         ell      acs_k3      avg_ed      meals
api00   1.00000000 -0.09112026  0.03072660  0.08883450 -0.3190889
ell    -0.13469956  1.00000000  0.06086724 -0.06173591  0.1626061
acs_k3  0.07245527  0.09709299  1.00000000 -0.13288465 -0.1367842
avg_ed  0.12079565 -0.05678795 -0.07662825  1.00000000 -0.2028836
meals  -0.29972194  0.10332189 -0.05448629 -0.14014709  1.0000000

$p.value
              api00        ell    acs_k3      avg_ed        meals
api00  0.000000e+00 0.07761805 0.5525340 0.085390280 2.403284e-10
ell    8.918743e-03 0.00000000 0.2390272 0.232377348 1.558141e-03
acs_k3 1.608778e-01 0.05998819 0.0000000 0.009891503 7.907183e-03
avg_ed 1.912418e-02 0.27203887 0.1380449 0.000000000 7.424903e-05
meals  3.041658e-09 0.04526574 0.2919775 0.006489783 0.000000e+00

$statistic
           api00       ell     acs_k3    avg_ed     meals
api00   0.000000 -1.769543  0.5945048  1.724797 -6.511264
ell    -2.628924  0.000000  1.1793030 -1.196197  3.187069
acs_k3  1.404911  1.886603  0.0000000 -2.592862 -2.670380
avg_ed  2.353309 -1.100002 -1.4862899  0.000000 -4.006914
meals  -6.075665  2.008902 -1.0552823 -2.737331  0.000000

$n
[1] 379

$gp
[1] 3

$method
[1] "pearson"

> spcor(da2)
$estimate
            api00         ell      avg_ed      meals
api00   1.0000000 -0.08988307  0.08485896 -0.3350062
ell    -0.1331295  1.00000000 -0.07018696  0.1557097
avg_ed  0.1164280 -0.06501592  1.00000000 -0.1967128
meals  -0.3170300  0.09948710 -0.13568145  1.0000000

$p.value
              api00        ell      avg_ed        meals
api00  0.000000e+00 0.08053412 0.099035570 2.160270e-11
ell    9.465897e-03 0.00000000 0.172705169 2.366624e-03
avg_ed 2.340012e-02 0.20663211 0.000000000 1.158869e-04
meals  2.698723e-10 0.05296417 0.008170237 0.000000e+00

$statistic
           api00       ell    avg_ed     meals
api00   0.000000 -1.752306  1.653628 -6.903560
ell    -2.608123  0.000000 -1.366153  3.060666
avg_ed  2.276102 -1.265057  0.000000 -3.895587
meals  -6.490414  1.941321 -2.659047  0.000000

$n
[1] 381

$gp
[1] 2

$method
[1] "pearson"

> 
</code>
====== e.g. mpg model in mtcars (in r) ======
{{:pasted:20201203-182515.png?700}}