# 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)

lm.y.x2 <- lm(y ~ x2, data=d)
summary(lm.y.x2)
anova(lm.y.x2)

lm.y.x1x2 <- lm(y ~ x1+x2, data=d)
summary(lm.y.x1x2)
anova(lm.y.x1x2)

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 * x1) + (b2 * x2)
y.pred 
# or 
y.pred2 <- predict(lm.y.x1x2)
head(y.pred == y.pred2)

y.real <- y
y.real
y.mean <- mean(y)
y.mean 

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
y==y2

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

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)
# 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

# 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)

# install.packages("ppcor")
library(ppcor)
partial.r <- pcor.test(y, x2, x1)
partial.r
str(partial.r)
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(y, x1, 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

#######################################################
#
# 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)
spr.1$estimate^2

lm.tmp.8 <- lm(y~res.x1.x2, data = d)
summary(lm.tmp.8)
spr.2$estimate^2

#######################################################
# 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

# OR 애초에 simple regression과 multiple 
# regression에서 얻은 R2을 가지고 
# 공통설명력을 알아볼 수도 있었다.
r2.x1 <- summary(lm.y.x1)$r.square
r2.x2 <- summary(lm.y.x2)$r.square
r2.x1x2 <- summary(lm.y.x1x2)$r.square
r2.x1 + r2.x2 - r2.x1x2