====== Beta coefficients in linear regression ======

{{:pasted:20190521-113150.png?200}}

\begin{align*}
\large{\beta = b * \frac{sd(x)}{sd(y)}} \
\end{align*}

<code>
# import test score data "tests_cor.csv"
tests <- read.csv("http://commres.net/wiki/_media/r/tests_cor.csv")
colnames(tests) <- c("ser", "sat", "clep", "gpa")
tests <- subset(tests, select=c("sat", "clep", "gpa"))
attach(tests)
</code>

<code>lm.gpa.clepsat <- lm(gpa ~ clep + sat, data = tests) 
summary(lm.gpa.clepsat)
Call:
lm(formula = gpa ~ clep + sat, data = tests)

Residuals:
      Min        1Q    Median        3Q       Max 
-0.197888 -0.128974 -0.000528  0.131170  0.226404 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)  
(Intercept)  1.1607560  0.4081117   2.844   0.0249 *
clep         0.0729294  0.0253799   2.874   0.0239 *
sat         -0.0007015  0.0012564  -0.558   0.5940  
---
Signif. codes:  
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.1713 on 7 degrees of freedom
Multiple R-squared:  0.7778,	Adjusted R-squared:  0.7143 
F-statistic: 12.25 on 2 and 7 DF,  p-value: 0.005175

> 
</code>

<code>> sd.clep <- sd(clep)
> sd.sat <- sd(sat)
> sd.gpa <- sd(gpa)
> lm.gpa.clepsat <- lm(gpa ~ clep + sat, data = tests) 
> summary(lm.gpa.clepsat)

Call:
lm(formula = gpa ~ clep + sat, data = tests)

Residuals:
      Min        1Q    Median        3Q       Max 
-0.197888 -0.128974 -0.000528  0.131170  0.226404 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)  
(Intercept)  1.1607560  0.4081117   2.844   0.0249 *
clep         0.0729294  0.0253799   2.874   0.0239 *
sat         -0.0007015  0.0012564  -0.558   0.5940  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.1713 on 7 degrees of freedom
Multiple R-squared:  0.7778,	Adjusted R-squared:  0.7143 
F-statistic: 12.25 on 2 and 7 DF,  p-value: 0.005175

> b.clep <- 0.0729294
> b.sat <- -0.0007015
> beta.clep <- b.clep * (sd.clep/sd.gpa)
> beta.sat <- b.sat * (sd.sat/sd.gpa)
> lm.beta(lm.gpa.clepsat)

Call:
lm(formula = gpa ~ clep + sat, data = tests)

Standardized Coefficients::
(Intercept)        clep         sat 
  0.0000000   1.0556486  -0.2051189 

> beta.clep
[1] 1.055648
> beta.sat
[1] -0.2051187
> 
</code>
====== e.g. ======

<code>
# get marketing data 
marketing <- read.csv("http://commres.net/wiki/_media/marketing_from_datarium.csv")
head(marketing)
# note that I need - X to get rid of X column in the marketing data
mod <- lm(sales ~ . - X, data=marketing)
summary(mod)
</code>

<code>
> marketing <- read.csv("http://commres.net/wiki/_media/marketing_from_datarium.csv")
> head(marketing)
  X youtube facebook newspaper sales
1 1  276.12    45.36     83.04 26.52
2 2   53.40    47.16     54.12 12.48
3 3   20.64    55.08     83.16 11.16
4 4  181.80    49.56     70.20 22.20
5 5  216.96    12.96     70.08 15.48
6 6   10.44    58.68     90.00  8.64
# note that I need - X to get rid of X column in the marketing data
> mod <- lm(sales ~ . - X, data=marketing) 
> summary(mod)

Call:
lm(formula = sales ~ . - X, 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
</code>



<code>
install.packages(lm.beta)
library(lm.beta)
lm.beta(mod)
</code>

<code>
lm.beta(mod)

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

Standardized Coefficients::
 (Intercept)      youtube     facebook    newspaper 
 0.000000000  0.753065912  0.536481550 -0.004330686 
> 
</code>

These beta coefficients also can be got from the coefficents from standardized data. 

<code>
mod.formula <- sales ~ youtube + facebook + newspaper
all.vars(mod.formula)
marketing.temp <- sapply(marketing[ , all.vars(mod.formula)], scale)
head(marketing.temp)
mod.scaled <- lm(sales ~ ., data=marketing.scaled)
head(marketing.scaled)
coefficients(mod.scaled)
</code>

<code>> mod.formula <- sales ~ youtube + facebook + newspaper
> all.vars(mod.formula)
[1] "sales"     "youtube"   "facebook"  "newspaper"
> marketing.temp <- sapply(marketing[ , all.vars(mod.formula)], scale)
> head(marketing.temp)
          sales     youtube   facebook newspaper
[1,]  1.5481681  0.96742460  0.9790656 1.7744925
[2,] -0.6943038 -1.19437904  1.0800974 0.6679027
[3,] -0.9051345 -1.51235985  1.5246374 1.7790842
[4,]  0.8581768  0.05191939  1.2148065 1.2831850
[5,] -0.2151431  0.39319551 -0.8395070 1.2785934
[6,] -1.3076295 -1.61136487  1.7267010 2.0408088
> mod.scaled <- lm(sales ~ ., data=marketing.scaled)
> head(marketing.scaled)
       sales     youtube   facebook newspaper
1  1.5481681  0.96742460  0.9790656 1.7744925
2 -0.6943038 -1.19437904  1.0800974 0.6679027
3 -0.9051345 -1.51235985  1.5246374 1.7790842
4  0.8581768  0.05191939  1.2148065 1.2831850
5 -0.2151431  0.39319551 -0.8395070 1.2785934
6 -1.3076295 -1.61136487  1.7267010 2.0408088
> coefficients(mod.scaled)
  (Intercept)       youtube      facebook     newspaper 
-5.034110e-16  7.530659e-01  5.364815e-01 -4.330686e-03 
> 
> </code>

check out that 
''lm.beta(mod) == coefficients(mod.scaled)''

and 
베타를 구하고 나면 서로의 계수값을 절대비교할 수 있다. 
''lm.beta(mod)''의 아웃풋을 보고 
''youtube'', ''facebook'', and ''newspaper'' 순으로 설명력을 갖는다고 말할 수 있다 (혹은 newspaper를 분석에서 제외하고 다시 분석하여 둘만의 설명력을 보는 것도 방법이다.