1 Objektif

Pada tulisan ini akan dibahas secara singkat langkah-langkah melakukan analisis korelasi dan regresi linier.

2 Package

3 Teori

3.1 Regresi Linier Sederhana

Regresi Linier Sederhana adalah sebuah metode untuk mendapatkan sebuah persamaan matematis dari sebuah data yang terdiri dari hanya satu buah variabel dependent (biasa disebut \(y\)) dan satu buah variabel independent (biasa disebut \(x\)). Metode regresi linier secara umum bertujuan untuk mencari nilai koefisien regresi (\(\beta\)) dari variabel \(x\) untuk dapat menduga nilai variabel \(y\), yaitu \(\hat y\), dengan meminimalisisr nilai error, \(\epsilon\).

Persamaan atau formula regresi linier sederhana adalah sebagai berikut. \[ y = \beta_0 + \beta_1 x + \epsilon\] dengan \(\epsilon\) adalah nilai error yang tidak dapat diketahui dan bersifat random.

Untuk mendapat nilai dugaan yang diinginkan mendekati nilai \(y\) yang sesunguhnya/aktual, maka digunakan persamaan matematis sebagai berikut.

\[ \hat y = b_0 + b_1 x\] Pada persamaan pendugaan tersebut tidak terdapat nilai error karena tidak dapat dihitung dan tidak diketahui, sehingga diharapkan nilai error yang ada adalah sekecil mungkin dan diabaikan dalam persamaan matematis.

3.2 Regresi Linier Berganda

Persamaan matematis di atas adalah ketika persamaan regresi menggunakan satu variabel independent saja. Jika variabel independent yang digunakan sebanyak \(k\) maka persamaan matematis untuk mendapatkan \(\hat y\) adalah sebagai berikut.

\[ \hat y = b_0 + b_1 x_1 + b_2x_2 + \cdots + b_kx_k\]

3.3 Notasi Matriks

Persamaan matematis untuk regresi linier sederhana atau berganda dapat dituliskan dalam notasi matriks sebagai berikut.

\[ \mathbf{\hat Y} = \mathbf{Xb}\]

dengan \[ \begin{aligned} \mathbf{\hat Y} &= \{\hat y_1, \hat y_2, \cdots, \hat y_n\}\\ \mathbf{b} &= \{b_0, b_1, \cdots, b_k\}\\ \mathbf{X} &= \left[\begin{array} {rrr} 1 & x_{11} & \cdots & x_{k1} \\ 1 & x_{12} & \cdots & x_{k2} \\ \cdots & \cdots & \cdots & \cdots \\ 1 & x_{1n} & \cdots & x_{kn} \end{array}\right] \end{aligned} \]

3.4 R-squared dan Adjusted R-squared

3.4.1 R-squared

\[ \begin{aligned} R^2 &= 1-\frac{SS Error}{SS Total} \\ &= 1-\frac{\sum_{i=1}^{n}(\hat y_i - \bar y)^2}{\sum_{i=1}^{n}(y_i - \bar y)^2} \end{aligned} \]

3.4.2 Adjusted R-sqaured

\[ \begin{aligned} R^2_{adj} &= 1 - \biggl[(1 - R^2)\biggl(\frac{n-1}{n - p - 1}\biggr)\biggr] \\ &= 1-\frac{p-1}{n-1}\biggl(\frac{SSE}{SST}\biggr)\\ &= 1 - \frac{MSE}{SST/p-1} \end{aligned} \] Cukup dengan teorinya # Data

Kita akan menggunakan data ?mtcars dengan varaiabel mpg sebagai dependent variable atau target. Data ini sudah tersedia di R. Data diambil dari majalah Motor Trend US tahun 1974.

head(mtcars)
mtcars2 <- within(mtcars, {
   vs <- factor(vs, labels = c("V-shaped", "Straight"))
   am <- factor(am, labels = c("Automatic", "Manual"))
   cyl  <- factor(cyl)
   gear <- factor(gear)
   carb <- factor(carb)
})

str(mtcars2)
'data.frame':   32 obs. of  11 variables:
 $ mpg : num  21 21 22.8 21.4 18.7 18.1 14.3 24.4 22.8 19.2 ...
 $ cyl : Factor w/ 3 levels "4","6","8": 2 2 1 2 3 2 3 1 1 2 ...
 $ disp: num  160 160 108 258 360 ...
 $ hp  : num  110 110 93 110 175 105 245 62 95 123 ...
 $ drat: num  3.9 3.9 3.85 3.08 3.15 2.76 3.21 3.69 3.92 3.92 ...
 $ wt  : num  2.62 2.88 2.32 3.21 3.44 ...
 $ qsec: num  16.5 17 18.6 19.4 17 ...
 $ vs  : Factor w/ 2 levels "V-shaped","Straight": 1 1 2 2 1 2 1 2 2 2 ...
 $ am  : Factor w/ 2 levels "Automatic","Manual": 2 2 2 1 1 1 1 1 1 1 ...
 $ gear: Factor w/ 3 levels "3","4","5": 2 2 2 1 1 1 1 2 2 2 ...
 $ carb: Factor w/ 6 levels "1","2","3","4",..: 4 4 1 1 2 1 4 2 2 4 ...

4 Eksplorasi

Kita lakukan eksplorasi terlebih dahulu.

summary(mtcars2)
      mpg        cyl         disp             hp             drat      
 Min.   :10.40   4:11   Min.   : 71.1   Min.   : 52.0   Min.   :2.760  
 1st Qu.:15.43   6: 7   1st Qu.:120.8   1st Qu.: 96.5   1st Qu.:3.080  
 Median :19.20   8:14   Median :196.3   Median :123.0   Median :3.695  
 Mean   :20.09          Mean   :230.7   Mean   :146.7   Mean   :3.597  
 3rd Qu.:22.80          3rd Qu.:326.0   3rd Qu.:180.0   3rd Qu.:3.920  
 Max.   :33.90          Max.   :472.0   Max.   :335.0   Max.   :4.930  
       wt             qsec              vs             am     gear   carb  
 Min.   :1.513   Min.   :14.50   V-shaped:18   Automatic:19   3:15   1: 7  
 1st Qu.:2.581   1st Qu.:16.89   Straight:14   Manual   :13   4:12   2:10  
 Median :3.325   Median :17.71                                5: 5   3: 3  
 Mean   :3.217   Mean   :17.85                                       4:10  
 3rd Qu.:3.610   3rd Qu.:18.90                                       6: 1  
 Max.   :5.424   Max.   :22.90                                       8: 1  
library(ggplot2)
ggplot(mtcars2, aes(x = mpg)) +
  geom_histogram(bins = 10, color = "white", fill = "pink")

ggplot(mtcars2, aes(x = disp, y = mpg)) +
  geom_point()

ggplot(mtcars2, aes(x = disp, y = mpg)) +
  geom_point() +
  geom_smooth(method = "lm")

cor(mtcars2[, c("mpg", "disp")])
            mpg       disp
mpg   1.0000000 -0.8475514
disp -0.8475514  1.0000000
cor.test(mtcars2$mpg, mtcars2$disp)

    Pearson's product-moment correlation

data:  mtcars2$mpg and mtcars2$disp
t = -8.7472, df = 30, p-value = 0.000000000938
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.9233594 -0.7081376
sample estimates:
       cor 
-0.8475514 
ggplot(mtcars2, aes(x = cyl, y = mpg)) +
  geom_boxplot()

ggplot(mtcars2, aes(x = hp, y = mpg)) +
  geom_point() +
  geom_smooth(method = "lm")

cor(mtcars2[, c("mpg", "hp")])
           mpg         hp
mpg  1.0000000 -0.7761684
hp  -0.7761684  1.0000000
cor.test(mtcars2$mpg, mtcars2$hp)

    Pearson's product-moment correlation

data:  mtcars2$mpg and mtcars2$hp
t = -6.7424, df = 30, p-value = 0.0000001788
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.8852686 -0.5860994
sample estimates:
       cor 
-0.7761684 
ggplot(mtcars2, aes(x = drat, y = mpg)) +
  geom_point()

ggplot(mtcars2, aes(x = drat, y = mpg)) +
  geom_point() +
  geom_smooth(method = "lm")

cor(mtcars2[, c("mpg", "drat")])
           mpg      drat
mpg  1.0000000 0.6811719
drat 0.6811719 1.0000000
cor.test(mtcars2$mpg, mtcars2$drat)

    Pearson's product-moment correlation

data:  mtcars2$mpg and mtcars2$drat
t = 5.096, df = 30, p-value = 0.00001776
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 0.4360484 0.8322010
sample estimates:
      cor 
0.6811719 
ggplot(mtcars2, aes(x = wt, y = mpg)) +
  geom_point()

ggplot(mtcars2, aes(x = wt, y = mpg)) +
  geom_point() +
  geom_smooth(method = "lm")

cor(mtcars2[, c("mpg", "wt")])
           mpg         wt
mpg  1.0000000 -0.8676594
wt  -0.8676594  1.0000000
cor.test(mtcars2$mpg, mtcars2$wt)

    Pearson's product-moment correlation

data:  mtcars2$mpg and mtcars2$wt
t = -9.559, df = 30, p-value = 0.0000000001294
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.9338264 -0.7440872
sample estimates:
       cor 
-0.8676594 
ggplot(mtcars2, aes(x = qsec, y = mpg)) +
  geom_point()

ggplot(mtcars2, aes(x = qsec, y = mpg)) +
  geom_point() +
  geom_smooth(method = "lm")

cor(mtcars2[, c("mpg", "qsec")])
          mpg     qsec
mpg  1.000000 0.418684
qsec 0.418684 1.000000
cor.test(mtcars2$mpg, mtcars2$qsec)

    Pearson's product-moment correlation

data:  mtcars2$mpg and mtcars2$qsec
t = 2.5252, df = 30, p-value = 0.01708
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 0.08195487 0.66961864
sample estimates:
     cor 
0.418684

5 Korelasi

library(corrplot)
corrmatrix <- cor(mtcars2[, c(1, 3:7)])

corrplot(corrmatrix, method = "number")

6 Regresi Linier

6.1 Variabel Numerik Independent

lm1 <- lm(mpg ~ disp, data = mtcars2)
summary(lm1)

Call:
lm(formula = mpg ~ disp, data = mtcars2)

Residuals:
    Min      1Q  Median      3Q     Max 
-4.8922 -2.2022 -0.9631  1.6272  7.2305 

Coefficients:
             Estimate Std. Error t value             Pr(>|t|)    
(Intercept) 29.599855   1.229720  24.070 < 0.0000000000000002 ***
disp        -0.041215   0.004712  -8.747       0.000000000938 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.251 on 30 degrees of freedom
Multiple R-squared:  0.7183,    Adjusted R-squared:  0.709 
F-statistic: 76.51 on 1 and 30 DF,  p-value: 0.000000000938
lm1 <- lm(mpg ~ hp, data = mtcars2)
summary(lm1)

Call:
lm(formula = mpg ~ hp, data = mtcars2)

Residuals:
    Min      1Q  Median      3Q     Max 
-5.7121 -2.1122 -0.8854  1.5819  8.2360 

Coefficients:
            Estimate Std. Error t value             Pr(>|t|)    
(Intercept) 30.09886    1.63392  18.421 < 0.0000000000000002 ***
hp          -0.06823    0.01012  -6.742          0.000000179 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.863 on 30 degrees of freedom
Multiple R-squared:  0.6024,    Adjusted R-squared:  0.5892 
F-statistic: 45.46 on 1 and 30 DF,  p-value: 0.0000001788
lm1 <- lm(mpg ~ drat, data = mtcars2)
summary(lm1)

Call:
lm(formula = mpg ~ drat, data = mtcars2)

Residuals:
    Min      1Q  Median      3Q     Max 
-9.0775 -2.6803 -0.2095  2.2976  9.0225 

Coefficients:
            Estimate Std. Error t value  Pr(>|t|)    
(Intercept)   -7.525      5.477  -1.374      0.18    
drat           7.678      1.507   5.096 0.0000178 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 4.485 on 30 degrees of freedom
Multiple R-squared:  0.464, Adjusted R-squared:  0.4461 
F-statistic: 25.97 on 1 and 30 DF,  p-value: 0.00001776
lm1 <- lm(mpg ~ wt, data = mtcars2)
summary(lm1)

Call:
lm(formula = mpg ~ wt, data = mtcars2)

Residuals:
    Min      1Q  Median      3Q     Max 
-4.5432 -2.3647 -0.1252  1.4096  6.8727 

Coefficients:
            Estimate Std. Error t value             Pr(>|t|)    
(Intercept)  37.2851     1.8776  19.858 < 0.0000000000000002 ***
wt           -5.3445     0.5591  -9.559       0.000000000129 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.046 on 30 degrees of freedom
Multiple R-squared:  0.7528,    Adjusted R-squared:  0.7446 
F-statistic: 91.38 on 1 and 30 DF,  p-value: 0.0000000001294
lm1 <- lm(mpg ~ qsec, data = mtcars2)
summary(lm1)

Call:
lm(formula = mpg ~ qsec, data = mtcars2)

Residuals:
    Min      1Q  Median      3Q     Max 
-9.8760 -3.4539 -0.7203  2.2774 11.6491 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)  
(Intercept)  -5.1140    10.0295  -0.510   0.6139  
qsec          1.4121     0.5592   2.525   0.0171 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 5.564 on 30 degrees of freedom
Multiple R-squared:  0.1753,    Adjusted R-squared:  0.1478 
F-statistic: 6.377 on 1 and 30 DF,  p-value: 0.01708

Dengan taraf nyata (\(\alpha\)) sebesar 5%, masing-masing variabel numerik berpengaruh signifikan terhadap mpg.

lm2 <- lm(mpg ~ disp + hp, data = mtcars2)
summary(lm2)

Call:
lm(formula = mpg ~ disp + hp, data = mtcars2)

Residuals:
    Min      1Q  Median      3Q     Max 
-4.7945 -2.3036 -0.8246  1.8582  6.9363 

Coefficients:
             Estimate Std. Error t value             Pr(>|t|)    
(Intercept) 30.735904   1.331566  23.083 < 0.0000000000000002 ***
disp        -0.030346   0.007405  -4.098             0.000306 ***
hp          -0.024840   0.013385  -1.856             0.073679 .  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.127 on 29 degrees of freedom
Multiple R-squared:  0.7482,    Adjusted R-squared:  0.7309 
F-statistic: 43.09 on 2 and 29 DF,  p-value: 0.000000002062

Ternyata jika disp dan hp digunakan bersamaan, hp tidak berpengaruh signifikan.

lm2 <- lm(mpg ~ disp + drat, data = mtcars2)
summary(lm2)

Call:
lm(formula = mpg ~ disp + drat, data = mtcars2)

Residuals:
    Min      1Q  Median      3Q     Max 
-5.1265 -2.2045 -0.5835  1.4497  6.9884 

Coefficients:
             Estimate Std. Error t value   Pr(>|t|)    
(Intercept) 21.844880   6.747971   3.237    0.00302 ** 
disp        -0.035694   0.006653  -5.365 0.00000919 ***
drat         1.802027   1.542091   1.169    0.25210    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.232 on 29 degrees of freedom
Multiple R-squared:  0.731, Adjusted R-squared:  0.7125 
F-statistic: 39.41 on 2 and 29 DF,  p-value: 0.000000005385

Variabel drat juga sama, tidak berpengaruh signifikan.

lm2 <- lm(mpg ~ disp + wt, data = mtcars2)
summary(lm2)

Call:
lm(formula = mpg ~ disp + wt, data = mtcars2)

Residuals:
    Min      1Q  Median      3Q     Max 
-3.4087 -2.3243 -0.7683  1.7721  6.3484 

Coefficients:
            Estimate Std. Error t value             Pr(>|t|)    
(Intercept) 34.96055    2.16454  16.151 0.000000000000000491 ***
disp        -0.01773    0.00919  -1.929              0.06362 .  
wt          -3.35082    1.16413  -2.878              0.00743 ** 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.917 on 29 degrees of freedom
Multiple R-squared:  0.7809,    Adjusted R-squared:  0.7658 
F-statistic: 51.69 on 2 and 29 DF,  p-value: 0.0000000002744

Jika disp dan wt diagunakan bersamaan dalam membentuk model regresi, wt lebih berpengaruh signifikan. Variabel disp menjadi tidak berpengaruh signifikan.

lm3 <- lm(mpg ~ disp + hp + drat, data = mtcars2)
summary(lm3)

Call:
lm(formula = mpg ~ disp + hp + drat, data = mtcars2)

Residuals:
    Min      1Q  Median      3Q     Max 
-5.1225 -1.8454 -0.4456  1.1342  6.4958 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)   
(Intercept) 19.344293   6.370882   3.036  0.00513 **
disp        -0.019232   0.009371  -2.052  0.04960 * 
hp          -0.031229   0.013345  -2.340  0.02663 * 
drat         2.714975   1.487366   1.825  0.07863 . 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.008 on 28 degrees of freedom
Multiple R-squared:  0.775, Adjusted R-squared:  0.7509 
F-statistic: 32.15 on 3 and 28 DF,  p-value: 0.00000000328

Hanya drat yang tidak berpengaruh signifikan.

6.2 Variabel Kategorik

lm1 <- lm(mpg ~ cyl, data = mtcars2)
summary(lm1)

Call:
lm(formula = mpg ~ cyl, data = mtcars2)

Residuals:
    Min      1Q  Median      3Q     Max 
-5.2636 -1.8357  0.0286  1.3893  7.2364 

Coefficients:
            Estimate Std. Error t value             Pr(>|t|)    
(Intercept)  26.6636     0.9718  27.437 < 0.0000000000000002 ***
cyl6         -6.9208     1.5583  -4.441             0.000119 ***
cyl8        -11.5636     1.2986  -8.905       0.000000000857 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.223 on 29 degrees of freedom
Multiple R-squared:  0.7325,    Adjusted R-squared:  0.714 
F-statistic:  39.7 on 2 and 29 DF,  p-value: 0.000000004979
lm1 <- lm(mpg ~ vs, data = mtcars2)
summary(lm1)

Call:
lm(formula = mpg ~ vs, data = mtcars2)

Residuals:
   Min     1Q Median     3Q    Max 
-6.757 -3.082 -1.267  2.828  9.383 

Coefficients:
            Estimate Std. Error t value             Pr(>|t|)    
(Intercept)   16.617      1.080  15.390 0.000000000000000885 ***
vsStraight     7.940      1.632   4.864 0.000034159372544199 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 4.581 on 30 degrees of freedom
Multiple R-squared:  0.4409,    Adjusted R-squared:  0.4223 
F-statistic: 23.66 on 1 and 30 DF,  p-value: 0.00003416
lm1 <- lm(mpg ~ am, data = mtcars2)
summary(lm1)

Call:
lm(formula = mpg ~ am, data = mtcars2)

Residuals:
    Min      1Q  Median      3Q     Max 
-9.3923 -3.0923 -0.2974  3.2439  9.5077 

Coefficients:
            Estimate Std. Error t value            Pr(>|t|)    
(Intercept)   17.147      1.125  15.247 0.00000000000000113 ***
amManual       7.245      1.764   4.106            0.000285 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 4.902 on 30 degrees of freedom
Multiple R-squared:  0.3598,    Adjusted R-squared:  0.3385 
F-statistic: 16.86 on 1 and 30 DF,  p-value: 0.000285
lm1 <- lm(mpg ~ gear, data = mtcars2)
summary(lm1)

Call:
lm(formula = mpg ~ gear, data = mtcars2)

Residuals:
    Min      1Q  Median      3Q     Max 
-6.7333 -3.2333 -0.9067  2.8483  9.3667 

Coefficients:
            Estimate Std. Error t value           Pr(>|t|)    
(Intercept)   16.107      1.216  13.250 0.0000000000000787 ***
gear4          8.427      1.823   4.621 0.0000725738200575 ***
gear5          5.273      2.431   2.169             0.0384 *  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 4.708 on 29 degrees of freedom
Multiple R-squared:  0.4292,    Adjusted R-squared:  0.3898 
F-statistic:  10.9 on 2 and 29 DF,  p-value: 0.0002948
lm1 <- lm(mpg ~ carb, data = mtcars2)
summary(lm1)

Call:
lm(formula = mpg ~ carb, data = mtcars2)

Residuals:
   Min     1Q Median     3Q    Max 
-7.243 -3.325  0.000  2.360  8.557 

Coefficients:
            Estimate Std. Error t value          Pr(>|t|)    
(Intercept)   25.343      1.854  13.670 0.000000000000221 ***
carb2         -2.943      2.417  -1.218           0.23435    
carb3         -9.043      3.385  -2.672           0.01285 *  
carb4         -9.553      2.417  -3.952           0.00053 ***
carb6         -5.643      5.243  -1.076           0.29174    
carb8        -10.343      5.243  -1.973           0.05927 .  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 4.905 on 26 degrees of freedom
Multiple R-squared:  0.4445,    Adjusted R-squared:  0.3377 
F-statistic: 4.161 on 5 and 26 DF,  p-value: 0.006546
lm2 <- lm(mpg ~ cyl + vs, data = mtcars2)
summary(lm2)

Call:
lm(formula = mpg ~ cyl + vs, data = mtcars2)

Residuals:
   Min     1Q Median     3Q    Max 
-5.201 -1.686  0.000  1.463  7.299 

Coefficients:
            Estimate Std. Error t value          Pr(>|t|)    
(Intercept)  27.2901     2.0856  13.085 0.000000000000188 ***
cyl6         -7.1535     1.7235  -4.151           0.00028 ***
cyl8        -12.1901     2.2616  -5.390 0.000009559631134 ***
vsStraight   -0.6891     2.0210  -0.341           0.73567    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.273 on 28 degrees of freedom
Multiple R-squared:  0.7336,    Adjusted R-squared:  0.705 
F-statistic:  25.7 on 3 and 28 DF,  p-value: 0.00000003412
lm2 <- lm(mpg ~ cyl + am, data = mtcars2)
summary(lm2)

Call:
lm(formula = mpg ~ cyl + am, data = mtcars2)

Residuals:
    Min      1Q  Median      3Q     Max 
-5.9618 -1.4971 -0.2057  1.8907  6.5382 

Coefficients:
            Estimate Std. Error t value             Pr(>|t|)    
(Intercept)   24.802      1.323  18.752 < 0.0000000000000002 ***
cyl6          -6.156      1.536  -4.009             0.000411 ***
cyl8         -10.068      1.452  -6.933          0.000000155 ***
amManual       2.560      1.298   1.973             0.058457 .  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.073 on 28 degrees of freedom
Multiple R-squared:  0.7651,    Adjusted R-squared:  0.7399 
F-statistic:  30.4 on 3 and 28 DF,  p-value: 0.000000005959
lm2 <- lm(mpg ~ cyl + gear, data = mtcars2)
summary(lm2)

Call:
lm(formula = mpg ~ cyl + gear, data = mtcars2)

Residuals:
    Min      1Q  Median      3Q     Max 
-5.3520 -1.7633 -0.3789  1.7393  7.1480 

Coefficients:
            Estimate Std. Error t value          Pr(>|t|)    
(Intercept)   25.428      1.881  13.517 0.000000000000155 ***
cyl6          -6.656      1.629  -4.086          0.000353 ***
cyl8         -10.542      1.958  -5.384 0.000010865640238 ***
gear4          1.324      1.928   0.687          0.498000    
gear5          1.500      1.855   0.809          0.425707    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.294 on 27 degrees of freedom
Multiple R-squared:  0.7398,    Adjusted R-squared:  0.7012 
F-statistic: 19.19 on 4 and 27 DF,  p-value: 0.0000001405
lm2 <- lm(mpg ~ cyl + carb, data = mtcars2)
summary(lm2)

Call:
lm(formula = mpg ~ cyl + carb, data = mtcars2)

Residuals:
   Min     1Q Median     3Q    Max 
-5.339 -1.354  0.000  2.068  7.061 

Coefficients:
            Estimate Std. Error t value            Pr(>|t|)    
(Intercept)  26.8391     1.4004  19.165 0.00000000000000047 ***
cyl6         -5.2370     2.0713  -2.528              0.0184 *  
cyl8        -10.2935     1.8498  -5.565 0.00001003182496434 ***
carb2        -0.3217     1.7737  -0.181              0.8576    
carb3        -0.2457     2.8136  -0.087              0.9312    
carb4        -2.7783     2.0787  -1.337              0.1939    
carb6        -1.9022     3.8829  -0.490              0.6287    
carb8        -1.5457     3.9287  -0.393              0.6975    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.358 on 24 degrees of freedom
Multiple R-squared:  0.7596,    Adjusted R-squared:  0.6895 
F-statistic: 10.84 on 7 and 24 DF,  p-value: 0.00000418
lm3 <- lm(mpg ~ cyl + vs + am, data = mtcars2)
summary(lm3)

Call:
lm(formula = mpg ~ cyl + vs + am, data = mtcars2)

Residuals:
    Min      1Q  Median      3Q     Max 
-6.2821 -1.4402  0.0391  1.8845  6.2179 

Coefficients:
            Estimate Std. Error t value     Pr(>|t|)    
(Intercept)   22.809      2.928   7.789 0.0000000224 ***
cyl6          -5.399      1.837  -2.938      0.00668 ** 
cyl8          -8.161      2.892  -2.822      0.00884 ** 
vsStraight     1.708      2.235   0.764      0.45135    
amManual       3.165      1.528   2.071      0.04805 *  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.097 on 27 degrees of freedom
Multiple R-squared:  0.7701,    Adjusted R-squared:  0.736 
F-statistic: 22.61 on 4 and 27 DF,  p-value: 0.00000002741

6.3 Semua Variabel

lm_all <- lm(mpg ~ ., data = mtcars2)
summary(lm_all)

Call:
lm(formula = mpg ~ ., data = mtcars2)

Residuals:
    Min      1Q  Median      3Q     Max 
-3.5087 -1.3584 -0.0948  0.7745  4.6251 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)  
(Intercept) 23.87913   20.06582   1.190   0.2525  
cyl6        -2.64870    3.04089  -0.871   0.3975  
cyl8        -0.33616    7.15954  -0.047   0.9632  
disp         0.03555    0.03190   1.114   0.2827  
hp          -0.07051    0.03943  -1.788   0.0939 .
drat         1.18283    2.48348   0.476   0.6407  
wt          -4.52978    2.53875  -1.784   0.0946 .
qsec         0.36784    0.93540   0.393   0.6997  
vsStraight   1.93085    2.87126   0.672   0.5115  
amManual     1.21212    3.21355   0.377   0.7113  
gear4        1.11435    3.79952   0.293   0.7733  
gear5        2.52840    3.73636   0.677   0.5089  
carb2       -0.97935    2.31797  -0.423   0.6787  
carb3        2.99964    4.29355   0.699   0.4955  
carb4        1.09142    4.44962   0.245   0.8096  
carb6        4.47757    6.38406   0.701   0.4938  
carb8        7.25041    8.36057   0.867   0.3995  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.833 on 15 degrees of freedom
Multiple R-squared:  0.8931,    Adjusted R-squared:  0.779 
F-statistic:  7.83 on 16 and 15 DF,  p-value: 0.000124

7 Conclusion

---
title:  "Basic Linear Regression"
author: "By Aep Hidayatuloh"
date:   "2019 Jun 21"
output: 
  html_notebook:
    number_sections: yes
    theme: spacelab
    df_print: paged
    toc: yes
    toc_depth: 4
    toc_float: true
---

<style type="text/css">

body{ /* Normal  */
      font-size: 12px;
  }
td {  /* Table  */
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}
h1.title {
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}
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```{r setup, include=FALSE}
#knitr::opts_chunk$set(echo = TRUE)
knitr::opts_chunk$set(echo=TRUE, fig.height=3.5, fig.width=9.2, results='hold', warning=FALSE, fig.show='hold', message=FALSE) 
options(scipen = 99)
```

<center>
<img src="regression.png">
</center>

# Objektif

Pada tulisan ini akan dibahas secara singkat langkah-langkah melakukan analisis korelasi dan regresi linier.

# Package


# Teori

## Regresi Linier Sederhana
Regresi Linier Sederhana adalah sebuah metode untuk mendapatkan sebuah persamaan matematis dari sebuah data yang terdiri dari hanya satu buah variabel dependent (biasa disebut $y$) dan satu buah variabel independent (biasa disebut $x$). Metode regresi linier secara umum bertujuan untuk mencari nilai koefisien regresi ($\beta$) dari variabel $x$ untuk dapat menduga nilai variabel $y$, yaitu $\hat y$, dengan meminimalisisr nilai error, $\epsilon$.

Persamaan atau formula regresi linier sederhana adalah sebagai berikut.
$$ y = \beta_0 + \beta_1 x + \epsilon$$
dengan $\epsilon$ adalah nilai error yang tidak dapat diketahui dan bersifat random.

Untuk mendapat nilai dugaan yang diinginkan mendekati nilai $y$ yang sesunguhnya/aktual, maka digunakan persamaan matematis sebagai berikut.

$$ \hat y = b_0 + b_1 x$$
Pada persamaan pendugaan tersebut tidak terdapat nilai error karena tidak dapat dihitung dan tidak diketahui, sehingga diharapkan nilai error yang ada adalah sekecil mungkin dan diabaikan dalam persamaan matematis.

## Regresi Linier Berganda

Persamaan matematis di atas adalah ketika persamaan regresi \textbf{hanya} menggunakan satu variabel independent saja. Jika variabel independent yang digunakan sebanyak $k$ maka persamaan matematis untuk mendapatkan $\hat y$ adalah sebagai berikut.

$$ \hat y = b_0 + b_1 x_1 + b_2x_2 + \cdots + b_kx_k$$

## Notasi Matriks

Persamaan matematis untuk regresi linier sederhana atau berganda dapat dituliskan dalam notasi matriks sebagai berikut.

$$ \mathbf{\hat Y} = \mathbf{Xb}$$

dengan 
$$
\begin{aligned}
\mathbf{\hat Y} &= \{\hat y_1, \hat y_2, \cdots, \hat y_n\}\\
\mathbf{b} &= \{b_0, b_1, \cdots, b_k\}\\
\mathbf{X} &= \left[\begin{array}
{rrr}
1 & x_{11} & \cdots & x_{k1} \\
1 & x_{12} & \cdots & x_{k2} \\
\cdots & \cdots & \cdots & \cdots \\
1 & x_{1n} & \cdots & x_{kn}
\end{array}\right]
\end{aligned}
$$

## R-squared dan Adjusted R-squared

### R-squared

$$
\begin{aligned}
R^2 &= 1-\frac{SS Error}{SS Total} \\
&= 1-\frac{\sum_{i=1}^{n}(\hat y_i - \bar y)^2}{\sum_{i=1}^{n}(y_i - \bar y)^2}
\end{aligned}
$$

### Adjusted R-sqaured

$$
\begin{aligned}
R^2_{adj} &= 1 - \biggl[(1 - R^2)\biggl(\frac{n-1}{n - p - 1}\biggr)\biggr] \\
&= 1-\frac{p-1}{n-1}\biggl(\frac{SSE}{SST}\biggr)\\
&= 1 - \frac{MSE}{SST/p-1}
\end{aligned}
$$
Cukup dengan teorinya
# Data

Kita akan menggunakan data `?mtcars` dengan varaiabel `mpg` sebagai dependent variable atau target. Data ini sudah tersedia di R. Data diambil dari majalah Motor Trend US tahun 1974.

```{r}
head(mtcars)
```

```{r}
mtcars2 <- within(mtcars, {
   vs <- factor(vs, labels = c("V-shaped", "Straight"))
   am <- factor(am, labels = c("Automatic", "Manual"))
   cyl  <- factor(cyl)
   gear <- factor(gear)
   carb <- factor(carb)
})

str(mtcars2)
```

# Eksplorasi

Kita lakukan eksplorasi terlebih dahulu.


```{r}
summary(mtcars2)
```

```{r}
library(ggplot2)
ggplot(mtcars2, aes(x = mpg)) +
  geom_histogram(bins = 10, color = "white", fill = "pink")
```

```{r}
ggplot(mtcars2, aes(x = disp, y = mpg)) +
  geom_point()
```

```{r}
ggplot(mtcars2, aes(x = disp, y = mpg)) +
  geom_point() +
  geom_smooth(method = "lm")
```
```{r}
cor(mtcars2[, c("mpg", "disp")])
cor.test(mtcars2$mpg, mtcars2$disp)
```

```{r}
ggplot(mtcars2, aes(x = cyl, y = mpg)) +
  geom_boxplot()
```

```{r}
ggplot(mtcars2, aes(x = hp, y = mpg)) +
  geom_point() +
  geom_smooth(method = "lm")
```

```{r}
cor(mtcars2[, c("mpg", "hp")])
cor.test(mtcars2$mpg, mtcars2$hp)
```


```{r}
ggplot(mtcars2, aes(x = drat, y = mpg)) +
  geom_point()
```

```{r}
ggplot(mtcars2, aes(x = drat, y = mpg)) +
  geom_point() +
  geom_smooth(method = "lm")
```

```{r}
cor(mtcars2[, c("mpg", "drat")])
cor.test(mtcars2$mpg, mtcars2$drat)
```


```{r}
ggplot(mtcars2, aes(x = wt, y = mpg)) +
  geom_point()
```

```{r}
ggplot(mtcars2, aes(x = wt, y = mpg)) +
  geom_point() +
  geom_smooth(method = "lm")
```

```{r}
cor(mtcars2[, c("mpg", "wt")])
cor.test(mtcars2$mpg, mtcars2$wt)
```


```{r}
ggplot(mtcars2, aes(x = qsec, y = mpg)) +
  geom_point()
```

```{r}
ggplot(mtcars2, aes(x = qsec, y = mpg)) +
  geom_point() +
  geom_smooth(method = "lm")
```

```{r}
cor(mtcars2[, c("mpg", "qsec")])
cor.test(mtcars2$mpg, mtcars2$qsec)
```

# Korelasi

```{r}
library(corrplot)
corrmatrix <- cor(mtcars2[, c(1, 3:7)])

corrplot(corrmatrix, method = "number")
```

# Regresi Linier

## Variabel Numerik Independent

```{r}
lm1 <- lm(mpg ~ disp, data = mtcars2)
summary(lm1)
```

```{r}
lm1 <- lm(mpg ~ hp, data = mtcars2)
summary(lm1)
```

```{r}
lm1 <- lm(mpg ~ drat, data = mtcars2)
summary(lm1)
```

```{r}
lm1 <- lm(mpg ~ wt, data = mtcars2)
summary(lm1)
```

```{r}
lm1 <- lm(mpg ~ qsec, data = mtcars2)
summary(lm1)
```

Dengan taraf nyata ($\alpha$) sebesar 5%, masing-masing variabel numerik berpengaruh signifikan terhadap `mpg`.

```{r}
lm2 <- lm(mpg ~ disp + hp, data = mtcars2)
summary(lm2)
```

Ternyata jika `disp` dan `hp` digunakan bersamaan, `hp` tidak berpengaruh signifikan. 

```{r}
lm2 <- lm(mpg ~ disp + drat, data = mtcars2)
summary(lm2)
```

Variabel `drat` juga sama, tidak berpengaruh signifikan.

```{r}
lm2 <- lm(mpg ~ disp + wt, data = mtcars2)
summary(lm2)
```

Jika `disp` dan `wt` diagunakan bersamaan dalam membentuk model regresi, `wt` lebih berpengaruh signifikan. Variabel `disp` menjadi tidak berpengaruh signifikan.

```{r}
lm3 <- lm(mpg ~ disp + hp + drat, data = mtcars2)
summary(lm3)
```

Hanya `drat` yang tidak berpengaruh signifikan.

## Variabel Kategorik

```{r}
lm1 <- lm(mpg ~ cyl, data = mtcars2)
summary(lm1)
```

```{r}
lm1 <- lm(mpg ~ vs, data = mtcars2)
summary(lm1)
```

```{r}
lm1 <- lm(mpg ~ am, data = mtcars2)
summary(lm1)
```

```{r}
lm1 <- lm(mpg ~ gear, data = mtcars2)
summary(lm1)
```

```{r}
lm1 <- lm(mpg ~ carb, data = mtcars2)
summary(lm1)
```

```{r}
lm2 <- lm(mpg ~ cyl + vs, data = mtcars2)
summary(lm2)
```

```{r}
lm2 <- lm(mpg ~ cyl + am, data = mtcars2)
summary(lm2)
```

```{r}
lm2 <- lm(mpg ~ cyl + gear, data = mtcars2)
summary(lm2)
```

```{r}
lm2 <- lm(mpg ~ cyl + carb, data = mtcars2)
summary(lm2)
```

```{r}
lm3 <- lm(mpg ~ cyl + vs + am, data = mtcars2)
summary(lm3)
```

## Semua Variabel
```{r}
lm_all <- lm(mpg ~ ., data = mtcars2)
summary(lm_all)
```


# Conclusion

```{r}

```

