Tugas Akhir Anreg

Data

Inisialisasi Library

library(readxl)

## Warning: package 'readxl' was built under R version 4.3.2

library(dplyr)

## 
## Attaching package: 'dplyr'

## The following objects are masked from 'package:stats':
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##     filter, lag

## The following objects are masked from 'package:base':
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##     intersect, setdiff, setequal, union

library(plotly)

## Warning: package 'plotly' was built under R version 4.3.2

## Loading required package: ggplot2

## Warning: package 'ggplot2' was built under R version 4.3.2

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library(lmtest)

## Warning: package 'lmtest' was built under R version 4.3.2

## Loading required package: zoo

## Warning: package 'zoo' was built under R version 4.3.2

## 
## Attaching package: 'zoo'

## The following objects are masked from 'package:base':
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##     as.Date, as.Date.numeric

library(car)

## Warning: package 'car' was built under R version 4.3.3

## Loading required package: carData

## 
## Attaching package: 'car'

## The following object is masked from 'package:dplyr':
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##     recode

library(randtests)
library(nortest)
library(MASS)

## 
## Attaching package: 'MASS'

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##     select

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library(olsrr)

## Warning: package 'olsrr' was built under R version 4.3.2

## 
## Attaching package: 'olsrr'

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library(ggcorrplot)

## Warning: package 'ggcorrplot' was built under R version 4.3.3

library(GGally)

## Warning: package 'GGally' was built under R version 4.3.2

## Registered S3 method overwritten by 'GGally':
##   method from   
##   +.gg   ggplot2

Import Data

dt <- read_xlsx("D:/Semester 4/Analisis Regresi/DATA TUGAS AKHIR.xlsx", sheet="Data")
dt

## # A tibble: 22 × 13
##        Y    X1    X2     X3    X4    X5    X6    X7    X8    X9   X10   X11
##    <dbl> <dbl> <dbl>  <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
##  1  27.2  65.2  6.92 152414 11.9   0.39  58.8  3.52 10199 12696  67.6  3.93
##  2  28.1  67.0  7.57 255498  5.87  0.3   61.8  2.21 16617 36053  65.8  2.02
##  3  21.8  65.8  7.42 376837  6.17  1.18  76.4  3.22 13793 59622  65.6  4.2 
##  4  25.2  63.6  6.97 474521  7.71  1.03  67.9  2.64 10953 57201  66.9  2.09
##  5  21.8  65.2  8.16 271277  3.98  0.45  80.6  1.96 11511 42634  67.6  1.2 
##  6  14.3  63.8  7.39 231008  6.34  0.71  84.0  5.45 13997 45693  65.6  2.23
##  7  20.0  63.0  8.45 221536  1.75  0.63  81.6  2.52  9900 22483  62.4  0.78
##  8  24.8  66.1  8.26 141391  3.95  1.2   87.4  2.55  8767 21220  67.9  2.04
##  9  11.8  65.8  8.04 288310  3.69  0.13  93.0  3.79 12818 35172  66.0  1.66
## 10  12.6  66.9  6.98 335360  5.02  0.5   80.7  2.62 10797 58147  68.3  0.93
## # ℹ 12 more rows
## # ℹ 1 more variable: X12 <dbl>

Reduksi Variabel X

Eksplorasi Korelasi Data

Hipotesis: H0 : Tidak ada korelasi antarpeubah H1 : Ada korelasi antarpeubah

ggpairs(dt,
        upper = list(continuous = wrap('cor', size = 2)),
        title = "Matriks Scatterplot Data")

Dengan minimal ditandai oleh bintang satu (*) yang berarti variabel X tersebut tolak H0 atau ada korelasi dengan Y, maka hanya X1, X2, X4, X6, X7, X8, X9, dan X12 yang mempunyai korelasi. Sedangkan variabel X3, X5, X10, dan X11 memiliki korelasi yang kecil dan telah terwakili oleh variabel X yang lainnya sehingga dapat dihilangkan untuk kemudian diperiksa model dan multikolinearitasnya.

Model Reduksi Hasil Korelasi

model2 = lm(formula = Y ~ X1+X2+X4+X6+X7+X8+X9+X12, data = dt)
summary(model2)

## 
## Call:
## lm(formula = Y ~ X1 + X2 + X4 + X6 + X7 + X8 + X9 + X12, data = dt)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -10.4006  -1.4476   0.5167   2.3660   6.7547 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)  
## (Intercept)  5.806e+01  5.405e+01   1.074   0.3023  
## X1          -7.717e-01  6.726e-01  -1.147   0.2719  
## X2           9.612e-01  3.126e+00   0.308   0.7633  
## X4           4.282e-01  6.137e-01   0.698   0.4976  
## X6          -1.058e-01  1.554e-01  -0.681   0.5078  
## X7          -2.572e+00  1.415e+00  -1.817   0.0923 .
## X8           9.913e-04  5.710e-04   1.736   0.1062  
## X9          -5.652e-05  7.422e-05  -0.762   0.4599  
## X12          1.318e-01  2.054e-01   0.642   0.5321  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.906 on 13 degrees of freedom
## Multiple R-squared:  0.6744, Adjusted R-squared:  0.4741 
## F-statistic: 3.366 on 8 and 13 DF,  p-value: 0.02559

Diperoleh model sebagai berikut
\[ \hat Y = 0.5806 - 0.7717X_1 - 0.9612X_2 + 0.4282X_4 - 0.1058X_6 - 2.572X_7 + 0.0009X_8 - 0.00005X_9 + 0.1318X_{12} + e \]

Pengecekan Multikolinearitas

vif(model2)

##        X1        X2        X4        X6        X7        X8        X9       X12 
##  6.429485  9.363917  4.320674  3.777043  2.107861 11.289087 10.675684 11.701402

model3 = lm(formula = Y ~ X1+X2+X4+X6+X7+X8+X9, data = dt)
summary(model3)

## 
## Call:
## lm(formula = Y ~ X1 + X2 + X4 + X6 + X7 + X8 + X9, data = dt)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -10.6906  -2.1485  -0.1154   2.9997   6.5588 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)  
## (Intercept)  7.950e+01  4.158e+01   1.912   0.0766 .
## X1          -9.080e-01  6.246e-01  -1.454   0.1681  
## X2           1.351e+00  3.001e+00   0.450   0.6594  
## X4           4.124e-01  6.001e-01   0.687   0.5033  
## X6          -1.562e-01  1.312e-01  -1.190   0.2538  
## X7          -2.598e+00  1.385e+00  -1.876   0.0817 .
## X8           9.174e-04  5.474e-04   1.676   0.1159  
## X9          -8.516e-05  5.805e-05  -1.467   0.1644  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.801 on 14 degrees of freedom
## Multiple R-squared:  0.6641, Adjusted R-squared:  0.4962 
## F-statistic: 3.954 on 7 and 14 DF,  p-value: 0.01375

vif(model3)

##        X1        X2        X4        X6        X7        X8        X9 
##  5.788236  9.009798  4.313712  2.813403  2.106158 10.829677  6.816299

Model Hasil Reduksi Multikolinearitas

model4 = lm(formula = Y ~ X1+X2+X4+X6+X7+X9, data = dt)
summary(model4)

## 
## Call:
## lm(formula = Y ~ X1 + X2 + X4 + X6 + X7 + X9, data = dt)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -10.2818  -2.6755  -0.1858   2.6222   8.0058 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)
## (Intercept)  4.727e+01  3.903e+01   1.211    0.245
## X1          -6.125e-01  6.343e-01  -0.966    0.350
## X2           3.486e+00  2.876e+00   1.212    0.244
## X4           8.014e-01  5.859e-01   1.368    0.192
## X6          -1.330e-01  1.381e-01  -0.963    0.351
## X7          -2.103e+00  1.432e+00  -1.468    0.163
## X9          -2.833e-05  4.987e-05  -0.568    0.578
## 
## Residual standard error: 5.083 on 15 degrees of freedom
## Multiple R-squared:  0.5967, Adjusted R-squared:  0.4354 
## F-statistic: 3.699 on 6 and 15 DF,  p-value: 0.01854

vif(model4)

##       X1       X2       X4       X6       X7       X9 
## 5.327040 7.386697 3.668411 2.782125 2.010498 4.489353

Diperoleh model sebagai berikut
\[ \hat Y = 47,27 - 0.6125X_1 + 3.486X_2 + 0.8014X_4 - 0.1330X_6 - 2.103X_7 - 0.00002833 \] ——————————————————————–

Pencilan, Titik Leverage, dan Amatan Berpengaruh

Perhitungan ri, ci, Hi, DFFITSi

index <- c(1:22)
ri <- studres(model4)
ci <- cooks.distance(model4)
DFFITSi <- dffits(model4)
hi <- ols_hadi(model4)
hii <- hatvalues(model4)
hasil <- data.frame(index,ri,ci,DFFITSi,hi,hii); round(hasil,4)

##    index      ri     ci DFFITSi    hadi potential residual    hii
## 1      1  0.4282 0.0165  0.3305  0.6870    0.5958   0.0912 0.3734
## 2      2  1.0002 0.0263  0.4287  0.6784    0.1838   0.4947 0.1552
## 3      3  0.5437 0.0068  0.2128  0.3005    0.1532   0.1474 0.1328
## 4      4  0.5689 0.0152  0.3191  0.4755    0.3145   0.1610 0.2393
## 5      5 -0.2444 0.0032 -0.1452  0.3826    0.3528   0.0298 0.2608
## 6      6 -0.1956 0.0038 -0.1587  0.6775    0.6584   0.0191 0.3970
## 7      7 -0.6673 0.0349 -0.4853  0.7492    0.5289   0.2202 0.3460
## 8      8  0.8353 0.0404  0.5264  0.7411    0.3971   0.3440 0.2842
## 9      9 -1.0517 0.0323 -0.4770  0.7514    0.2057   0.5457 0.1706
## 10    10 -1.1523 0.1654 -1.0878  1.5267    0.8912   0.6355 0.4712
## 11    11  1.4434 0.0926  0.8339  1.3381    0.3337   1.0044 0.2502
## 12    12 -0.8916 0.0679 -0.6849  0.9794    0.5901   0.3892 0.3711
## 13    13 -0.4006 0.0061 -0.2008  0.3313    0.2512   0.0801 0.2008
## 14    14  1.7955 0.0528  0.6515  1.7015    0.1316   1.5698 0.1163
## 15    15  0.4072 0.0102  0.2593  0.4882    0.4056   0.0826 0.2886
## 16    16  0.2896 0.0055  0.1896  0.4704    0.4286   0.0419 0.3000
## 17    17 -0.5523 0.0246 -0.4056  0.6908    0.5395   0.1513 0.3504
## 18    18 -1.1497 0.0486 -0.5894  0.9110    0.2629   0.6482 0.2081
## 19    19 -0.1593 0.0047 -0.1746  1.2136    1.2009   0.0127 0.5456
## 20    20  1.2520 0.1391  1.0053  1.3956    0.6448   0.7508 0.3920
## 21    21 -2.6233 0.1469 -1.1963  3.3803    0.2080   3.1723 0.1722
## 22    22  0.2283 0.2990  1.4002 37.6507   37.6248   0.0260 0.9741

Plot ri, ci, Hi, DFFITSi, dan Potential Residual

Internally Studentized Residuals Plot (ri)

ggplot(hasil) +
  geom_point(aes(y = `ri`, x = `index`),
             color="blue",size=2) +
  ylab("ri") +
  xlab("index") +
  ggtitle("Plot ri") +
  theme_classic() +
  theme(plot.title = element_text(hjust = 0.5))

Cook Distance Plot (ci)

ggplot(hasil) +
  geom_point(aes(y = `ci`, x = `index`),
             color="blue",size=2) +
  ylab("ci") +
  xlab("index") +
  ggtitle("Plot ci") +
  theme_classic() +
  theme(plot.title = element_text(hjust = 0.5))

Difference in Fits Plot (DFFITSi)

ggplot(hasil) +
  geom_point(aes(y = `DFFITSi`, x = `index`),
             color="blue",size=2) +
  ylab("DFFITSi") +
  xlab("index") +
  ggtitle("Plot DFFITSi") +
  theme_classic() +
  theme(plot.title = element_text(hjust = 0.5))

Hadi’s Influence Measure (Hi)

ggplot(hasil) +
  geom_point(aes(y = `hadi`, x = `index`),
             color="blue",size=2) +
  ylab("Hi") +
  xlab("index") +
  ggtitle("Plot Hi") +
  theme_classic() +
  theme(plot.title = element_text(hjust = 0.5))

Potential Residual Plot (P-R Plot)

ggplot(hasil) +
  geom_point(aes(y = `potential`, x = `residual`),
             color="blue",size=2) +
  ylab("potential") +
  xlab("residual") +
  ggtitle("P-R Plot") +
  theme_classic() +
  theme(plot.title = element_text(hjust = 0.5))

Pendeteksian Nilai Tidak Biasa

Pencilan atau Outlier

for (i in 1:dim(hasil)[1]){
  absri <- abs(hasil$ri)
  pencilan <- which(absri > 2)
}
pencilan

## [1] 21

Pencilan adalah titik amatan yang menjauh pada tren Y walau X nya masih berada pada selang. Dalam data tersebut terdapat satu titik observasi yang terdeteksi sebagai sebuah pencilan, yaitu amatan ke-21 dengan ri bernilai -2.6233 yang absolutnya lebih besar dari 2..

Titik Leverage

titik_leverage <- vector("list", dim(hasil)[1])
for (i in 1:dim(hasil)[1]) {
  cutoff <- 2 * 7 / 22
  titik_leverage[[i]] <- which(hii > cutoff)
}
leverages <- unlist(titik_leverage)
titik_leverage <- sort(unique(leverages))
titik_leverage

## [1] 22

Titik leverage adalah nilai amatan yang X nya jauh lebih besar dari X lainnya dan terletak hampir dalam garis regresi tetapi juga menjauh. Terdapat satu titik observasi yang termasuk sebagai titik leverage pada data tersebut, yaitu amatan ke-22 dengan nilai hii 0.9741. Kedua titik ini memiliki nilai hii yang lebih besar dari 2*p/n = 0,636, dengan p = 7 adalah banyaknya parameter dan n = 22 adalah banyaknya amatan.

Amatan Berpengaruh

amatan_berpengaruh <- vector("list", dim(hasil)[1])
for (i in 1:dim(hasil)[1]) {
  cutoff <- 2 * sqrt((7 / 22))
  amatan_berpengaruh[[i]] <- which(abs(DFFITSi) > cutoff)
}
berpengaruh <- unlist(amatan_berpengaruh)
amatan_berpengaruh <- sort(unique(berpengaruh))
amatan_berpengaruh

## [1] 21 22

Amatan berpengaruh adalah nilai amatan yang memengaruhi hasil dugaan parameter regresi, R-square, dan uji hipotesis jika disisihkan. Hasil pada data tersebut menunjukkan bahwa amatan ke-10, 21, dan 22 termasuk sebagai amatan berpengaruh yang memiliki nilai DFFITSi -1.1963 dan 1,400. Absolut dari nilai tersebut lebih besar dari 2*sqrt(p/n) = 1,128.

#gambaran amatan berpengaruh dengan pencilan dan laverage
ols_plot_resid_lev(model4)

data1 <- dt[-c(21, 22),]
data2 <- dt[-c(21),]
data3 <- dt[-c(22),]

model_berpengaruh1 <- lm(Y~X1+X2+X4+X6+X7+X9, data=data1)
model_berpengaruh2 <- lm(Y~X1+X2+X4+X6+X7+X9, data=data2)
model_berpengaruh3 <- lm(Y~X1+X2+X4+X6+X7+X9, data=data3)

summary(model4)

## 
## Call:
## lm(formula = Y ~ X1 + X2 + X4 + X6 + X7 + X9, data = dt)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -10.2818  -2.6755  -0.1858   2.6222   8.0058 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)
## (Intercept)  4.727e+01  3.903e+01   1.211    0.245
## X1          -6.125e-01  6.343e-01  -0.966    0.350
## X2           3.486e+00  2.876e+00   1.212    0.244
## X4           8.014e-01  5.859e-01   1.368    0.192
## X6          -1.330e-01  1.381e-01  -0.963    0.351
## X7          -2.103e+00  1.432e+00  -1.468    0.163
## X9          -2.833e-05  4.987e-05  -0.568    0.578
## 
## Residual standard error: 5.083 on 15 degrees of freedom
## Multiple R-squared:  0.5967, Adjusted R-squared:  0.4354 
## F-statistic: 3.699 on 6 and 15 DF,  p-value: 0.01854

summary(model_berpengaruh1)

## 
## Call:
## lm(formula = Y ~ X1 + X2 + X4 + X6 + X7 + X9, data = data1)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -5.658 -2.782 -0.081  2.215  6.994 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)  
## (Intercept)  6.321e+01  4.101e+01   1.541   0.1472  
## X1          -9.305e-01  5.932e-01  -1.569   0.1408  
## X2           3.616e+00  4.856e+00   0.745   0.4697  
## X4           1.027e+00  7.174e-01   1.431   0.1759  
## X6          -7.489e-02  1.478e-01  -0.507   0.6208  
## X7          -2.482e+00  1.387e+00  -1.789   0.0969 .
## X9          -3.013e-05  8.413e-05  -0.358   0.7259  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.463 on 13 degrees of freedom
## Multiple R-squared:  0.6597, Adjusted R-squared:  0.5026 
## F-statistic:   4.2 on 6 and 13 DF,  p-value: 0.01441

summary(model_berpengaruh2)

## 
## Call:
## lm(formula = Y ~ X1 + X2 + X4 + X6 + X7 + X9, data = data2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -5.6614 -2.5983  0.1445  2.3220  7.2611 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)  
## (Intercept)  5.877e+01  3.337e+01   1.761   0.1000  
## X1          -9.608e-01  5.537e-01  -1.735   0.1047  
## X2           4.448e+00  2.465e+00   1.804   0.0927 .
## X4           1.124e+00  5.116e-01   2.198   0.0453 *
## X6          -9.156e-02  1.181e-01  -0.775   0.4512  
## X7          -2.366e+00  1.218e+00  -1.942   0.0725 .
## X9          -1.571e-05  4.254e-05  -0.369   0.7174  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.308 on 14 degrees of freedom
## Multiple R-squared:  0.7177, Adjusted R-squared:  0.5968 
## F-statistic: 5.933 on 6 and 14 DF,  p-value: 0.002919

summary(model_berpengaruh3)

## 
## Call:
## lm(formula = Y ~ X1 + X2 + X4 + X6 + X7 + X9, data = data3)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -10.2528  -2.6639  -0.4331   3.0228   7.6474 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)
## (Intercept)  5.323e+01  4.802e+01   1.108    0.286
## X1          -5.730e-01  6.778e-01  -0.845    0.412
## X2           2.380e+00  5.684e+00   0.419    0.682
## X4           6.725e-01  8.277e-01   0.812    0.430
## X6          -1.106e-01  1.731e-01  -0.639    0.533
## X7          -2.258e+00  1.629e+00  -1.387    0.187
## X9          -4.753e-05  9.865e-05  -0.482    0.637
## 
## Residual standard error: 5.251 on 14 degrees of freedom
## Multiple R-squared:  0.523,  Adjusted R-squared:  0.3186 
## F-statistic: 2.559 on 6 and 14 DF,  p-value: 0.06921

Tetep model terbaiknya saat data ke 21 dikeluarkan

#pembentukan data baru tanpa amatan 9 dan 19
dt2 <- subset(dt, select = -c(X3, X5, X8, X10, X11, X12))
dt2 <- dt2[-c(21),]
#pemodelan rlb terbaru
model5 <- lm(Y~.,data= dt2)
summary(model5)

## 
## Call:
## lm(formula = Y ~ ., data = dt2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -5.6614 -2.5983  0.1445  2.3220  7.2611 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)  
## (Intercept)  5.877e+01  3.337e+01   1.761   0.1000  
## X1          -9.608e-01  5.537e-01  -1.735   0.1047  
## X2           4.448e+00  2.465e+00   1.804   0.0927 .
## X4           1.124e+00  5.116e-01   2.198   0.0453 *
## X6          -9.156e-02  1.181e-01  -0.775   0.4512  
## X7          -2.366e+00  1.218e+00  -1.942   0.0725 .
## X9          -1.571e-05  4.254e-05  -0.369   0.7174  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.308 on 14 degrees of freedom
## Multiple R-squared:  0.7177, Adjusted R-squared:  0.5968 
## F-statistic: 5.933 on 6 and 14 DF,  p-value: 0.002919

anova(model5)

## Analysis of Variance Table
## 
## Response: Y
##           Df Sum Sq Mean Sq F value  Pr(>F)    
## X1         1 325.62  325.62 17.5467 0.00091 ***
## X2         1  26.04   26.04  1.4033 0.25590    
## X4         1 131.44  131.44  7.0830 0.01861 *  
## X6         1  97.50   97.50  5.2538 0.03791 *  
## X7         1  77.48   77.48  4.1754 0.06031 .  
## X9         1   2.53    2.53  0.1364 0.71745    
## Residuals 14 259.80   18.56                    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Eksplorasi Kondisi

Plot Sisaan Vs Y duga

plot(model5,1)

1. Sisaan menyebar di sekitar 0, sehingga nilai harapan galat sama dengan nol
2. Lebar pita sama untuk setiap nilai dugaan, sehingga ragam homogen
3. Plot sisaan vs y duga tidak berpola, model pas

plot(model5,2)

Plot Sisaan Vs Urutan

plot(x = 1:dim(dt2)[1],
     y = model5$residuals,
     type = 'b', 
     ylab = "Residuals",
     xlab = "Observation")

Kondisi Gaus Markov

1. Nilai Harapan Sisaan sama dengan Nol

H0: Nilai harapan sisaan sama dengan nol
H1: Nilai harapan sisaan tidak sama dengan nol

t.test(model5$residuals,mu = 0,conf.level = 0.95)

## 
##  One Sample t-test
## 
## data:  model5$residuals
## t = -2.6763e-17, df = 20, p-value = 1
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
##  -1.640606  1.640606
## sample estimates:
##     mean of x 
## -2.104901e-17

Uji t.tes tersebut menunjukkan hasil p-value = 1 > alpha = 0.05, maka tak tolak H0, nilai harapan sisaan sama dengan nol pada taraf nyata 5%. Asumsi terpenuhi.

2. Ragam Sisaan Homogen

\(H0:var[e]=sigma2I\) (ragam sisan homogen)
\(H1:var[e] != sigma2I\) (ragam siaan tidak homogen)

bptest(model5)

## 
##  studentized Breusch-Pagan test
## 
## data:  model5
## BP = 7.4095, df = 6, p-value = 0.2846

Uji ini sering disebut dengan uji homokesdasitas yang dilakukan dengan yang dilakukan dengan uji Breusch-Pagan. Karena p-value = 0.5789 > alpha = 0.05, maka tak tolak H0, ragam sisaan homogen pada taraf nyata 5%. Asumsi terpenuhi.

3. Sisaan Saling Bebas

\(H0:E[ei,ej]=0\) (sisaan saling bebas/tidak ada autokorelasi)
\(H1:E[ei,ej] != 0\) (sisaan tidak saling bebas/ada autokorelasi)

dwtest(model5)

## 
##  Durbin-Watson test
## 
## data:  model5
## DW = 1.9585, p-value = 0.3501
## alternative hypothesis: true autocorrelation is greater than 0

Uji ini sering disebut dengan uji autokorelasi yang dilakukan dengan Durbin watson. Karena p-value = 0.634 (pada DW test) > alpha = 0.05, maka tak tolak H0, sisaan saling bebas pada taraf nyata 5%, sehingga asumsi terpenuhi.

Uji Formal Normalitas Sisaan

\(H_0 : N\) (sisaan menyebar Normal) \(H_1 : N\) (sisaan tidak menyebar Normal)

shapiro.test(model5$residuals)

## 
##  Shapiro-Wilk normality test
## 
## data:  model5$residuals
## W = 0.96194, p-value = 0.5559

bs <- ols_step_best_subset(model5)
bs

##     Best Subsets Regression     
## --------------------------------
## Model Index    Predictors
## --------------------------------
##      1         X6                
##      2         X4 X7             
##      3         X1 X4 X7          
##      4         X1 X2 X4 X7       
##      5         X1 X2 X4 X6 X7    
##      6         X1 X2 X4 X6 X7 X9 
## --------------------------------
## 
##                                                     Subsets Regression Summary                                                    
## ----------------------------------------------------------------------------------------------------------------------------------
##                        Adj.        Pred                                                                                            
## Model    R-Square    R-Square    R-Square     C(p)        AIC        SBIC        SBC         MSEP        FPE       HSP       APC  
## ----------------------------------------------------------------------------------------------------------------------------------
##   1        0.4434      0.4141      0.3404    10.6049    132.6765    72.0095    135.8100    566.4822    29.5295    1.4979    0.6737 
##   2        0.6017      0.5575      0.4955     4.7542    127.6493    68.4243    131.8274    429.2228    23.2752    1.1980    0.5310 
##   3        0.6394      0.5757      0.3606     4.8873    127.5646    69.3007    132.7872    412.9505    23.2452    1.2204    0.5304 
##   4        0.7034      0.6293      0.5274     3.7096    125.4571    69.7724    131.7243    362.2283    21.1228    1.1374    0.4819 
##   5        0.7150      0.6200      0.4941     5.1364    126.6224    72.3819    133.9341    372.9779    22.4857    1.2492    0.5130 
##   6        0.7177      0.5968      0.4386     7.0000    128.4189    75.3234    136.7750    397.7938    24.7432    1.4275    0.5645 
## ----------------------------------------------------------------------------------------------------------------------------------
## AIC: Akaike Information Criteria 
##  SBIC: Sawa's Bayesian Information Criteria 
##  SBC: Schwarz Bayesian Criteria 
##  MSEP: Estimated error of prediction, assuming multivariate normality 
##  FPE: Final Prediction Error 
##  HSP: Hocking's Sp 
##  APC: Amemiya Prediction Criteria

null<-lm(Y~ 1, data=dt2) # 1 here means the intercept 
full<-lm(Y~., data=dt2)
 
step(full, scope=list(lower=null, upper=full),data=datanew, direction='backward', trace=0)

## 
## Call:
## lm(formula = Y ~ X1 + X2 + X4 + X7, data = dt2)
## 
## Coefficients:
## (Intercept)           X1           X2           X4           X7  
##      59.761       -1.042        4.181        1.289       -2.955

step(null, scope=list(lower=null, upper=full),data=datanew, direction='forward', trace=0)

## 
## Call:
## lm(formula = Y ~ X6 + X1, data = dt2)
## 
## Coefficients:
## (Intercept)           X6           X1  
##     84.3127      -0.2551      -0.6719

step(null, scope=list(lower=null, upper=full),data=datanew, direction='both', trace=0)

## 
## Call:
## lm(formula = Y ~ X6 + X1, data = dt2)
## 
## Coefficients:
## (Intercept)           X6           X1  
##     84.3127      -0.2551      -0.6719

modell1 <-  lm(Y~X1+X2+X4+X7, data = dt2)
modell2 <- lm(Y~X1+X6, data = dt2)
summary(modell1)$adj.r.squared

## [1] 0.6292841

summary(modell2)$adj.r.squared

## [1] 0.5351652

ols_aic(modell1)

## [1] 125.4571

ols_aic(modell2)

## [1] 128.6817

model <- modell1
summary(model)

## 
## Call:
## lm(formula = Y ~ X1 + X2 + X4 + X7, data = dt2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -6.5916 -3.4506  0.4097  1.8438  7.0582 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept)  59.7614    18.7337   3.190  0.00570 **
## X1           -1.0423     0.4450  -2.342  0.03243 * 
## X2            4.1812     2.2490   1.859  0.08150 . 
## X4            1.2889     0.3934   3.276  0.00475 **
## X7           -2.9545     0.9519  -3.104  0.00683 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.13 on 16 degrees of freedom
## Multiple R-squared:  0.7034, Adjusted R-squared:  0.6293 
## F-statistic: 9.487 on 4 and 16 DF,  p-value: 0.0003966

summary(model)

## 
## Call:
## lm(formula = Y ~ X1 + X2 + X4 + X7, data = dt2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -6.5916 -3.4506  0.4097  1.8438  7.0582 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept)  59.7614    18.7337   3.190  0.00570 **
## X1           -1.0423     0.4450  -2.342  0.03243 * 
## X2            4.1812     2.2490   1.859  0.08150 . 
## X4            1.2889     0.3934   3.276  0.00475 **
## X7           -2.9545     0.9519  -3.104  0.00683 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.13 on 16 degrees of freedom
## Multiple R-squared:  0.7034, Adjusted R-squared:  0.6293 
## F-statistic: 9.487 on 4 and 16 DF,  p-value: 0.0003966