data = read.table(file.choose(),header = TRUE)
data
## Y X1 X2 X3
## 1 57.5 78 2.75 29.5
## 2 52.8 69 2.15 26.3
## 3 61.3 77 4.41 32.2
## 4 67.0 88 5.52 36.5
## 5 53.5 67 3.21 27.2
## 6 62.7 80 4.32 27.7
## 7 56.2 74 2.31 28.3
## 8 68.5 94 4.30 30.3
## 9 69.2 102 3.71 28.7
library(ggplot2)
ggplot(data, aes(x = X1, y = Y)) +
geom_point() +
geom_smooth(method = "lm", se = FALSE) +
labs(title = "Scatterplot X1 vs Y",
x = "X1", y = "Y") +
theme_minimal()
## `geom_smooth()` using formula = 'y ~ x'
ggplot(data, aes(x = X2, y = Y)) +
geom_point() +
geom_smooth(method = "lm", se = FALSE) +
labs(title = "Scatterplot X2 vs Y",
x = "X2", y = "Y") +
theme_minimal()
## `geom_smooth()` using formula = 'y ~ x'
ggplot(data, aes(x = X3, y = Y)) +
geom_point() +
geom_smooth(method = "lm", se = FALSE) +
labs(title = "Scatterplot X3 vs Y",
x = "X3", y = "Y") +
theme_minimal()
## `geom_smooth()` using formula = 'y ~ x'
reg=lm(Y~X1,data=data)
summary(reg)
##
## Call:
## lm(formula = Y ~ X1, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.6440 -1.9128 -0.2151 2.2513 2.4075
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 19.01108 5.42272 3.506 0.009915 **
## X1 0.51797 0.06635 7.807 0.000106 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.174 on 7 degrees of freedom
## Multiple R-squared: 0.897, Adjusted R-squared: 0.8823
## F-statistic: 60.95 on 1 and 7 DF, p-value: 0.0001065
library(lmtest)
## Loading required package: zoo
##
## Attaching package: 'zoo'
## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numeric
library(car)
## Loading required package: carData
library(zoo)
reg=lm(formula=Y~X1, data=data)
summary(reg)
##
## Call:
## lm(formula = Y ~ X1, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.6440 -1.9128 -0.2151 2.2513 2.4075
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 19.01108 5.42272 3.506 0.009915 **
## X1 0.51797 0.06635 7.807 0.000106 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.174 on 7 degrees of freedom
## Multiple R-squared: 0.897, Adjusted R-squared: 0.8823
## F-statistic: 60.95 on 1 and 7 DF, p-value: 0.0001065
bptest(reg, varformula = ~fitted.values(reg), studentize = FALSE)
##
## Breusch-Pagan test
##
## data: reg
## BP = 0.35178, df = 1, p-value = 0.5531
dwtest(reg, alternative = 'two.sided')
##
## Durbin-Watson test
##
## data: reg
## DW = 1.785, p-value = 0.5744
## alternative hypothesis: true autocorrelation is not 0
shapiro.test(resid(reg))
##
## Shapiro-Wilk normality test
##
## data: resid(reg)
## W = 0.87986, p-value = 0.1565
resid(reg)
## 1 2 3 4 5 6 7
## -1.9127561 -1.9510242 2.4052142 2.4075419 -0.2150838 2.2513035 -1.1408752
## 8 9
## 0.7997207 -2.6440410
abs(resid(reg))
## 1 2 3 4 5 6 7 8
## 1.9127561 1.9510242 2.4052142 2.4075419 0.2150838 2.2513035 1.1408752 0.7997207
## 9
## 2.6440410
reg22=lm(abs(resid(reg))~X1,data=data)
summary(reg22)
##
## Call:
## lm(formula = abs(resid(reg)) ~ X1, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.3200 -0.4062 0.2952 0.5324 0.7722
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.57193 2.04979 -0.279 0.788
## X1 0.02864 0.02508 1.142 0.291
##
## Residual standard error: 0.8219 on 7 degrees of freedom
## Multiple R-squared: 0.157, Adjusted R-squared: 0.03657
## F-statistic: 1.304 on 1 and 7 DF, p-value: 0.2911
bgtest(Y~X1, data=data, order=1)
##
## Breusch-Godfrey test for serial correlation of order up to 1
##
## data: Y ~ X1
## LM test = 0.033549, df = 1, p-value = 0.8547
reg2=lm(Y~X2,data=data)
summary(reg2)
##
## Call:
## lm(formula = Y ~ X2, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.6496 -2.1172 -1.2392 0.9339 7.8929
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 45.298 5.254 8.621 5.64e-05 ***
## X2 4.315 1.390 3.105 0.0172 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.394 on 7 degrees of freedom
## Multiple R-squared: 0.5793, Adjusted R-squared: 0.5192
## F-statistic: 9.64 on 1 and 7 DF, p-value: 0.0172
bptest(reg2, varformula = ~fitted.values(reg2), studentize = FALSE)
##
## Breusch-Pagan test
##
## data: reg2
## BP = 0.061595, df = 1, p-value = 0.804
dwtest(reg2, alternative = 'two.sided')
##
## Durbin-Watson test
##
## data: reg2
## DW = 0.50193, p-value = 0.005426
## alternative hypothesis: true autocorrelation is not 0
shapiro.test(resid(reg2))
##
## Shapiro-Wilk normality test
##
## data: resid(reg2)
## W = 0.93927, p-value = 0.5742
resid(reg2)
## 1 2 3 4 5 6 7
## 0.3353100 -1.7757020 -3.0275565 -2.1171842 -5.6495807 -1.2392084 0.9339012
## 8 9
## 4.6470912 7.8929294
abs(resid(reg2))
## 1 2 3 4 5 6 7 8
## 0.3353100 1.7757020 3.0275565 2.1171842 5.6495807 1.2392084 0.9339012 4.6470912
## 9
## 7.8929294
reg23=lm(abs(resid(reg2))~X2,data=data)
summary(reg23)
##
## Call:
## lm(formula = abs(resid(reg2)) ~ X2, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.3452 -1.7837 -0.6405 1.2837 4.7895
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.4690 3.1460 0.467 0.655
## X2 0.4406 0.8321 0.529 0.613
##
## Residual standard error: 2.631 on 7 degrees of freedom
## Multiple R-squared: 0.03851, Adjusted R-squared: -0.09885
## F-statistic: 0.2804 on 1 and 7 DF, p-value: 0.6128
bgtest(Y~X2, data=data, order=1)
##
## Breusch-Godfrey test for serial correlation of order up to 1
##
## data: Y ~ X2
## LM test = 5.4817, df = 1, p-value = 0.01922
reg3=lm(Y~X3,data=data)
summary(reg3)
##
## Call:
## lm(formula = Y ~ X3, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.693 -3.315 -2.592 3.937 9.297
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 27.1873 18.9650 1.434 0.195
## X3 1.1399 0.6369 1.790 0.117
##
## Residual standard error: 5.611 on 7 degrees of freedom
## Multiple R-squared: 0.314, Adjusted R-squared: 0.216
## F-statistic: 3.204 on 1 and 7 DF, p-value: 0.1166
bptest(reg3, varformula = ~fitted.values(reg3), studentize = FALSE)
##
## Breusch-Pagan test
##
## data: reg3
## BP = 0.32297, df = 1, p-value = 0.5698
dwtest(reg3, alternative = 'two.sided')
##
## Durbin-Watson test
##
## data: reg3
## DW = 1.117, p-value = 0.1651
## alternative hypothesis: true autocorrelation is not 0
shapiro.test(resid(reg3))
##
## Shapiro-Wilk normality test
##
## data: resid(reg3)
## W = 0.81761, p-value = 0.03238
resid(reg3)
## 1 2 3 4 5 6 7 8
## -3.314678 -4.366959 -2.592442 -1.794065 -4.692880 3.937164 -3.246783 6.773392
## 9
## 9.297252
abs(resid(reg3))
## 1 2 3 4 5 6 7 8
## 3.314678 4.366959 2.592442 1.794065 4.692880 3.937164 3.246783 6.773392
## 9
## 9.297252
reg33=lm(abs(resid(reg3))~X3,data=data)
summary(reg33)
##
## Call:
## lm(formula = abs(resid(reg3)) ~ X3, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.6039 -1.0955 -1.0751 -0.4915 4.5679
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 13.4360 7.5924 1.77 0.120
## X3 -0.3034 0.2550 -1.19 0.273
##
## Residual standard error: 2.246 on 7 degrees of freedom
## Multiple R-squared: 0.1682, Adjusted R-squared: 0.0494
## F-statistic: 1.416 on 1 and 7 DF, p-value: 0.2729
bgtest(Y~X3, data=data, order=1)
##
## Breusch-Godfrey test for serial correlation of order up to 1
##
## data: Y ~ X3
## LM test = 0.80317, df = 1, p-value = 0.3701
RLB=lm(Y~X1+X2+X3,data=data)
summary(RLB)
##
## Call:
## lm(formula = Y ~ X1 + X2 + X3, data = data)
##
## Residuals:
## 1 2 3 4 5 6 7 8 9
## -0.2981 -0.2142 0.4714 -0.4746 -0.9524 0.4761 0.9274 0.7466 -0.6823
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 21.87353 4.07389 5.369 0.00302 **
## X1 0.41277 0.03067 13.460 4.05e-05 ***
## X2 2.20267 0.47199 4.667 0.00550 **
## X3 -0.07895 0.15551 -0.508 0.63330
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.8484 on 5 degrees of freedom
## Multiple R-squared: 0.9888, Adjusted R-squared: 0.9821
## F-statistic: 147.1 on 3 and 5 DF, p-value: 2.696e-05
vif(RLB)
## X1 X2 X3
## 1.403219 3.093549 2.607991
bptest(RLB, varformula = ~fitted.values(RLB), studentize = FALSE)
##
## Breusch-Pagan test
##
## data: RLB
## BP = 0.031878, df = 1, p-value = 0.8583
dwtest(RLB, alternative = 'two.sided')
##
## Durbin-Watson test
##
## data: RLB
## DW = 1.6446, p-value = 0.233
## alternative hypothesis: true autocorrelation is not 0
shapiro.test(resid(RLB))
##
## Shapiro-Wilk normality test
##
## data: resid(RLB)
## W = 0.93208, p-value = 0.5014
bptest(RLB)
##
## studentized Breusch-Pagan test
##
## data: RLB
## BP = 0.52402, df = 3, p-value = 0.9136
resid(RLB)
## 1 2 3 4 5 6 7
## -0.2980857 -0.2141744 0.4714083 -0.4745650 -0.9524106 0.4760829 0.9274363
## 8 9
## 0.7466056 -0.6822975
abs(resid(RLB))
## 1 2 3 4 5 6 7 8
## 0.2980857 0.2141744 0.4714083 0.4745650 0.9524106 0.4760829 0.9274363 0.7466056
## 9
## 0.6822975
RLB2=lm(formula=abs(resid(RLB))~X1+X2+X3, data=data)
summary(RLB2)
##
## Call:
## lm(formula = abs(resid(RLB)) ~ X1 + X2 + X3, data = data)
##
## Residuals:
## 1 2 3 4 5 6 7 8
## -0.24968 -0.37078 -0.06495 -0.01987 0.35744 -0.17650 0.37477 0.12608
## 9
## 0.02347
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.000238 1.548904 0.646 0.547
## X1 0.002391 0.011660 0.205 0.846
## X2 0.035130 0.179452 0.196 0.853
## X3 -0.024936 0.059127 -0.422 0.691
##
## Residual standard error: 0.3226 on 5 degrees of freedom
## Multiple R-squared: 0.04467, Adjusted R-squared: -0.5285
## F-statistic: 0.07793 on 3 and 5 DF, p-value: 0.9692
bgtest(RLB)
##
## Breusch-Godfrey test for serial correlation of order up to 1
##
## data: RLB
## LM test = 0.51249, df = 1, p-value = 0.4741
dwtest(RLB, alternative='two.sided')
##
## Durbin-Watson test
##
## data: RLB
## DW = 1.6446, p-value = 0.233
## alternative hypothesis: true autocorrelation is not 0
shapiro.test(resid(RLB))
##
## Shapiro-Wilk normality test
##
## data: resid(RLB)
## W = 0.93208, p-value = 0.5014