Objective

To perform multivariate regression analysis using the attitude dataset and test assumptions.

Load Data

data(attitude)
head(attitude)
##   rating complaints privileges learning raises critical advance
## 1     43         51         30       39     61       92      45
## 2     63         64         51       54     63       73      47
## 3     71         70         68       69     76       86      48
## 4     61         63         45       47     54       84      35
## 5     81         78         56       66     71       83      47
## 6     43         55         49       44     54       49      34

Full Model

model_full <- lm(rating ~ ., data = attitude)
summary(model_full)
## 
## Call:
## lm(formula = rating ~ ., data = attitude)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -10.9418  -4.3555   0.3158   5.5425  11.5990 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 10.78708   11.58926   0.931 0.361634    
## complaints   0.61319    0.16098   3.809 0.000903 ***
## privileges  -0.07305    0.13572  -0.538 0.595594    
## learning     0.32033    0.16852   1.901 0.069925 .  
## raises       0.08173    0.22148   0.369 0.715480    
## critical     0.03838    0.14700   0.261 0.796334    
## advance     -0.21706    0.17821  -1.218 0.235577    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 7.068 on 23 degrees of freedom
## Multiple R-squared:  0.7326, Adjusted R-squared:  0.6628 
## F-statistic:  10.5 on 6 and 23 DF,  p-value: 1.24e-05

Stepwise Model

model_best <- step(model_full)
## Start:  AIC=123.36
## rating ~ complaints + privileges + learning + raises + critical + 
##     advance
## 
##              Df Sum of Sq    RSS    AIC
## - critical    1      3.41 1152.4 121.45
## - raises      1      6.80 1155.8 121.54
## - privileges  1     14.47 1163.5 121.74
## - advance     1     74.11 1223.1 123.24
## <none>                    1149.0 123.36
## - learning    1    180.50 1329.5 125.74
## - complaints  1    724.80 1873.8 136.04
## 
## Step:  AIC=121.45
## rating ~ complaints + privileges + learning + raises + advance
## 
##              Df Sum of Sq    RSS    AIC
## - raises      1     10.61 1163.0 119.73
## - privileges  1     14.16 1166.6 119.82
## - advance     1     71.27 1223.7 121.25
## <none>                    1152.4 121.45
## - learning    1    177.74 1330.1 123.75
## - complaints  1    724.70 1877.1 134.09
## 
## Step:  AIC=119.73
## rating ~ complaints + privileges + learning + advance
## 
##              Df Sum of Sq    RSS    AIC
## - privileges  1     16.10 1179.1 118.14
## - advance     1     61.60 1224.6 119.28
## <none>                    1163.0 119.73
## - learning    1    197.03 1360.0 122.42
## - complaints  1   1165.94 2328.9 138.56
## 
## Step:  AIC=118.14
## rating ~ complaints + learning + advance
## 
##              Df Sum of Sq    RSS    AIC
## - advance     1     75.54 1254.7 118.00
## <none>                    1179.1 118.14
## - learning    1    186.12 1365.2 120.54
## - complaints  1   1259.91 2439.0 137.94
## 
## Step:  AIC=118
## rating ~ complaints + learning
## 
##              Df Sum of Sq    RSS    AIC
## <none>                    1254.7 118.00
## - learning    1    114.73 1369.4 118.63
## - complaints  1   1370.91 2625.6 138.16
summary(model_best)
## 
## Call:
## lm(formula = rating ~ complaints + learning, data = attitude)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -11.5568  -5.7331   0.6701   6.5341  10.3610 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   9.8709     7.0612   1.398    0.174    
## complaints    0.6435     0.1185   5.432 9.57e-06 ***
## learning      0.2112     0.1344   1.571    0.128    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 6.817 on 27 degrees of freedom
## Multiple R-squared:  0.708,  Adjusted R-squared:  0.6864 
## F-statistic: 32.74 on 2 and 27 DF,  p-value: 6.058e-08

Final Model

model_best <- lm(rating ~ complaints + learning, data = attitude)
summary(model_best)
## 
## Call:
## lm(formula = rating ~ complaints + learning, data = attitude)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -11.5568  -5.7331   0.6701   6.5341  10.3610 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   9.8709     7.0612   1.398    0.174    
## complaints    0.6435     0.1185   5.432 9.57e-06 ***
## learning      0.2112     0.1344   1.571    0.128    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 6.817 on 27 degrees of freedom
## Multiple R-squared:  0.708,  Adjusted R-squared:  0.6864 
## F-statistic: 32.74 on 2 and 27 DF,  p-value: 6.058e-08

Diagnostic Plots

par(mfrow = c(2,2))
plot(model_best)

VIF Test

library(car)
## Warning: package 'car' was built under R version 4.5.3
## Loading required package: carData
## Warning: package 'carData' was built under R version 4.5.3
vif(model_best)
## complaints   learning 
##   1.553021   1.553021

Durbin-Watson Test

library(lmtest)
## Warning: package 'lmtest' was built under R version 4.5.3
## Loading required package: zoo
## Warning: package 'zoo' was built under R version 4.5.3
## 
## Attaching package: 'zoo'
## The following objects are masked from 'package:base':
## 
##     as.Date, as.Date.numeric
dwtest(model_best)
## 
##  Durbin-Watson test
## 
## data:  model_best
## DW = 1.9582, p-value = 0.439
## alternative hypothesis: true autocorrelation is greater than 0

Interpretation