Question

13.27 Perform a residual analysis for these data. Tear 0.00 0.00 0.45 0.85 0.35 0.30 0.70 1.90 0.25 0.10 0.15 3.90 0.00 0.55 0.00 0.05 0.40 4.30 0.00 Viscosity 350.00 350.00 319.00 380.00 350.00 300.00 400.00 350.00 350.00 319.00 380.00 350.00 380.00 350.00 350.00 319.00 319.00 380.00 350.00 Pressure 180.00 170.00 186.00 174.00 180.00 180.00 180.00 190.00 180.00 186.00 186.00 180.00 174.00 180.00 180.00 174.00 174.00 186.00 180.00 Plate Gap 0.00 0.00 1.80 1.80 0.00 0.00 0.00 0.00 0.00 -1.80 -1.80 3.00 -1.80 0.00 -3.00 -1.80 1.80 1.80 0.00 Based on these results, evaluate whether the assumptions of regression have been seriously violated.

Answer 1

We use R to perform the residual analysis. Here tear is taken to be the response variable and the rest are taken to be the predictors.

^- the above image is for Model 1

the above image is for Model 2the above image is for Model 3

To check for linearity we observe the residual versus fitted values plot:

Ideally, the residual plot will show no fitted pattern. That is, the red line should be approximately horizontal at zero. The presence of a pattern may indicate a problem with some aspect of the linear model.But here in all the three models this is not so. So the relationship between the response and the predictors cannot be assumed to be linear.

To check for homoscedasticity we observe the spread versus location plot:

This plot shows if residuals are spread equally along the ranges of predictors. It’s good if there is a horizontal line with equally spread points. In our example, this is not the case.

In model 1 and model 3 it can be seen that the variability (variances) of the residual points increases with the value of the fitted outcome variable, suggesting non-constant variances in the residuals errors (or heteroscedasticity). But in model 2 the variability decreases after increasing. All of these indicate the models being heteroscedastic

To check for normality of the errors we observe the QQ plot:

The QQ plot of residuals can be used to visually check the normality assumption. The normal probability plot of residuals should approximately follow a straight line. But in all the three cases all the points do not fall on the line . So the normality assumption is violated.

To check for outliers we observe the Residuals versus leverages plot:

The red dotted lines are Cook's distance lines. In all the three models few observations lie on and beyond those red dotted lines. This indicates that those observations are outliers and they are influential. So the regression is affected by their presence.