Section 19 Two-way ANOVA: Model Assumptions


Residual

\[ \Huge \hat{\epsilon_{ijk}} = y_{ijk} - \hat{y_{ij}} \]


19.1 Assumptions

  • The data are independent samples from normal distributions

  • The variability within each treatment group is identical (\(\sigma^2\))

  • Properties of residual as \(\large \epsilon_{ij} \sim NID(0, \sigma^2)\)

  • Various diagnostic tools are available in order to check these assumptions:

    - Histogram of residuals: should be approximately $\sim NID(0, \sigma^2)$

- QQ plot should show the distribution of residual as Normal

- Residuals versus fitted values

- Residuals versus explanatory variables

- Residuals versus any other variables not included in the model

  • All these plots should show no patterns or trend

  • Test for homogeneity of variance

    - `fligner.test(SBP ~ Group, data=BP)`

- `bartlett.test(SBP ~ Group, data=BP)`

- Also, `var.test` to perform an F test to compare the variances of two samples from normal populations.

19.2 Standard residual plots using lm object

plot(fm)

19.3 Customised residual plots

19.3.1 Calculate predicted values and residuals

pred <- predict(object = fm, data=BP)
BP$pred <- pred

res <- fm$residuals
BP$res <- res


19.3.2 Histogram of residuals

hist(x=res, freq = TRUE, 
     main = '', 
     xlab = 'Residuals', 
     ylab = 'Frequency', 
     axes = TRUE, 
     col = 'lightblue', 
     lty = 1, border = 'purple')

hist(x=res, breaks=10, freq = FALSE, 
     xlab = 'Residuals', 
     ylab = 'Density', 
     axes = TRUE, 
     col = 'lightblue', 
     lty = 1, border = 'purple')
lines(density(res), col = 'red', lwd = 2, lty = 1)

19.3.3 QQ plot of residuals

qqnorm(y=res, main='Normal QQ of residuals',
     xlab='Theoretical Quantiles',
     ylab='Residuals',
     col='blue', pch=20)
qqline(y=res, lty=1, lwd=2, col='red')


19.3.4 Scatter plot of residual against fitted value

plot(x=pred, y=res, pch=20, col='purple')
# lo <- loess(res ~ pred)
# lines(predict(lo), col='red', lty=1, lwd=2)
abline(a = 0, b = 0, lty=2, col='red')


19.3.5 Residuals for each treatment group

# plot(res ~ Group, data=BP)

plot(x = as.factor(BP$Group), y = res)
abline(a = 0, b = 0, lty=2, col='red', lwd=2)

19.3.6 Residuals for each DM level

# plot(res ~ DM, data=BP)
plot(x = as.factor(BP$DM), y = res)
abline(a = 0, b = 0, lty=2, col='red', lwd=2)