3.5 Model adequacy checking

The analysis of variance rests on the assumptions that the errors are independent and normally distributed with mean zero and a common variance. These are checked through the residuals \[ e_{ij} = y_{ij} - \hat y_{ij} = y_{ij} - \bar y_{i\cdot } , \] which estimate the errors. They are not quite the errors, as the next result shows.

So the residuals within a treatment are slightly negatively correlated (they sum to zero) and have slightly less variance than the errors. For plotting, they are usually scaled: the standardised residuals \(d_{ij} = e_{ij}/\sqrt {MSE}\), or the studentised residuals \(r_{ij} = e_{ij}/\sqrt {MSE(1 - 1/n_i)}\), which have variance close to \(1\). If the errors are normal, about \(95\%\) of them lie within \(\pm 2\), and a value beyond \(\pm 3\) marks a possible outlier.

Residual plots

(i)

A normal probability plot of the residuals checks normality. The ordered residuals \(e_{(1)} \le \dots \le e_{(N)}\) are plotted against the normal scores \(\Phi ^{-1}\bigl ((k - 0.5)/N\bigr )\); under normality the points lie near a straight line. Curvature suggests skewness, and an S shape suggests tails heavier or lighter than normal.

(ii)

Plotting the residuals against the fitted values \(\bar y_{i\cdot }\) checks that the variance is constant. The vertical spread should be the same at every fitted value. A spread that grows with the fitted value, a funnel, is the commonest violation, and is treated by transformation below.

(iii)

A plot of the residuals against time order checks independence. Runs of residuals of the same sign, or a trend, indicate correlation between successive runs, which randomisation of the run order is meant to prevent but cannot always.

Tests for equality of variances

Formal tests of \(H_0 : \sigma _1^2 = \dots = \sigma _t^2\) supplement the plot.

Bartlett’s test. With \(S_i^2\) the sample variance of treatment \(i\) and \(S_p^2 = MSE\) the pooled variance, compute \begin {gather*} q = (N - t)\log _{10}S_p^2 - \sum _{i=1}^t(n_i - 1)\log _{10}S_i^2 , \\ c = 1 + \frac {1}{3(t - 1)}\Bigl (\sum _{i=1}^t\frac {1}{n_i - 1} - \frac {1}{N - t}\Bigr ) , \end {gather*} and \(\chi _0^2 = 2.3026\,q/c\). Under \(H_0\) and normality, \(\chi _0^2\) is approximately \(\chi ^2_{t-1}\), and \(H_0\) is rejected if \(\chi _0^2 > \chi ^2_{\alpha ,\,t-1}\).

The statistic is a likelihood ratio statistic. Up to the use of \(n_i - 1\) in place of \(n_i\), \(-2\log \Lambda = \sum (n_i - 1)\log (S_p^2/S_i^2)\), which is \(2.3026\,q\); it is zero when all the \(S_i^2\) are equal, because a weighted arithmetic mean exceeds the corresponding geometric mean unless the terms are equal, and it grows as they spread. The divisor \(c\) is Bartlett’s correction, which makes the \(\chi ^2\) approximation accurate even for small \(n_i\).

Levene’s test. Bartlett’s test is sensitive to non-normality: with heavy-tailed errors it rejects too often. Levene’s test, in the modified form of Brown and Forsythe, replaces each observation by its absolute deviation from the treatment median, \[ z_{ij} = |y_{ij} - \tilde y_i| , \] and applies the one-way analysis of variance \(F\) test to the \(z_{ij}\). Unequal variances show up as unequal mean absolute deviations. The test is robust to departures from normality.