1 Assumptions
For the ANOVA model for a CRD design, yij =µ+αi+εij, the assumptions about the εij terms are:
1) independence, 2) normality, and 3) constant variance. Of these assumptions, the most important (and
hardest to check) in terms of the effect of a violation is 1). In terms of importance of a violation the
next most important assumption is 3), and finally 2) . Transformations are a common classical remedy for
problems with assumptions 2) and 3) . The ANOVA null hypothesis is not affected by transformation, but
confidence intervals for the mean are affected, and can instead be interpreted as confidence intervals for the
median on the original scale.
1.1 Assessment of problems
We attempt to assess problems with assumptions (particularly 2 and 3) via the sample residuals. We can
use one of several types of residuals:
Name of residual Formula
raw residual rij =yij −byij
standardized residual zij =rij/sε
(Internally) studentized residual sij =rij/(sεp1−Hij )
(Externally) studentized or jackknife residual tij =rij/(sε(−ij )p1−Hij)
where sε=√MSE,Hiis called the leverage of the ith observation and satisfies 0 ≤Hi≤1, and sε(−ij)is
sεcomputed without the ijth observation. The last three types (standardized, internally studentized, and
jackknife) are fairly similar if the model assumptions are satisfied. The raw residuals are obtained in SAS
Proc GLM on the Output statement with the R option, while the jackknife residuals use the RSTUDENT
option. One advantage of the jackknife residuals over the raw residuals is that they are easier to use to spot
outliers.
1.1.1 Normality
To assess normality, we can create a normal probability plot (or quantile plot). If normality is satisfied, the
plot should look more or less like a straight line.
1.1.2 Homogeneity of variance
To assess HOV, we can use a residual-by-predicted plot. If HOV is ok, then the residuals should form a
mostly horizontal band about zero.
1.1.3 Independence
Departures from independence due to either time or spatial effects can be assessed via residual-by-time plots
or variogram plots, respectively. In a residual-by-time plot, autocorrelation can be detected if successive
residuals are either too close together (positive autocorrelation) or too far apart (negative autocorrelation).
If there is no spatial association then the variogram should be relatively flat.
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