## Residual Analysis

The assumption for an analysis of variance (AOV) test is described as follows: The random variable is called a residual, and it is assumed that all 's are independent and approximately normally distributed with mean 0 and common variance .

The data from k groups are arranged
either (a) all in a single variable with another categorical variable
indicating ``levels,''
or (b) in multiple columns each of whose variables represents
a ``level.''
In either case,
(i) the original variables 's are
converted to new variables 's
via the
transformation if necessary; otherwise, leave it to ```no change`.''
Then, (ii) they are moved to the column `Variable`
in the table below,
and (iii) the residual
is calculated.

When the nonnormality cannot be eliminated by the use of transformation, the Kruskal-Wallis test is appropriate for the hypothesis testing. Here the null hypothesis is that k population distributions (not necessarily normal) are identical. It calculates the test statistic and the p-value . By rejecting we can find some evidence supporting that not all the distributions are the same.