e-Statistics

## 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.