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Worked example 1
Our first worked example uses data from Cobo et al.
Draw boxplots and assess shape of distributionsWe first examine box plots to assess whether a parametric or non-parametric ANOVA would be more appropriate.
The plot of the raw data strongly suggests that distributions are non-normal (at least for B and C) and variances are not homogeneous. However, a log transformation appears to make distributions more symmetrical and variances more similar. Hence the most powerful approach to analyzing these data would be to 1. test homogeneity of variances before and after a log transformation, and 2. if the transformation successfully stabilized
An alternative approach (which we will do here) is to analyze the data with Kruskal-Wallis ANOVA - but bear in mind that heterogeneous variances will make interpretation of the result more complex. Note that a log transformation would have no effect at all on the value of the Kruskal-Wallis statistic.
Pool groups and rank observations
Note we have also worked out the sum of ranks (Si) for each group.
Calculate the Kruskal-Wallis statistic
The corrected value of K is given by:
As we pointed out above, how we interpret this difference is complicated by the differences between the variances. Ideally we would use the Fligner-Policello robust-rank order test - but even this is not appropriate in this instance becase it requires symmetrical distributions. Instead we will use the non-parametric equivalent of Tukey's honestly significant difference test for unequal numbers of replicates (Dunn's test) and accept that the test may be too liberal. For this we need to compute mean ranks (rather than sum of ranks) for each group by simply dividing the sum of ranks (Si) by the number of observations (ni).
The standard errors for comparing each pair of groups are:
Honestly significant differences and actual differences in mean rank (from table above) are therefore:
The actual differences between levels in groups C and A, and between groups C and B were markedly larger than the honestly significant differences. We therefore accept these differences were unlikely to have arisen by chance. There was also a big difference in variability, with levels in groups C much more variable. Differences in variability are commonly ignored - but they could have important clinical implications (for example very high levels may be toxic). The obseved differences between groups may have resulted from the different treatment regimens - but since allocation was consecutive (not random), they may also have resulted from changes in protocols or in patients over time.