Adjustment for tied ranks in the Kruskal-Wallis test

Exercise 10.2.b.

These notes are concerned with the Kruskal-Wallis test (Section 10.2). The notes show you how to adjust your test statistic when you have tied values, and so tied ranks.

• Introduction
• Calculating the adjustment factor
• Applying the correction to H
• Example Data
  • Example Ranks
  • Original H value
  • Tied ranks
  • Final H adjustment

The Kruskal-Wallis test is appropriate when you have non-parametric data and one predictor variable (Section 10.2). It is analogous to a one-way ANOVA but uses ranks of items in various groups to determine the likely significance.

When you have tied values, you will get tied ranks. In these circumstances you should apply a correction to your calculated test statistic. The notes show you how this can be done. The calculations are simple but in Excel it can be difficult to get the process “automated”. In R the kruskal.test() command computes the adjustment for you.

In order to correct for tied ranks you first need to know which values are tied. Then you need to know how many ties there are for each rank value. Finally, you’ll need to know how many replicates there are in the dataset.

Once you’ve ascertained these things you can use the following formula to work out a correction, or adjustment, factor:

In the formula t is the number of ties for each rank value. For each value of t, you evaluate the t-cubed minus t part. This is then summed for all the tied values (values without ties can also be evaluated but 1³ – 1 = 0). Once you have the numerator you work out the denominator using n, the number of replicates in the original dataset.

The final value of D is then 1 – your fraction.

Once you have the value of D, the correction factor, you can use it to adjust the original Kruakal-Wallis test statistic (H).

Formula for adjustment of the Kruskal-Wallis statistic in the case of tied ranks.

The correction is simple: H/D.

You then use the Hadj value in place of the original to determine the final test significance (using critical values tables – see exercise 10.2a).

Have a look at the correction in action using the following example:

Example data. Three non-parametric samples for use in Kruskal-Wallis test.

Here there are three samples and you can see that there are tied values, so you know there will be tied ranks.

The first step is to evaluate the ranks. Each value is replaced by its rank in the overall dataset.

Sample data shown as ranks instead of original values. Samples are combined for ranking.

The Kruskal-Wallis test looks at the sum of the ranks from each sample. If the ∑rank is different between samples there is a good chance that differences are statistically significant. If the ∑rank are close, then differences are less likely to be significant.

In this example the rank sums are:

Rank sums for three samples.

You can now calculate the Kruskal-Wallis test statistic, H.

Once you have the rank sums you can compute the Kruskal-Wallis test statistic:

Formula for calculating the Kruskal-Wallis statistic for non-parametric samples.

The Kruskal–Wallis formula looks pretty horrendous but actually it is not that bad. The numbers 12 and 3 are constants. Uppercase N is the total number of observations. The R refers to the ranks of the observations in each sample and n is the number of observations per sample.

In the example the final value of the Kruskal-Wallis statistic works out to be H = 6.403.

When you paste the ranks you only want the numbers and don’t want the formula (which would give an error).

Once you have the column of ranks you can simply re-arrange them in ascending order (use the Home > Sort & Filter button or the Data > Sort button). In the following table the second column shows the sorted ranks (you don’t need to make a second column, I have done this to show the effect).

Rearranging ranked data to make calculation of tied ranks easier.

Once you have the ranks in order it is easy to work out the number of repeats by inspection. You can simply fill in the values as you work down the column. In the preceding table the 3rd column shows the tied rank repeats. So, for example the rank 1.5 is repeated 2 times. The rank 9.5 has 4 repeats.

The final column shows the T³ – T values. In other words, you take the number of repeats and cube it, then subtract the number of repeats.

Once you have these values you can sum them to get an overall value, in this case 102.

Once you have your final sum of T³ -3 values you can work out the value of D using the formula given earlier. You need to know the total number of replicates in the original dataset (in this case 17).

The final value of D works out at: 0.9792.

The adjusted H value is then H/D = 6.403 / 0.9792 = 6.540.

You can now use the adjusted H-value to compare to critical value tables (see exercise 10.2a) to see if your result is statistically significant.