# Slight inconsistency between the Kruskal-Wallis built-in R function and manual calculation

I'm confused by the following, and I haven't been able to dig up the answer elsewhere.

I'm trying to learn R while doing some statistics, and, as an exercise, I try to double-check the results of the built-in R functions by also doing these 'by hand', as it were, in R. However, for the Kruskal-Wallis test I keep getting different results, and I can't figure out why.

For example, I'm looking at the following data handed out in an exercise

activity <- c(2, 4, 3, 2, 3, 3, 4, 0, 4, 3, 4, 0, 0, 1, 3, 1, 2, 0, 3, 1, 0, 3, 4, 0, 1, 2, 2, 2, 3, 2)
group <- c(rep("A", 11), rep("B", 10), rep("C", 9))
group <- factor(group)
data.raw <- data.frame(activity, group)


And I want to analyse activity by group. First I run a Kruskal-Wallis test using the in-built R function

kruskal.test(activity ~ group, data = data.raw)


Which returns $H = 8.9056$.

To double-check, I try doing the same 'by hand' in R, with the following (no doubt helpless) code

rank <- rank(activity)
data.rank <- data.frame(rank, group)
rank.sum <- aggregate(rank ~ group, data = data.rank, sum)

x <- rank.sum[1,2]^2 / 11 + rank.sum[2,2]^2 / 10 + rank.sum[3,2]^2 / 9
H <- (12 / (length(activity) * (length(activity) + 1))) * x - 3 * (length(activity) + 1)
H


Which is meant to reflect the following formula:

$$H =\frac{12}{N(N+1)}\sum_{i = 1}^g \left(\frac{R^2_i}{n_i} \right) - 3(N + 1)$$

Where $N$ is the total number of observations, $g$ is the number of groups, $n_i$ is the number of observations in the $i$th group, and $R_i$ is the sum of ranks of the $i$th group.

And now I get $H = 8.499$, which, adding to my confusion, is also the answer given for the exercise in question. I've tried this for a couple of different data sets, and I tend to get a slightly higher value for $H$ using the in-built function.

I've tried searching to figure out what I'm doing wrong or failing to understand, but to no avail. Can anyone help me understand why the inbuilt kruskal.test function returns a different value from the one I get by spelling things out?

kruskal.test applies a correction for ties as described in this Wikipedia article (point 4):

A correction for ties if using the short-cut formula described in the previous point can be made by dividing H by $1 - \frac{\sum_{i=1}^G (t_i^3 - t_i)}{N^3-N}$, ...

TIES <- table(activity)
H / (1 - sum(TIES^3 - TIES)/(length(activity)^3 - length(activity)))
# 8.9056


You can find out what the R function does by carefully studying the code, which you can see using getAnywhere(kruskal.test.default).

• @MichaelChernick No, it's not. The point is that OP has been taught a simplification of the test that should be used only if there are no ties. Jan 9 '17 at 15:21
• @MichaelChernick I'm not saying it wouldn't fit at Stack Overflow. But I'd argue that it fits equally well at CV. Obviously, it would have been helpful if OP had not only shared their code but also the formulas they are using. Jan 9 '17 at 15:45
• @Michael The status of this thread is an easy call: it's squarely within our purview because it seeks to understand a statistical test.
– whuber
Jan 9 '17 at 16:16
• Edited to include the formula reflected in the code. Should've thought to do so the first time around. Apologies.
– MSR
Jan 9 '17 at 18:31
• See also the R Hmisc package spearman2 function which uses midranks for ties and an F test to get Kruskal-Wallis. I think this is more accurate than some methods. Jan 9 '17 at 20:28