The Kruskal-Wallis test is an alternative for a one-way ANOVA if the assumptions of the latter are violated. We'll show in a minute why that's the case with creatine.sav, the data we'll use in this tutorial. But let's first take a quick look at what's in the data anyway.
Quick Data Description
Our data contain the result of a small experiment regarding creatine, a supplement that's popular among body builders. These were divided into 3 groups: some didn't take any creatine, others took it in the morning and still others took it in the evening. After doing so for a month, their weight gains were measured. The basic research question is
does the average weight gain depend on
the creatine condition to which people were assigned?
That is, we'll test if three means -each calculated on a different group of people- are equal. The most likely test for this scenario is a one-way ANOVA but using it requires some assumptions. Some basic checks will tell us that these assumptions aren't satisfied by our data at hand.
Data Check 1 - Histogram
A very efficient data check is to run histograms on all metric variables. The fastest way for doing so is by running the syntax below.
frequencies gain
/formats notable
/histogram.
Histogram Result
First, our histogram looks plausible with all weight gains between -1 and +5 kilos, which are reasonable outcomes over one month. However, our outcome variable is not normally distributed as required for ANOVA. This isn't an issue for larger sample sizes of, say, at least 30 people in each group. The reason for this is the central limit theorem. It basically states that for reasonable sample sizes the sampling distribution for means and sums are always normally distributed regardless of a variable’s original distribution. However, for our tiny sample at hand, this does pose a real problem.
Data Check 2 - Descriptives per Group
Right, now after making sure the results for weight gain look credible, let's see if our 3 groups actually have different means. The fastest way to do so is a simple MEANS command as shown below.
means gain by group.
SPSS MEANS Output
First, note that our evening creatine group (4 participants) gained an average of 961 grams as opposed to 120 grams for “no creatine”. This suggests that creatine does make a real difference.
But don't overlook the standard deviations for our groups: they are very different but ANOVA requires them to be equal.The assumption of equal population standard deviations for all groups is known as homoscedasticity. This is a second violation of the ANOVA assumptions.
Kruskal-Wallis Test
So what should we do now? We'd like to use an ANOVA but our data seriously violates its assumptions. Well, a test that was designed for precisely this situation is the Kruskal-Wallis test which doesn't require these assumptions. It basically replaces the weight gain scores with their rank numbers and tests whether these are equal over groups. We'll run it by following the screenshots below.
Running a Kruskal-Wallis Test in SPSS
We use
if we compare 3 or more groups of cases. They are “independent” because our groups don't overlap (each case belongs to only one creatine condition).Depending on your license, your SPSS version may or may have the option shown below. It's fine to skip this step otherwise.
SPSS Kruskal-Wallis Test Syntax
Following the previous screenshots results in the syntax below. We'll run it and explain the output.
NPAR TESTS
/K-W=gain BY group(1 3)
/MISSING ANALYSIS.
SPSS Kruskal-Wallis Test Output
We'll skip the “RANKS” table and head over to the “Test Statistics” shown below.
Our test statistic -incorrectly labeled as “Chi-Square” by SPSS- is known as Kruskal-Wallis H. A larger value indicates larger differences between the groups we're comparing. For our data it's roughly 3.87. We need to know its sampling distribution for evaluating whether this is unusually large.
Exact Sig. uses the exact (but very complex) sampling distribution of H. However, it turns out that if each group contains 4 or more cases, this exact sampling distribution is almost identical to the (much simpler) chi-square distribution.
We therefore usually approximate the p-value with a chi-square distribution. If we compare k groups, we have k - 1 degrees of freedom, denoted by df in our output.
Asymp. Sig. is the p-value based on our chi-square approximation. The value of 0.145 basically means there's a 14.5% chance of finding our sample results if creatine doesn't have any effect in the population at large. So if creatine does nothing whatsoever, we have a fair (14.5%) chance of finding such minor weight gain differences just because of random sampling. If p > 0.05, we usually conclude that our differences are not statistically significant.
Note that our exact p-value is 0.146 whereas the approximate p-value is 0.145. This supports the claim that H is almost perfectly chi-square distributed.
Kruskal-Wallis Test - Reporting
The official way for reporting our test results includes our chi-square value, df and p as in
“this study did not demonstrate any effect from creatine,
H(2) = 3.87, p = 0.15.”
So that's it for now. I hope you found this tutorial helpful. Please let me know by leaving a comment below. Thanks!
THIS TUTORIAL HAS 71 COMMENTS:
By Ludovico on May 30th, 2018
Hi,
So I have 4 independent samples of different sizes (N= 20, 32, 17, 12). Most of the dependent variables measured showed significant Shapiro-Wilk and Levene's tests.
Moreover, I've found an effect of the Age.
I have to compare these 4 groups on several dependent variables (i.e. cognitive tests scores).
I tried using the non-parametric test (i.e. Kruskal-Wallis, Multiple comparisons = All pairwise) but, as you said, SPSS doesn't allow me to enter Age as a covariate in any of the non-parametric test tabs.
Looking at the literature I've understood that there is still a relevant controversy surrounding transformations (Natural log or square root), I've also already tried and it doesn't seem to change the results I get.
A proposed solution was to perform Linear Regressions on each dependent variable and then perform the Kruskal-Wallis test, but I really don't know if doing this could really help.
I was also thinking about performing Bootstrapping, so that I could then perform parametric tests on the bootstrapped data but neither I nor my tutor are familiar with this technique.
Finally, it was proposed to use ANCOVAs and MANCOVAs so that I could use Age as a covariate, acknowledging the violations of the assumptions and the probable influence of the sample size making it crystal clear later on in the discussion section of my thesis.
It's quite frustrating because both parametric and non-parametric tests shows very similar results.
What would you do in this situation ?
Thanks again for your patience
By Ruben Geert van den Berg on May 30th, 2018
Hi Ludovico!
Go for ANCOVA then and be open and honest about violating assumptions.
Perhaps be more restrictive regarding statistical significance as well and use alpha = 0.025 or 0.01. Even if your p-values are somewhat biased, with p < 0.025 or p < 0.01 it seems reasonable that the "unbiased" p can't exceed 0.05. I'm sceptical towards transformations. The basic reason is that your research questions relate to your variables as measured but your tests only relate to transformed variables -which is not what you're interested in.
Hope that helps!
By finess on June 1st, 2018
How do you deal with a stratified sample in spss? For example I have 10 strata with a total of 318 participants and am subjecting them all to the same likert questionnaire?
By Ruben Geert van den Berg on June 1st, 2018
Most users will analyze such data as if they're dealing with a simple random sample -which is quietly assumed by basically all standard statistical tests.
If you're very strict, however, you must correct your test results (standard errors) when you're analyzing a stratified sample. Precisely, the true standard errors are smaller in this case so you'll tend to have more statistically significant results when you do it properly.
How? You'll need the SPSS complex samples option -or an entirely different statistical package- for doing so.
Hope that helps!
By James K. Agyen on June 7th, 2018
I find this very useful and informative. It is a piece that touches on the very basic aspects of the Kruskal-Wallis test in simple ''self-explanable'' terms. You can make it better by adding where to write the syntax in the SPSS.