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Kruskal Wallis test
Kruskal-Wallis Stacked
Kruskal Post-Hoc test
Studentized Range Q
Selecting sub-sets
Friedman test
Friedman post-hoc
Rank data ANOVA

Using R for statistical analyses - Non-parametric stats

This page is intended to be a help in getting to grips with the powerful statistical program called R. It is not intended as a course in statistics. If you have an analysis to perform I hope that you will be able to find the commands you need here and copy/paste them into R to get going.

On this page learn how to perform some non-parametric statistics on multiple variables. Routines covered include Kruskal-Wallis and Friedman tests. Learn also how to carry out a post-hoc analysis on the Kruskal-Wallis test. Learn about the studentized range - a useful statistic used in many post-hoc testing situations.

R is Open Source

R is Free

Get R at the R Project Page

What is R?

R is an open-source (GPL) statistical environment modeled after S and S-Plus. The S language was developed in the late 1980s at AT&T labs. The R project was started by Robert Gentleman and Ross Ihaka (hence the name, R) of the Statistics Department of the University of Auckland in 1995. It has quickly gained a widespread audience. It is currently maintained by the R core-development team, a hard-working, international team of volunteer developers. The R project web page is the main site for information on R. At this site are directions for obtaining the software, accompanying packages and other sources of documentation.

R is a powerful statistical program but it is first and foremost a programming language. Many routines have been written for R by people all over the world and made freely available from the R project website as "packages". However, the basic installation (for Linux, Windows or Mac) contains a powerful set of tools for most purposes.

Because R is a programming language it can seem a bit daunting; you have to type in commands to get it to work. However, it does have a Graphical User Interface (GUI) to make things easier. You can also copy and paste text from other applications into it (e.g. word processors). So, if you have a library of these commands it is easy to pop in the ones you need for the task at hand. That is the purpose of this web page; to provide a library of basic commands that the user can copy and paste into R to perform a variety of statistical analyses.

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Kruskal-Wallis test

When you have more than two samples to compare you would usually attempt to use analysis of variance (see the section on anova). However, if the data are not normally distributed (i.e. not parametric) then an alternative must be sought. This is where the Kruskal-Wallis test comes in. It is designed to test for significant differences in population medians when you have more than two samples (otherwize you would use the U-test). You can think of the K-W test as a non-parametric version of one-way anova.

In order to carry out a Kruskal-Wallis test we need some data. The simplest form of data would be one containing several columns, one for each sample. Usually it is best to create your data in a spreadsheet and then save as a CSV file for reading into R. See the sections on creating data and reading data into R in the introduction for more details.

Here is a data file. The numbers represent the growth of an insect fed upon a variety of sugar diets. Each column is a sample of a single diet.

carbs

The Kruskal-Wallis test is carried out using the kruskal.test() function. In this case we type:

carbs.kw = kruskal.test(carbs)
carbs.kw

Kruskal-Wallis rank sum test

data: carbs
Kruskal-Wallis chi-squared = 46.0217, df = 5, p-value = 8.99e-09

We can see from the result that there is a significant difference in growth with diet. However, at this stage we cannot tell which of the treatments is different from which. We can get an overview by creating a simple boxplot of the data:

boxplot(carbs)

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Kruskal-Wallis and stacked data

The Kruskal-Wallis test can be run on data that are arranged by sample in columns. However, your data may be arranged in a different configuration. The most useful configuration for our data would be two columns, one containing the numeric values and the other containing the group (or factor). This configuration is usually used for analysis of variance.

It is generally best to create your data in a spreadsheet and then save as a CSV file for reading into R. See the sections on creating data and reading data into R in the introduction for more details.

Here is an example of part of a data file (it is the same as the one used above). The data represent the growth of an insect when fed upon a range of diets containing different sugars. The first column gives the growth whilst the second gives the grouping.

carbs

The Kruskal-Wallis test is carried out using the kruskal.test() function as before. We can type the command in one of two forms. For the above data we might use:

attach(carbs)
carbs.kw = kruskal.test(growth, sugar)

Alternatively we can use the model syntax that is also used for anova and regression amongst others.

carbs.kw = kruskal.test(growth ~ sugar, data= carbs)

The second method is to be preferred as it is not necessary to attach the data file. The variables are found by the data= carbs parameter in the model syntax but this will not work for the first method. Whichever method is chosen the output is the same:

carbs.kw

Kruskal-Wallis rank sum test

data: growth by sugar
Kruskal-Wallis chi-squared = 46.0217, df = 5, p-value = 8.99e-09

Of course we get an identical result to that obtained before. If we want to create a boxplot of the data this time we use a slightly different syntax:

boxplot(growth ~ sugar, data= carbs)

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Post-Hoc test for Kruskal Wallis

Opinions vary amongst statisticians about the best way to conduct a post-hoc test on non-parametric data. Here I will describe an approach based upon the Tukey method and described in Sokal and Rohlf (1995).

There is no in-built function in R that will conduct the post-hoc analysis. You will have to do it 'long-hand'. The basic idea is that you conduct a pairwize U-test and then compare to a new statistic based upon the Studentized range (this is used in 'regular' post-hoc analyses e.g. Tukey HSD).

After you have carried out the pairwize U-test you calculate a critical U value using the studentized range (Q) in the formula below:

If your value of U is greater than this calculated critical value then the pairwize comparison is significant. If you have unequal sample sizes then you must calculate the harmonic mean of the samples sizes (n) like so:

Of course you may easily re-arrange the equation to work out a critical value of Q:

Now that we know the maths to use we need to work out how to obtain values of Q.

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Studentized Range - Q

The studentized range statistic is commonly used in post-hoc analyses. The distribution function is built-in to R and we may access it in one of two ways.

ptukey(q, nmeans, df)
qtukey(p, nmeans, df)

In the first case we input a confidence level and get the corresponding Q value. In the second case we input a Q value and get the corresponding confidence level. In both cases the nmeans parameter is a numerical value that corresponds to the number of samples that were in the original analysis. The df parameter is the degrees of freedom, for the post-hoc test we use infinity (i.e. df= Inf).

With respect to our Kruskal-Wallis post-hoc test the easiest way to proceed would be to calculate the value of Q that results when using the U value calculated by the pairwize U-test (using the formula as shown above).

Next enter the values into the ptukey() command using:

q= the value of Q you just found.
nmeans= the number of samples in the original K-W test (e.g. 6 for our carbs data set)
df= Inf

The result is the Confidence Interval not a p-value. We need to use 1-our.result to get a p-value.

An alternative method would be to work out the critical value of Q first of all. Use:

qtukey(CI, nmeans, df= Inf)

Where CI is your chosen confidence interval (e.g. 0.95) and nmeans= the number of samples in the original K-W test (e.g. 6 for our carbs data).

Then we would calculate the critical value of U using the equation as shown above.

Finally we would carry out the pairwize U-test and compare the U value to our critical value (of course we would ignore the actual result of the U-test as we are only interested in the U value).

Which U-value?

However, there is a potential problem. When a U-test is carried out there are two possible U values. When calculating U by hand you ignore the larger one and compare the smaller against a table of critical values. For our post-hoc test we actually want the larger of the two U values. However, R only displays one value and whether it is the largest or the smallest depends upon which order you entered the variables for comparison.

If you used a data frame with multiple sample columns then you will have to repeat the test with the variables in the reverse order so that you can decide which is the largest U value (alternatively you can work it out by the fact that U1 + U2 = n1 * n2).

If however, your data are in the stacked form then the wilcox.test() displays the U value that arises from the order of variables in the data set. Since you cannot repeat the test with the variables in any other order you will have to work out the other one and select the largest. Since the stacked data set contains all the samples you will have to select which ones to include in the pairwize comparison. How to do this is detailed below.

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Selecting two samples from a larger data set

If you have a stacked data set containing several samples you may wish to analyze only two of them. You need to select a subset for analysis e.g. in the carbs example above we had 6 samples but to perform post-hoc tests we need to carry out U-tests on pairs of samples.

The basic wilcox.test() function allows us to select a subset quite easily. For our carb example we might use something like the following:

wilcox.test(growth ~ sugar, data= carbs, subset= sugar %in% c("test", "C"))

To get the subset we use the subset= parameter. Here we have used a subset of the variable sugar. The list of samples follows the %in% part and is in the c(item1, item2) format that we have met before. The variable names must be in double quotes.

The subset= parameter works with other functions too, for example we may wish to run our Kruskal-Wallis test on all the monosaccharides e.g.

kruskal.test(growth ~ sugar, data= carbs, subset= sugar %in% c("G", "F", F.G"))

This command would run the test and analyze differences between the three samples (G, F and F.G) only.

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Friedman test (in lieu of two-way anova)

The Friedman test is essentially a 2-way analysis of variance used on non-parametric data. The test only works when you have completely balanced design. Here is an example of a data file.

survey

What we have here are data on surveys of amphibians. The first column (count) represents the number of individuals captured. The final column is the year that the survey was conducted. The middle column (month) shows that for each year there were 5 survey events in each year. What we have here is a replicated block design. Each year is a "treatment" (or "group") whilst the month variable represents a "block". This is a common sort of experimental design; the blocks are set up to take care of any possible variation and to provide replication for the treatment. In this instance we wish to know if there is any significant difference due to year.

The Friedman test allows us to carry out a test on these data. We need to determine which variable is the group and which the block. The friedman.test() function allows us to perform the test, there are two ways to specify it:

attach(survey)
friedman.test(count, year, month)

Friedman rank sum test

data: count, year and month
Friedman chi-squared = 7.6, df = 2, p-value = 0.02237

>

Alternatively we can use a model syntax:

friedman.test(count ~ year | month, data= survey)

Friedman rank sum test

data: count and year and month
Friedman chi-squared = 7.6, df = 2, p-value = 0.02237

>

In the first case we had to attach(data) the data so that the variables could be read (e.g. data, groups, blocks). In the second case this is not necessary, we specify the data ~ groups | blocks as a formula and add data= data at the end.

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Post-hoc testing for Friedman tests

There is a real drop in statistical power when using a Friedman test compared to anova. Although there are methods that enable post-hoc tests (similar to the K-W post-hoc test discussed above) the power is such that obtaining significance is well nigh impossible. The best you can do is to present a boxplot of the data (dependent ~ group).

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Ranked data anova

There may be occasions when you simply need to run an ANOVA but the data don't quite fit with a Friedman or Kruskal test. What you may consider is to replace all the original data with the ranks instead. Simply create a new dependent variable using new.variable= rank(old.variable) and then perform your analysis of variance from there. This is far from ideal but may be the only thing you can do. Watch this space...