Friedman rank sum test (or simply Friedman test) is a non-parametric test that may be used as an alternative to repeated measures ANOVA when, for example, you suspect or you know that the assumption of normality of distribution is not met.

In this test, the null hypothesis H0 states that there is no difference between the means of the groups.

The function to use in R is friedman.test(). It is possible to use the following two syntaxes friedman.test(response, predictor-group, predictor-block) and friedman.test(response ~ predictor-group | predictor-block). In both cases, response is response variable, predictor-group is the predictor variable corresponding to the grouping factor and predictor-block is the predictor variable corresponding to the blocking factor. Note that the position of the variables in the function is important; inverting predictor-group and predictor-block will most likely result in a different test outcome.

Let’s take an example where 5 rats are weighed 4 times with intervals of 4 weeks (week8 to 20). Here is the code for the dataframe:

# response variable
rat.weight <- c(164,164,158,159,155,220,230,226,227,222,261,275,264,280,272,306,326,320,330,312)

# predictor variables
rat.ID <- as.factor(rep(c("rat1", "rat2", "rat3", "rat4", "rat5"),4))
time.point <- as.factor(c(rep("week08",5), rep("week12",5), rep("week16",5), rep("week20",5)))

# dataframe
my.dataframe <- data.frame(rat.ID,time.point,rat.weight)

Let’s visualize the data with a multiple line chart:

Running the test

In such an experimental design where subjects (rats) are being tested in a time series, the time points are the groups and the subjects are the blocks. We thus write:

friedman.test(rat.weight, time.point, rat.ID)

## 
##  Friedman rank sum test
## 
## data:  rat.weight, time.point and rat.ID
## Friedman chi-squared = 15, df = 3, p-value = 0.001817

The null hypothesis H0 stating that there is no difference between the means of the groups is rejected due to the low p-value.

As mentioned earlier, there is an alternative syntax for the function. Here is the code:

friedman.test(rat.weight ~ time.point | rat.ID)

## 
##  Friedman rank sum test
## 
## data:  rat.weight and time.point and rat.ID
## Friedman chi-squared = 15, df = 3, p-value = 0.001817

As you see, the result is identical.

---