Statistics Toolbox

Friedman's Test

In Two-Way Analysis of Variance (ANOVA) we used two-way analysis of variance to study the effect of car model and factory on car mileage. We tested whether either of these factors had a significant effect on mileage, and whether there was an interaction between these factors. We concluded that there was no interaction, but that each individual factor had a significant effect. Now we will see if a nonparametric analysis will lead to the same conclusion.

Friedman's test is a nonparametric test for data having a two-way layout (data grouped by two categorical factors). Unlike two-way analysis of variance, Friedman's test does not treat the two factors symmetrically and it does not test for an interaction between them. Instead, it is a test for whether the columns are different after adjusting for possible row differences. The test is based on an analysis of variance using the ranks of the data across categories of the row factor. Output includes a table similar to an anova table.

We can run Friedman's test as follows.

• p = friedman(mileage, 3)
  ans =
    7.4659e-004
  

Recall the classical analysis of variance gave a p-value to test column effects, row effects, and interaction effects. This p-value is for column effects. Using either this p-value or the p-value from ANOVA (p < 0.0001), we conclude that there are significant column effects.

In order to test for row effects, we need to rearrange the data to swap the roles of the rows in columns. For a data matrix x with no replications, we could simply transpose the data and type

• p = friedman(x')
  

With replicated data it is slightly more complicated. A simple way is to transform the matrix into a three-dimensional array with the first dimension representing the replicates, swapping the other two dimensions, and restoring the two-dimensional shape.

• x = reshape(mileage, [3 2 3]);
  x = permute(x, [1 3 2]);
  x = reshape(x, [9 2])
  x =
     33.3000   32.6000
     33.4000   32.5000
     32.9000   33.0000
     34.5000   33.4000
     34.8000   33.7000
     33.8000   33.9000
     37.4000   36.6000
     36.8000   37.0000
     37.6000   36.7000
  friedman(x, 3)
  ans =
      0.0082
  

Again, the conclusion is similar to the conclusion from the classical analysis of variance. Both this p-value and the one from ANOVA (p = 0.0039) lead us to conclude there are significant row effects.

You cannot use Friedman's test to test for interactions between the row and column factors.

Kruskal-Wallis Test

Nonlinear Regression Models