The command compares multiple related samples using the Friedman test (nonparametric alternative to the one-way ANOVA with repeated measures) and calculates the Kendall's coefficient of concordance (also known as Kendall’s W).  Kendall's W makes no assumptions about the underlying probability distribution and allows to handle any number of outcomes, unlike the standard Pearson correlation coefficient. Friedman test is similar to the Kruskal-Wallis one-way analysis of variance with the difference that Friedman test is an alternative to the repeated measures ANOVA with balanced design.

How To

For unstacked data (each column is a sample):

o   Run the Statistics→Nonparametric Statistics → Compare Multiple Related Samples [Friedman ANOVA, Concordance] command.

o   Select variables to compare.

For stacked data (with a group variable):

o   Run the Statistics→Nonparametric Statistics → Compare Multiple Related Samples (with group variable) command.

o   Select a variable with observations (Variable) and a text or numeric variable with the group names (Groups).

Results

The report includes Friedman ANOVA and Kendall’s W test results.

 

The Friedman ANOVA tests the null hypothesis that the samples are from identical populations. If the p-value is less than the selected  level the null-hypothesis is rejected.

If there are no ties, Friedman test statistic F_t is defined as:

where n is the number of rows, or subjects; k is the number of columns or conditions, and R_i is the sum of the ranks of i^th column.

If ranking results in any ties, the Friedman test statistic F_t is defined as:

where n is the number rows, or subjects,  k is the number of columns, and R_i is the sum of the ranks from column, or condition I; C_F is the ties correction (Corder et al., 2009).  When n > 15 or k > 4  the test statistic approximately follows chi-square distribution with d.f.= k – 1.

 

Kendall’s W is used to assess the agreement between samples, it is a normalization of the Friedman test statistic and ranges from 0 (no agreement) to 1 (complete agreement).

Kendall’s W is defined by:

where m is a number of raters, rating k subjects in a rank order from 1 to k. R is a squared deviation.

References

Conover, W. J. (1999), Practical Nonparametric Statistics, Third Edition, New York: John Wiley & Sons.

Corder, Gregory W., Foreman, Dale I. (2009). Nonparametric Statistics for Non-Statisticians: A Step-by-Step Approach.

Friedman, Milton (March 1940). A comparison of alternative tests of significance for the problem of m rankings. The Annals of Mathematical Statistics 11 (1): 86–92.