Compare Multiple Related Samples

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*column.

^{th}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*is the ties correction (Corder et al., 2009). When

_{F}*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.