In SPSS, these tests assume that the Exact Tests add-on module be installed.

Key Concepts and Terms

• Dependent samples, also called related samples or correlated samples, are ones in which the response of the nth subject in one sample is partly a function of the response of the nth subject in an earlier sample. Examples of dependent samples include before-during-after samples of the same people or matched triplets of similar people.

• Type of significance estimate. The Exact button in the SPSS dialog above allows the researcher to select among asymptotic, exact, or Monte-Carlo estimates of the significance of the particular test value. These three types of estimates are discussed separately in the section on significance testing. This requires that the SPSS Exact Tests add-on module be installed.

  
  

• The Friedman Test

  • Purpose: The Friedman test, also known as Friedman two-way analysis of variance, tests the null hypothesis that measures from k dependent samples come from the same population. It is based on the rationale that if the groups do not differ on the criterion variable, then the rankings of each subject will be random and there will be no difference in mean ranks between groups on the criterion variable.

  • In SPSS, select Analyze, Nonparametric Tests, K Related Samples; select Test Variables; under Test Type, select Friedman. OK.

  • Calculation. The Friedman test statistic is distributed approximately as chi-square, with (k - 1) degrees of freedom, where k is the number of groups in the criterion variable, from i = 1 to k. Let n be the number of subjects and let T_i be the sum of ranks for each group. Friedman chi-square is then computed by this formula:
    Chi-square_Friedman = ([12/nk(k + 1)]*[SUM(T_i²] - 3n(k + 1))

    SPSS prints out Friedman chi-square, degrees of freedom, n, and the corresponding significance level.

  • Interpretation: The Friedman test can be seen as a two-way analysis of variance with one observation per cell. It can also be seen as a repeated measures analysis of variance for one group. A typical use of the Friedman test is to test inter-rater reliability: cases (rows) are judges, variables (columns) are items being judged, and the Friedman statistic tests the hypothesis that there is no systematic difference in the ratings. If the significance of Friedman chi-square is less than .05, the researcher concludes that the groups do not differ on the criterion variable.

  
  

• Kendall's W Test

  • Purpose; Kendall's W, referred to as a coefficient of concordance, is a normalization of the Friedman test to vary from 0 to 1.

  • In SPSS, select Analyze, Nonparametric Tests, K Related Samples; select Test Variables; under Test Type, select Kendall's W. OK.

  • Interpretation: Kendall's W can be interpreted as a coefficient of agreement among raters. Each case (row) is a rater and each variable (column) is an item being rated. The coefficient W ranges from 0 to 1, with 1 indicating complete inter-rater agreement, and 0 indicating complete disagreement among raters.

  
  

  Cochran's Q Test

  • In SPSS, select Analyze, Nonparametric Tests, K Related Samples; select Test Variables; under Test Type, select Cochran's Q. OK.

  • Purpose: Used to test similar distributions among several dependent samples when variables are dichotomous. It tests the null hypothesis that the dependent samples have the same mean on a dichotomous variable. For dichotomouz items, it is a test of equal proportions.

  • Interpretation: Cochran's Q is an extension of the McNemar test to more than two dependent samples. If sign(Q) < .05, then the researcher rejects the null hypothesis that the proportion of subjects having low (or high) values on a set of dichotomous items is the same across items. Cochran's Q is discussed further in the section on reliability.

  
  

Assumptions

• Random sampling is assumed, as in all significance tests.

• Dependent samples. The three tests discussed in this section all permit multiple dependent samples.

• Data distribution. The three tests discussed in this section are nonparametric tests which do not assume the normal distribution.

Frequently Asked Questions

• Where are these tests found in SPSS?
  
  
  
  
  
  
  
• Where are these tests found in SPSS?
  From the SPSS menu, select Statistics, Nonparametric Tests, K Related Samples. In the "Tests for Several Related Samples" dialog box, check Friedman, Kendall's W, and/or Cochran's Q under "Test Type."

Bibliography

• Siegel, Sidney (1956). Nonparametric statistics for the behavioral sciences. NY: McGraw-Hill. A standard reference work.