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These WINKS statistics tutorials explain the use and interpretation of standard statistical analysis techniques for Medical, Pharmaceutical, Clinical Trials, Marketing or Scientific Research. The examples include how-to instructions for WINKS SDA Version 6.0 Software. Download evaluation copy of WINKS.

Friedman's Test
(Non-Parametric Repeated Measures Comparisons)

Definition: A non-parametric test (distribution-free) used to compare observations repeated on the same subjects. This is also called a non-parametric randomized black analysis of variance.

Assumptions: Unlike the parametric repeated measures ANOVA or paired t-test, this non-parametric makes no assumptions about the distribution of the data (e.g., normality).

Characteristics: This test is an alternative to the repeated measures ANOVA, when the assumption of normality or equality of variance is not met. This, like many non-parametric tests, uses the ranks of the data rather than their raw values to calculate the statistic. Since this test does not make a distribution assumption, it is not as powerful as the ANOVA. If there are only two measures for this test, it is equivalent to the sign test. (See the Zar reference for more information.)

Test: The hypotheses for the comparison across repeated measures are:

Ho: The distributions are the same across repeated measures.

Ha: The distributions across repeated measures are different

Notice that the hypothesis makes no assumptions about the distribution of the populations. These hypotheses could also be expressed as comparing mean ranks across measures.

The test statistic for the Friedman's test is a Chi-square with a-1 degrees of freedom, where a is the number of repeated measures. When the p-value for this test is small (usually <0.05) you have evidence to reject the null hypothesis.

Example: Friedman's non-parametric repeated measures comparisons

Five people were given four different drugs (in random order) and with a washout period. Reaction time to a test was measured. The data are in the file DRUG.DBF. The output is:

Friedman's Test for Repeated Measures

Number of repeated measures= 4 Number of subjects = 5

1 )DRUG1 Rank sum = 13.0 Mean rank = 2.6
2 )DRUG2 Rank sum = 12.0 Mean rank = 2.4
3 )DRUG3 Rank sum = 5.0 Mean rank = 1.0
4 )DRUG4 Rank sum = 20.0 Mean rank = 4.0

Ho:There is no difference in mean ranks for repeated measures.
Ha:A difference exists in the mean ranks for repeated measures.

Friedman's Chi-Square = 14.13 with d.f. = 3 p < 0.001

Kendall's coefficient of concordance = 0.942

When the p-value is low, there is evidence to reject Ho,
and conclude that there is a difference between mean ranks.

Error term used for comparisons = 2.89

Critical q
Tukey Multiple Comp. Difference Q (.05)
----------------------------------------------------------------
Rank(4)-Rank(3) = 15.0 5.196 3.63 *
Rank(4)-Rank(2) = 8.0 2.771 3.63
Rank(4)-Rank(1) = 7.0 (Do not test)
Rank(1)-Rank(3) = 8.0 2.771 3.63
Rank(1)-Rank(2) = 1.0 (Do not test)
Rank(2)-Rank(3) = 7.0 (Do not test)

Homogeneous Populations, repeated measures ranked

Gp 1 refers to DRUG1
Gp 2 refers to DRUG2
Gp 3 refers to DRUG3
Gp 4 refers to DRUG4

Gp Gp Gp Gp
3  2  1  4
--------
--------

This is a graphical representation of the Tukey multiple comparisons test.
At the 0.05 significance level, the ranks of any two groups underscored by
the same line are not significantly different.

This analysis indicates that there is a difference in reaction times across drugs with p < 0.01. In this case, the multiple comparisons indicate (at the 0.05 significance level) that the reaction time for drug 3 was shorter than from drug 4. However, it was not shown to be significantly shorter than the time for drugs 2 or 1. Note that the p-value for the Chi-square test is reported even though the sample size is small. In this case, the tabled value agrees with the Chi-square value. However, the p-value becomes less accurate for small values as indicated in the note above.

Exercise: Friedman's non-parametric repeated measures comparison

Six welders with different expertise were asked to weld two pipes together using 5 different welding torches. Torches were used in random order for each welder. Finished pipes were measured on a variety of quality factors, and rated from 1 to 10, where 10 represents a perfect weld. The data are:

Welder Torch1 Torch2 Torch3 Torch4 Torch5
1 3.9 4.1 4.2 4.1 3.3
2 9.4 9.5 9.4 9.0 8.6
3 9.7 9.3 9.3 9.2 8.4
4 8.3 8.0 7.9 8.6 7.4
5 9.8 8.9 9.0 9.0 8.3
6 9.9 10.0 9.7 9.6 9.1

1. Perform a Friedman's test on this data. From the results, can you select a torch that you believe is the "best" of the 5 when measured on this test?

2. Perform the test using parametric methods, and compare the difference.

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