Saturday, January 10, 2015

Friedman Test For Repeated-Measures ANOVA in 3 Steps in Excel

Friedman Test For

Repeated-Measure ANOVA

in 3 Steps in Excel

This is one of the following four articles on Repeated-Measures ANOVA

Single-Factor Repeated-Measures ANOVA in 4 Steps in Excel

Sphericity Testing For Repeated-Measures ANOVA in 9 Steps in Excel

Effect Size For Repeated-Measures ANOVA in Excel

Friedman Testing For Repeated-Measures ANOVA in 3 Steps in Excel

Overview of Friedman Test

All types of ANOVA tests require that the any data sample that will undergo an individual F Test be normally distributed. The reason for the ANOVA requirement of sample group normality is that the F Test is particularly susceptible to deviation from normality. As with many other types of ANOVA, repeated-measure ANOVA has an alternative nonparametric test that can be substituted if sample group normality cannot be confirmed. This is the Friedman Test.

The Friedman Test is a nonparametric alternative for single-factor, repeated-measures ANOVA when sample groups are not normally distributed. The Friedman Test is also an alternative for single-factor, repeated-measures ANOVA when the dependent variable is ordinal instead of continuous as required by ANOVA. An ordinal variable is a categorical variable whose value indicates a rank or order among other data points. The Likert scale is an example of an ordinal data scale. The Likert scale is often associated with survey in which a respondent rates something with ratings such as “good,” “very good,” and “excellent.”

The Friedman Test somewhat resembles the Kruskal-Wallis test that is used as a nonparametric substitute test for single-factor ANOVA expect that the Friedman Test has fewer required assumption.

The Kruskal-Wallis Test requires that sample groups have similar distributions. A histogram of each sample group will quickly show whether that condition has been met. The Friedman Test does not have that required assumption.

Friedman Test Required

Assumptions

The Freidman Test has only the following three assumptions:

1) Sample Data Are Continuous or Ordinal Sample group data (the dependent variable’s measured value) can be ratio or interval data, which are the two major types of continuous data. sample data can be ordinal which is a type of categorical data in which the data labels indicate a ranking or order of the data. Sample group data cannot be nominal data, which is a type of categorical data in which order has no meaning because data labeling does not indicate any data order.

2) Independent Variable is Categorical The determinant of which group each data observation belongs to is a categorical, independent variable. Repeated-measures ANOVA uses a single categorical variable that has at least two levels. All data observations associated with each variable level represent a unique data group and will occupy a separate column on the Excel worksheet.

3) Sample Data Are Randomly Sampled From a Population

The Friedman test procedure can be summarized as follows:

1) Samples are placed next to each other and ranked. Repeated-measures ANOVA consists of sample group taken from the same subjects at different times or in different conditions. The data points that were taken from one subject are ranked only against each other. This is very different than the Kruskal-Wallis Test which ranks each data point against all other data points in all samples.

2) The rankings of all points in each sample group are summed. Each sum of rankings is then squared. Each of these terms is then added together to produce a sum of the squares of the sum of rankings for each group.

3) Test Statistic F is calculated from this information.

4) The significance of Test Statistic F is determined. If the Friedman test is performed on small groups of samples, Test Statistic F is compared to critical F Values in a table. If samples groups are larger, the distribution of Test Statistic F can be approximated by the Chi-Square distribution with k-1 degrees of freedom. A p Value can be calculated in Excel with the following formula:

p Value = CHISQ.DIST.RT(Test Statistic F, k - 1)

The Friedman Test result is determined to be significant if the p Value is smaller than the specified alpha, which is often set at 0.05.

The Null Hypothesis for the Friedman tests states that the sums of ranks for all sample groups are the same. This is a one-tailed test due to the Chi-Square distribution which states that the sum of ranks of at least one sample group is greater than the sum of ranks of any other sample group.

If the test result is found to be significant, the Null Hypothesis can be rejected.

Here is a detailed description of each step in Excel:

Step 1 - Rank the Values of Data

Points For Each Subject

Here is the raw data matrix:

Repeated-Measures ANOVA Friedman Test - Raw Data
(Click On Image To See a Larger Version)

Note that data values are ranked only against other data values taken from the same subject. Repeated-measures ANOVA takes measurements from the same group of subjects at different time intervals or in different conditions. Ranking must only be performed on data values taken from the same subjects.

Data values that are the same are assigned the average of the rankings that they occupy. The Excel command RANK.AVG() should be used to create those average rankings as follows:

Repeated-Measures ANOVA Friedman Test - Ranking Data
(Click On Image To See a Larger Version)

Step 2 - Calculate Σ(R2)

The sum of the squares of the sum of each sample group’s rankings is calculated in Excel as shown below:

Repeated-Measures ANOVA Friedman Test - Sum of R Square
(Click On Image To See a Larger Version)

Step 3 - Calculate Test Statistic F

The formula for the Friedman Test Statistic F when the Friedman Test is used as a nonparametric alternative to single-factor repeated-measure ANOVA is as follows:

Repeated-Measures ANOVA Friedman Test - Test Statistic Formula

These calculations are performed in Excel as follows:

Repeated-Measures ANOVA Friedman Test - F Calculation
(Click On Image To See a Larger Version)

Step 4 - Determine if the Test

Result Is Significant

The distribution of F can be approximated by the Chi-Square distribution with (k-1) degrees of freedom if either k = 5 OR n > 15. If both of these two conditions is not met, the small-sample procedure is used to determine the test result is significant.

The test result is deemed significant if the small-sample procedure indicates that Test Statistic F is greater than the critical F value that will be looked up on a chart.

If k = 5 OR n > 15 the large-sample procedure will be used. This procedure calculates a p value. If the p Value is smaller than the designated alpha (usually set at 0.05), the test result is deemed to be significant.

A significant test result indicates that the Friedman Test’s Null Hypothesis can be rejected.

The Null Hypothesis for the Friedman tests states that the sums of ranks for all sample groups are the same.

Small-Sample Procedure

If either of these two conditions does not occur, Test Statistic F must be compared with the following table of Critical F Values for k = 3 and 4.

Repeated-Measures ANOVA Friedman Test - Small Sample Critical Values
(Click On Image To See a Larger Version)

F = 7.98

n = 5

k = 4

a = 0.05

Critical F = 7.800

The test result is determined to be significant if the Test Statistic F is greater than the Critical F value for the respective n, k, and a.

The Friedman test therefore does indicate a difference between the average ranks of each group.

It should be noted that the sample size is very small. Small sample size greatly reduces the power of the Friedman test.

Large-Sample Procedure

If either k = 5 OR n > 15, then the distribution of test Statistic F can be approximated by the Chi-Square distribution with (k-1) degrees of freedom.

If that is the case, the test's p Value can be determined by the following Excel equation:

p Value = CHISQ.DIST.RT(Test Statistic F, k - 1)

p Value = 0.050331098 =CHISQ.DIST.RT(7.98,4-1)

This result is very close to the result found by comparing the Test Statistic to the Critical F Value. Both methods show that the test has just barely achieved significance. This result should not be viewed with great confidence because such a small sample size greatly reduces the power of this already relatively weak nonparametric test.

 

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