Provide Information Regarding Statistics & Econometrics : Friedman Test Lecture 56

Friedman Test

Lecture 56

The Friedman test is a non-parametric version of two-way ANOVA or randomized complete block designs. The Friedman test is applicable when the assumptions of normality and homogeneity of variances are not met. It is an extension of the sign test when there may be more than two treatments. The Friedman test assumes that there are k experimental treatments (k ≥ 2). The observations are arranged in b blocks, that is

Procedure to perform the Friedman test:

1. Data Preparation:

The data will be available in a b x k table.

2. Ranking Within blocks

Assign the rank rij within a block to Xij from smallest to largest value denoted by rij. Assign the average rank if there are tied observations.

3. Total Ranks

Computed the sum of ranks by adding rij.

Calculating the Friedman test statistic:

Where:

K is the number of treatments.

Ri is the sum of ranks for treatment i.

The critical region will be based on chi-square distribution with (k – 1) degrees of freedom.

Reject H0 when F ≥ χ2α (k-1)d.f.

Assumptions

There will be no interaction between blocks and treatments.

All the k sampled populations have the same variability.

All the k sampled populations have the same shape.

The observed data constitute at least an ordinal scale of measurement within each block.

The variable of interest (dependent variable) is continuous.

The Dunn’s test is used as a post hoc test when the null hypothesis is significant.

Example 13.14: A pharmaceutical investigator claims that the reaction time of subjects is the same on three different drugs. To test the claim, a sample of 6 subjects of different ages and measure each of their reaction times (in seconds) on three different drugs. The following information collected from the subjects is given below:

Age	Drug A	Drug B	Drug C
10	27	20	34
15	2	8	31
20	4	14	3
35	18	36	23
40	7	21	30
45	9	22	6

Use the Friedman test for the hypothesis that the reaction time of the three drugs is identical for all ages.

Solution:

i. State the null and alternative hypotheses.

H0: Median A = Median B = Median C vs. H1: Median A ≠ Median B ≠ Median C

ii. The significance level; α = 0.05

iii. The test statistic:

iv. Reject H0 when F ≥ χ² 0.05 (2) d.f. = 5.991

v. Computation:

Age	Drug A	Drug B	Drug C
10	27	20	34
Rank	2	1	3
15	2	8	31
Rank	1	2	3
20	4	14	3
Rank	2	3	1
35	18	36	23
Rank	1	3	2
40	7	21	30
Rank	1	2	3
45	9	22	6
Rank	2	3	1
Ri	9	14	13

vi. Remarks: The Friedman test calculated value falls in the acceptance region; the sample data does not provide sufficient evidence to reject the null hypothesis that the reaction time is identical. Thus, it is concluded that the reaction time of the three drugs is the same.

Example 13.15: A manager is interested in improving the employee timelines. The manager suggest the two courses of action to improve the employee’s timeline. In the first week she deducts $10 for each day that they do not arrive on time. In the second week she deducts $15. The following data is collected from payroll.

Employee	Baseline Salary	Action 1	Action 2
1	12	5	1
2	13	7	4
3	12	8	5
4	11	7	4
5	12	8	3
6	13	9	2
7	14	7	4
8	12	6	5
9	15	5	4
10	11	6	3

Use the Friedman test to test the hypothesis that employees' timelines are affected by pay cheque reductions.