Friedman two way analysis of variance by

Genaro C. Reyes III, RN
Master in Public Health

Friedman two way ANOVA By Rank
 is a test for comparing three or more related samples
and which makes no assumptions about the
underlying distribution of the data. The data is set out
in a table comprising n rows and k columns.
 The data is ranked horizontally or across the rows and
the mean rank for each column is compared.
 This test is very useful when the data are ordinal
(i.e., ranked)

History
 Friedman test is a non parametric statistical method
developed by Dr. Milton Friedman

Friedman Formula
2
2
1
12 ( 1)
( 1) 2
k
r j
j
b k
R
bk k


 
    

2 2
1
12
3 ( 1)
( 1)
k
r j
j
R b k
bk k


  


EQUATION 1
EQUATION 2
EQUATION 3

Example
 A water company sought evidence the measures taken to
clean up a river were effective. Biological Oxygen Demand
(BOD) at 12 sites on the river were compared before clean
up, 1 month later and a year after clean up.
Aqualytic sensor system AL606

Hypothesis Testing Steps
 1. Data
Site BOD (biological oxygen demand)
Before After 1
month
After 1
year
1 17.4 13.6 13.2
2 15.7 10.1 9.8
3 12.9 9.7 9.7
4 9.8 9.2 9.0
5 13.4 11.1 10.7
6 18.7 20.4 19.6
7 13.9 10.4 10.2
8 11 11.4 11.5
9 5.4 4.9 5.2
10 10.4 8.9 9.2
11 16.4 11.2 11.0
12 5.6 4.8 4.6

 1. Data
Before After 1
month
After 1
year
1 17.4 13.6 13.2
2 15.7 10.1 9.8
3 12.9 10.3 9.7
4 9.8 9.2 9.0
5 13.4 11.1 10.7
6 18.7 20.4 19.6
7 13.9 10.4 10.2
8 11 11.4 11.5
9 5.4 4.9 5.2
10 10.4 8.9 9.2
11 16.4 11.2 11.0
12 5.6 4.8 4.6
Before After 1
month
After 1
year
1 17.4 3 13.6 2 13.2 1
2 15.7 3 10.1 2 9.8 1
3 12.9 3 9.7 1.5 9.7 1.5
4 9.8 3 9.2 2 9.0 1
5 13.4 3 11.1 2 10.7 1
6 18.7 1 20.4 3 19.6 2
7 13.9 3 10.4 2 10.2 1
8 11 1 11.4 2 11.5 3
9 5.4 3 4.9 1 5.2 2
10 10.4 3 8.9 1 9.2 2
11 16.4 3 11.2 2 11.0 1
12 5.6 3 4.8 2 4.6 1
Rj 32 22.5 17.5

 1. Data
 2. Assumption
The observations appearing in a given block are independent of the observations appearing in
each of the other blocks, and within each block measurement on at least an ordinal scale is
achieved.
 3. Hypothesis
H0 : The clean up procedure has had no effect on the BOD.
HA : The clean up procedure has affected the BOD.
 4. Decision Rule: Reject H0 if M > critical value at 5% level of
significance
5. Calculation of Test Statistic

Calculating of test statistic……
 Friedman’s magic formula!!!!
Where, k = number of columns (treatments)
n = number of rows (blocks)
Rj = sum of the ranks

BOD (biological oxygen demand)
Site Before After 1 month After 1 year
Sum of ranks 32 22.5 17.5
2
(sum of ranks) 1024 506.25 306.25
Number of columns, k 3
 Solution:Number of rows, n 12
1836.5 = (1024 + 506.25 + 306.25)
__12__
nk(k+1)
0.083 = ___12___
12 x 3 x 4
3n(k+1) 144 = 3 x 12 x 4
Test Statistic M 8.43 = 0.083 x 1836.5 - 144

 6. Statistical decision
Compare computed M value to critical value at 5% level of significance.
M(computed value) = 8.43
critical value at 5% level of significance is = 6.17
• 7. Conclusion M is > than critical value
Reject the null hypothesis
Alternative hypothesis:
HA : The clean up procedure has affected the BOD.

Critical Values for Friedman’s two way ANOVA by
Ranks
k n =0.10 =0.05 =0.1
3 3 6.00 6.00 ---
4 6.00 6.50 8.00
5 5.20 6.40 8.40
6 5.33 7.00 9.00
7 5.43 7.14 8.86
8 5.25 6.25 9.00
9 5.56 6.22 8.67
10 5.00 6.20 9.60
11 4.91 6.54 8.91
12 5.17 6.17 8.67
13 4.77 6.00 9.39
-- 4.61 5.99 9.21

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