Genaro C. Reyes III, RNMaster 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

History Friedman test is a non parametric statistical method

developed by Dr. Milton Friedman

Friedman Formula

Friedman Formula2

2

1

12 ( 1)

( 1) 2

k

r j

j

b kR

bk k

2 2

1

123 ( 1)

( 1)

k

r j

j

R b kbk k

EQUATION 1

EQUATION 2

EQUATION 3

Friedman Formula

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 1month

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

Hypothesis Testing Steps

1. Data

Site BOD (biological oxygen demand)

Before After 1month

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

Site BOD (biological oxygen demand)

Before After 1month

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

Hypothesis Testing Steps

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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