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# Friedman two way analysis of variance by

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### Friedman two way analysis of variance by

1. 1. Genaro C. Reyes III, RN Master in Public Health
2. 2. 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)
3. 3. History  Friedman test is a non parametric statistical method developed by Dr. Milton Friedman
4. 4. History  Friedman test is a non parametric statistical method developed by Dr. Milton Friedman
5. 5. Friedman Formula
6. 6. 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
7. 7. Friedman Formula
8. 8. 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
9. 9. 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
10. 10. 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 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 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
11. 11. 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
12. 12. Calculating of test statistic……  Friedman’s magic formula!!!! Where, k = number of columns (treatments) n = number of rows (blocks) Rj = sum of the ranks
13. 13. 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
14. 14.  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.
15. 15. 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
16. 16. Friedman test online calculator!
17. 17. Offline version Statistic calculator
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