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Rank-Sum Test STAT 401 - Statistical Methods for Research Workers Jarad Niemi Iowa State University 6 September 2013 Jarad Niemi (Iowa State) Rank-Sum Test 6 September 2013 1 / 9 Rank sum test Do these data look normal? Raw histogram of

  1. Rank-Sum Test STAT 401 - Statistical Methods for Research Workers Jarad Niemi Iowa State University 6 September 2013 Jarad Niemi (Iowa State) Rank-Sum Test 6 September 2013 1 / 9

  2. Rank sum test Do these data look normal? Raw histogram of mpg for US and Japanese cars. 30 20 count 10 0 10 20 30 40 50 mpg Jarad Niemi (Iowa State) Rank-Sum Test 6 September 2013 2 / 9

  3. Rank sum test Do these data look normal? Compared to best fitting normal curves. 0.125 0.100 0.075 density 0.050 0.025 0.000 10 20 30 40 50 mpg Jarad Niemi (Iowa State) Rank-Sum Test 6 September 2013 3 / 9

  4. Rank sum test Rank-sum test Rank-sum test Also referred to as the Wilcoxon rank-sum test and the Mann-Whitney U test: Jarad Niemi (Iowa State) Rank-Sum Test 6 September 2013 4 / 9

  5. Rank sum test Rank-sum test Rank-sum test Also referred to as the Wilcoxon rank-sum test and the Mann-Whitney U test: Transform the data to ranks Jarad Niemi (Iowa State) Rank-Sum Test 6 September 2013 4 / 9

  6. Rank sum test Rank-sum test Rank-sum test Also referred to as the Wilcoxon rank-sum test and the Mann-Whitney U test: Transform the data to ranks Calculate U , the sum of ranks of the group with a smaller sample size Jarad Niemi (Iowa State) Rank-Sum Test 6 September 2013 4 / 9

  7. Rank sum test Rank-sum test Rank-sum test Also referred to as the Wilcoxon rank-sum test and the Mann-Whitney U test: Transform the data to ranks Calculate U , the sum of ranks of the group with a smaller sample size Calculate E [ U ] = n 1 R Jarad Niemi (Iowa State) Rank-Sum Test 6 September 2013 4 / 9

  8. Rank sum test Rank-sum test Rank-sum test Also referred to as the Wilcoxon rank-sum test and the Mann-Whitney U test: Transform the data to ranks Calculate U , the sum of ranks of the group with a smaller sample size Calculate E [ U ] = n 1 R n 1 : sample size of the smaller group Jarad Niemi (Iowa State) Rank-Sum Test 6 September 2013 4 / 9

  9. Rank sum test Rank-sum test Rank-sum test Also referred to as the Wilcoxon rank-sum test and the Mann-Whitney U test: Transform the data to ranks Calculate U , the sum of ranks of the group with a smaller sample size Calculate E [ U ] = n 1 R n 1 : sample size of the smaller group R : average rank Jarad Niemi (Iowa State) Rank-Sum Test 6 September 2013 4 / 9

  10. Rank sum test Rank-sum test Rank-sum test Also referred to as the Wilcoxon rank-sum test and the Mann-Whitney U test: Transform the data to ranks Calculate U , the sum of ranks of the group with a smaller sample size Calculate E [ U ] = n 1 R n 1 : sample size of the smaller group R : average rank � n 1 n 2 Calculate SD ( U ) = s R ( n 1 + n 2 ) Jarad Niemi (Iowa State) Rank-Sum Test 6 September 2013 4 / 9

  11. Rank sum test Rank-sum test Rank-sum test Also referred to as the Wilcoxon rank-sum test and the Mann-Whitney U test: Transform the data to ranks Calculate U , the sum of ranks of the group with a smaller sample size Calculate E [ U ] = n 1 R n 1 : sample size of the smaller group R : average rank � n 1 n 2 Calculate SD ( U ) = s R ( n 1 + n 2 ) n 2 : sample size of the larger group Jarad Niemi (Iowa State) Rank-Sum Test 6 September 2013 4 / 9

  12. Rank sum test Rank-sum test Rank-sum test Also referred to as the Wilcoxon rank-sum test and the Mann-Whitney U test: Transform the data to ranks Calculate U , the sum of ranks of the group with a smaller sample size Calculate E [ U ] = n 1 R n 1 : sample size of the smaller group R : average rank � n 1 n 2 Calculate SD ( U ) = s R ( n 1 + n 2 ) n 2 : sample size of the larger group s R : standard deviation of the ranks Jarad Niemi (Iowa State) Rank-Sum Test 6 September 2013 4 / 9

  13. Rank sum test Rank-sum test Rank-sum test Also referred to as the Wilcoxon rank-sum test and the Mann-Whitney U test: Transform the data to ranks Calculate U , the sum of ranks of the group with a smaller sample size Calculate E [ U ] = n 1 R n 1 : sample size of the smaller group R : average rank � n 1 n 2 Calculate SD ( U ) = s R ( n 1 + n 2 ) n 2 : sample size of the larger group s R : standard deviation of the ranks Calculate Z = ( U + c − E [ U ]) / SD ( U ) where c, the continuity correction, is either 0.5 or -0.5. Jarad Niemi (Iowa State) Rank-Sum Test 6 September 2013 4 / 9

  14. Rank sum test Rank-sum test Rank-sum test Also referred to as the Wilcoxon rank-sum test and the Mann-Whitney U test: Transform the data to ranks Calculate U , the sum of ranks of the group with a smaller sample size Calculate E [ U ] = n 1 R n 1 : sample size of the smaller group R : average rank � n 1 n 2 Calculate SD ( U ) = s R ( n 1 + n 2 ) n 2 : sample size of the larger group s R : standard deviation of the ranks Calculate Z = ( U + c − E [ U ]) / SD ( U ) where c, the continuity correction, is either 0.5 or -0.5. Determine the pvalue using a standard normal distribution. Jarad Niemi (Iowa State) Rank-Sum Test 6 September 2013 4 / 9

  15. Rank sum test Rank-sum test Example on a small dataset mpg country rank 1 13 US 1 2 15 US 2 3 17 US 3 4 22 US 4 5 26 Japan 5.5 6 26 US 5.5 7 28 US 7 8 32 Japan 8 9 33 Japan 9 Jarad Niemi (Iowa State) Rank-Sum Test 6 September 2013 5 / 9

  16. Rank sum test Rank-sum test Example on a small dataset mpg country rank 1 13 US 1 2 15 US 2 3 17 US 3 4 22 US 4 5 26 Japan 5.5 6 26 US 5.5 7 28 US 7 8 32 Japan 8 9 33 Japan 9 U = 22 . 5 Jarad Niemi (Iowa State) Rank-Sum Test 6 September 2013 5 / 9

  17. Rank sum test Rank-sum test Example on a small dataset mpg country rank 1 13 US 1 2 15 US 2 3 17 US 3 4 22 US 4 5 26 Japan 5.5 6 26 US 5.5 7 28 US 7 8 32 Japan 8 9 33 Japan 9 U = 22 . 5 E [ U ] = 15 Jarad Niemi (Iowa State) Rank-Sum Test 6 September 2013 5 / 9

  18. Rank sum test Rank-sum test Example on a small dataset mpg country rank 1 13 US 1 2 15 US 2 3 17 US 3 4 22 US 4 5 26 Japan 5.5 6 26 US 5.5 7 28 US 7 8 32 Japan 8 9 33 Japan 9 U = 22 . 5 E [ U ] = 15 SD [ U ] = 3 . 86 Jarad Niemi (Iowa State) Rank-Sum Test 6 September 2013 5 / 9

  19. Rank sum test Rank-sum test Example on a small dataset mpg country rank 1 13 US 1 2 15 US 2 3 17 US 3 4 22 US 4 5 26 Japan 5.5 6 26 US 5.5 7 28 US 7 8 32 Japan 8 9 33 Japan 9 U = 22 . 5 E [ U ] = 15 SD [ U ] = 3 . 86 z = 1 . 81 (appropriate continuity correction is -0.5) Jarad Niemi (Iowa State) Rank-Sum Test 6 September 2013 5 / 9

  20. Rank sum test Rank-sum test Example on a small dataset mpg country rank 1 13 US 1 2 15 US 2 3 17 US 3 4 22 US 4 5 26 Japan 5.5 6 26 US 5.5 7 28 US 7 8 32 Japan 8 9 33 Japan 9 U = 22 . 5 E [ U ] = 15 SD [ U ] = 3 . 86 z = 1 . 81 (appropriate continuity correction is -0.5) p = 0 . 07 Jarad Niemi (Iowa State) Rank-Sum Test 6 September 2013 5 / 9

  21. Rank sum test Full data Transform data to ranks ● ● ● ● ● Japan ● ● ● ● ● ● ● ● ● ● 300 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● US ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 250 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 200 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Rank ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 150 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 100 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 50 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 ● 10 20 30 40 MPG Jarad Niemi (Iowa State) Rank-Sum Test 6 September 2013 6 / 9

  22. Rank sum test Full data SAS code for Wilcoxon rank sum test DATA mpg; INFILE ’mpg.csv’ DELIMITER=’,’ FIRSTOBS=2; INPUT mpg country $; PROC NPAR1WAY DATA=mpg WILCOXON; CLASS country; VAR mpg; RUN; Jarad Niemi (Iowa State) Rank-Sum Test 6 September 2013 7 / 9

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