NONPARAMETRIC STATISTICS Assoc.Prof.Dr. Fikri GÖKPINAR

7.1 OBJECTIVES In this lecture, you will learn the following items: TESTS FOR MORE THAN TWO DEPENDENT SAMPLES 7.1 OBJECTIVES In this lecture, you will learn the following items: How to perform Friedman’s S test for more than two dependent samples. How to use SPSS® perform Friedman’s S test for more than two dependent samples. How to perform a pairwise comparison for Test for Friedman’s S test. How to use SPSS® perform a pairwise comparison for Test for Friedman’s S test.

LECTURE 6: TESTS FOR MORE THAN TWO DEPENDENT SAMPLES 7.2 INTRODUCTION While comparing treatments, if same units or same type of units are used in every groups, these unit are called block and this kind of design are called Randomized Complete Block Design. Friedman S test can be described as a nonparametric version of Randomized Complete Block Design.  

7.3 FRIEDMAN S TEST Assumptions: There are c treatments and n blocks. LECTURE 6: TESTS FOR MORE THAN TWO DEPENDENT SAMPLES 7.3 FRIEDMAN S TEST Assumptions: There are c treatments and n blocks. There are no interaction between blocks and treatments. Hypotheses: The Hypotheses can be given as follows Ho: 𝑀 1 = 𝑀 2 =… 𝑀 𝑘 H1: 𝑀 𝑖 ≠ 𝑀 𝑗 i𝑗   Test Statistics: Let Xij, i=1,2,…c, j=1,2,…n, be a random sample from c dependent population

7.3 FRIEDMAN S TEST Treatment 1 Treatment 2 … Treatment c 𝑋 11 𝑋 21 LECTURE 6: TESTS FOR MORE THAN TWO DEPENDENT SAMPLES 7.3 FRIEDMAN S TEST Treatment 1 Treatment 2 … Treatment c 𝑋 11 𝑋 21 𝑋 𝑐1 𝑋 12 𝑋 22 𝑋 𝑐2 𝑋 1𝑛 𝑋 2𝑛 𝑋 𝑐𝑛

LECTURE 6: TESTS FOR MORE THAN TWO DEPENDENT SAMPLES 7.3 FRIEDMAN S TEST Each blocks are ranked from 1 to c and these are indicated with 𝑅 𝑖𝑗 . The ranks of the sample can be given as follows:   Treatment 1 Treatment 2 … Treatment c 𝑅 11 𝑅 21 𝑅 𝑐1 𝑅 12 𝑅 22 𝑅 𝑐2 𝑅 1𝑛 𝑅 2𝑛 𝑅 𝑐𝑛

LECTURE 6: TESTS FOR MORE THAN TWO DEPENDENT SAMPLES 7.3 FRIEDMAN S TEST By using these definitons, Friedman’s S statistics can be given as follows: 𝑆= 12𝑛 𝑐 𝑐+1 𝑖=1 𝑐 𝑅 𝑖. − 𝑅 .. 2 or 𝑆= 12 𝑛𝑐 𝑐+1 𝑖=1 𝑐 𝑅 𝑖. 2 −3𝑛(𝑐+1)

7.3 FRIEDMAN S TEST Decision Rule: LECTURE 6: TESTS FOR MORE THAN TWO DEPENDENT SAMPLES 7.3 FRIEDMAN S TEST Decision Rule: Observed value of S based on sample data can be given as 𝑆 ℎ . Also let 𝑆 𝛼 be the critical value obtained table.   By using these definitions the decision rules can be given as follows: 𝐻 1 𝐻 0 is rejected 𝑀 𝑖 ≠ 𝑀 𝑗 i𝑗 𝑆 ℎ ≥ 𝑆 𝛼

LECTURE 6: TESTS FOR MORE THAN TWO DEPENDENT SAMPLES 7.3 FRIEDMAN S TEST  Example: Six raters evaluated four restaurants. The results of the experiment are displayed in Table. Under nonnormality assumption of the service ratings, test the equality of the service rating points of the restaurants at significance level %10.   Rater R1 R2 R3 R4 1 70(2) 61(1) 82(4) 81(3) 2 77(3) 66(1) 74(2) 84(4) 3 76(3) 69(1) 73(2) 91(4) 4 80(3) 63(1) 75(2) 96(4) 5 71(1) 78(2) 98(4) 6 83(3) 68(1) 79(2) 𝑅 𝑖. 15 7 23 𝑅 𝑖. 2.50 1.17 3.87

7.4 LARGE SAMPLE APPROXIMATION LECTURE 6: TESTS FOR MORE THAN TWO DEPENDENT SAMPLES 7.4 LARGE SAMPLE APPROXIMATION  When ni>15 Large sample approximation can be used for Kruskal Wallis H statistic. When ni>15, H statistic asymptotically distributed as Chi-Square distribution with degrees of freedom c-1. Decision rule can be given as follows: 𝐻 1 𝐻 0 is rejected 𝑀 𝑖 ≠ 𝑀 𝑗 i𝑗 𝐻 ℎ ≥ χ 𝑐−1,𝛼 2

7.4 LARGE SAMPLE APPROXIMATION LECTURE 6: TESTS FOR MORE THAN TWO DEPENDENT SAMPLES 7.4 LARGE SAMPLE APPROXIMATION  Example: Sixteen experts rated five brands of Colombian coffee in a taste-testing experiment. A rating on a 7-point scale (1=extremely unpleasing, 7 == extremely pleasing) is given for each of the following four characteristics: taste, aroma, richness, and acidity. The following table displays the summated ratings over all four characteristics. a. At the 0.10 level of significance, is there evidence of a difference in the median summated ratings of the five brands of Colombian coffee?

7.4 LARGE SAMPLE APPROXIMATION LECTURE 6: TESTS FOR MORE THAN TWO DEPENDENT SAMPLES 7.4 LARGE SAMPLE APPROXIMATION Expert A B C D E 1 20(3) 14(2) 10(1) 27(4) 28(5) 2 13(2) 11(1) 18(3) 26(5) 25(4) 3 15(1) 16(2) 22(3) 4 18(2) 21(3) 5 15(2) 24(3) 6 14(1) 7 17(3) 19(4) 25(5) 8 19(3) 16(1) 17(2) 24(5) 9 16(3) 22(4) 10 21(4) 11 12 11(2) 9(1) 27(5) 13 21(2) 24(4) 14 26(4) 15 17(1) 16 𝑅 𝑖. 33 28 36 69 74 𝑅 𝑖. 2.0625 1.75 2.25 4.3125 4.6250  

LECTURE 6: TESTS FOR MORE THAN TWO DEPENDENT SAMPLES 7.5 PAIRWISE COMPARISON  When null hypothesis are rejected, we also may interest in which group/groups create the difference between groups. To do this we can use pairwise comparison. The hypotheses are as follows:   Ho: 𝑀 𝑠 = 𝑀 𝑡 H1: 𝑀 𝑠 ≠ 𝑀 𝑡 Also the test statistics can be given as follows: 𝑆 𝑠𝑡 = 𝑅 𝑠. − 𝑅 𝑡. 𝑐 𝑐+1 6𝑛

7.5 PAIRWISE COMPARISON Decision rule can be given as follows: LECTURE 6: TESTS FOR MORE THAN TWO DEPENDENT SAMPLES 7.5 PAIRWISE COMPARISON  Decision rule can be given as follows: When ni>15 Large sample approximation can be also used for pairwise comparison. Decision rule can be given as follows: 𝐻 1 𝐻 0 is rejected 𝑀 𝑠 ≠ 𝑀 𝑡 𝑆 𝑠𝑡 ≥ 𝑆 𝛼 𝐻 1 𝐻 0 is rejected 𝑀 𝑠 ≠ 𝑀 𝑡 𝐻 𝑠𝑡 ≥ 𝜒 𝑐−1,𝛼 2

LECTURE 6: TESTS FOR MORE THAN TWO DEPENDENT SAMPLES 7.5 PAIRWISE COMPARISON  Example 3(Example 1 Cont. ): Compare these four restaurant by using pairwise comparison and find the reason of difference.  

LECTURE 6: TESTS FOR MORE THAN TWO DEPENDENT SAMPLES 7.5 PAIRWISE COMPARISON  Example 4: (Example 2 Cont.): Compare these five Coffees by using pairwise comparison and find the reason of difference.