KENDALL’S COEFFICIENT OF CONCORDANCE () PRETEST- Problem Kendall’s Coefficient of Concordance () Jegie 177 Problem Three judges rated eight essays with the following results. Calculate the coefficient of concordance for these data. Judge Essay 1 2 3 1 2 3 4 5 6 7 8 8 6 4 1 3 2 5 7 7 5 6 2 3 1 4 8 8 6 5 1 2 3 4 7 Solve the problem by getting the following requirements: 1). Specific Problem 2). Null Hypothesis 3). Alternative Hypothesis 4). Statistical Method Used 5). Level of Significance 6). Region of Rejection 7). Computation of the value of W 8). Decision 9). Conclusion PRETEST- Problem Kendall’s Coefficient of Concordance () I - Jegie 178 Specific Problem Is there an agreement among the three judges in rating the eight essays? II - Null Hypothesis There is no agreement among the three judges in rating the eight judges. III - Alternative Hypothesis There is an agreement among the three judges in rating the eight judges. IV - Level of Significance: = 0.01 V - Statistical Method Used: Kendall’s Coefficient of Concordance W VI - Rejection Region Df = n – 1 VII- Df =7 X2 = 18.475 Computation W = 0.94 VIII- Df = 8 – 1 The computed X2 = 19.74 Decision Since the computed value of W = 0.94. the verbal interpretation is very high correlation and the computed X2 = 19.74 is greater than the 2 X = 18.475 at 0.01 level of significance, therefore accept the alternative hypothesis. IX- Conclusion There is an agreement among the three judges in rating the eight judges. KENDALL’S COEFFICIENT OF CONCORDANCE Jegie 179 KENDALL’S COEFFICIENT OF CONCORDANCE () Definition ❖ If we wish to determine the relationship among three or more sets of ranks, KENDALL’S COEFFICIENT OF CONCORDANCE () is used. Formula 12 D2 = m2 (n) ( n2 - 1 ) Where: m n D X 12 = = = = = number of judges number of individuals to be judged (Rank per row - x) is the average is constant To check: Total number of ranks = m ( n ) ( n + 1 ) 2 Note: Perfect agreement is indicated by by = 0. = 1 and lack of agreement by To test the significance: Formula: X2 = m (2-1) Where: m n Degrees of Freedom = number of judges = number of projects being rated = Kendall’s coefficient of concordance df = n - 1 KENDALL’S COEFFICIENT OF CONCORDANCE Jegie 180 Problem Calculate the value of the coefficient of concordance by using the data consisting of the ranking of 10 projects by five judges. Individual Project Project Project Project Project Project Project Project Project Project Project 1 2 3 4 5 6 7 8 9 10 Judge 1 2 1 3 5 4 7 6 8 9 10 Judges Ranks Judge 2 Judge 3 1 3 4 5 2 8 6 7 10 9 2 1 4 5 6 3 8 7 10 9 Judge 4 Judge 5 3 2 1 5 7 4 6 8 9 10 Solution I Specific Problem Is there an agreement among these five judges in the ranking of the 10 projects? II Null Hypothesis There is no agreement among these five judges in the ranking of the 10 projects. III Alternative Hypothesis There is an agreement among these five judges in the ranking of the 10 projects. IV Level of Significance = 0.05 4 2 3 1 6 7 5 9 8 10 KENDALL’S COEFFICIENT OF CONCORDANCE Jegie 181 V Statistical Method Used Kendall Coefficient of Concordance VI Rejection Region Degrees of freedom (df) (df) = = = n - 1 10 - 1 9 The critical value at 5 % level of significance: VII Individual ProJect P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 X2 0.05 = 16.919 Computation Judges Ranks J1 J2 2 1 3 5 4 7 6 8 9 10 1 3 4 5 2 8 6 7 10 9 J3 2 1 4 5 6 3 8 7 10 9 J4 3 2 1 5 7 4 6 8 9 10 J5 4 2 3 1 6 7 5 9 8 10 Sum of Ranks (1) ( Difference) D (4) 12 9 15 21 25 29 31 39 46 48 12-27.5 =15.5 9-27.5 =18.5 15-27.5 =12.5 21-27.5 = 6.5 25-27.5 = 2.5 29-27.5 = 1.5 31-27.5 = 3.5 39-27.5 =11.5 46-27.5 =18.5 48-27.5 =20.5 Ranks = 275 (2) n = 10 Get the Sum of Ranks per project or per row. P1 P2 = 2+1+2+3+4 = 1+3+1+2+2 = = 12 9 D2 (5) 240.25 342.25 156.25 42.25 6.25 2.25 12.25 132.25 342.25 420.25 D2 =1,696.50 Procedure 1. (D x D) ( Ranks – Mean) etc. KENDALL’S COEFFICIENT OF CONCORDANCE Jegie 182 ) 2. Sum all the ranks of the projects ( Ranks) Ranks = 12 + 9 + 15 + 21 + 25 + 29 + 31 + 39 + 46 + 48 Ranks = 275 3. Find the mean by dividing ( Ranks) by the total number of projects Mean Ranks = Total Number of Projects being rated 4. 5. Mean = 275 10 Mean = 27.50 Get the difference (D) by subtracting the Mean from the total ranks per project as shown in Column with number (4). (Disregard the negative sign. ) D = Sum of Ranks - Mean D1 D2 D3 Etc. = = = 12 – 27.50 9 – 27.50 15 – 27.50 = = = 15.50 18.50 12.50 Square the difference ( D2 ) as shown in column with number (5). (D1)2 = (15.50) (15.50) = 240.25 (D2)2 = (18.50) (18.50) = 342.25 (D3)2 = (12.50) (12.50) = 156.25 KENDALL’S COEFFICIENT OF CONCORDANCE Jegie 183 6. Find the total of D2 ( D2 ) . D2 = 240.25 + 342.25 + 156.25 + 42.25 + 6.25 + 2.25 + 12.25 + 132.25 + 342.25 + 420.25 D2 = 1,696.50 7. Substituting the computed values to the given formula = 12 D2 m (n) ( n2 - 1) 2 Where: m n D2 = number of judges = no. of projects to be judged = 1696.50 = 12 ( 1 696.50 ) (5)2 ( 10 ) ( 10 2 - 1 ) = 12 ( 1 696.50 ) 25 ( 10 ) ( 100 -1 ) = 20 358 25 (10) (99) = 20 358 24 750 = 0.82 = = = = 5 10 20 358 25 (10) (99) 0.82 It means there is a high agreement among these five judges in the ranking of the ten projects since 0.82 falls between 0.71 to 0.90 as shown on page 27 which means high correlation or marked relationship. 8. Check your solution by getting the Total sum of ranks To check: Total sum of ranks = m ( n ) ( n + 1 ) 2 Total sum of ranks = (5) (10) (10 + 1) 2 KENDALL’S COEFFICIENT OF CONCORDANCE Jegie 184 Total sum of ranks = (5) (10) (11) 2 Total sum of ranks = Total sum of ranks = 275 ✓ Note: page 26. 550 2 is same as rs in the analization of the interpretation. Please refer to ❖ To test the significance: Formula: X2 = m (n-1) m n number of judges number of projects being rated Kendall’s coefficient of concordance Where: = = = = = = 5 10 0.82 9. Substitute the values to the given formula. X2 = m (n-1) W X2 = (5) (10-1) (0.82) X2 = (5) (9) (0.82) X2 = 36.90 VIII Decision Since the computed value X2 = 36.90 is greater than X2 0.05 = 16.919, therefore, accept the alternative hypothesis that it is significant and there exist relationship in the judging of the ten projects by the five judges. IX Conclusion There is a significant agreement among these five judges in the ranking of the ten projects. POST TEST- Problem Kendall’s Coefficient of Concordance (W) Jegie 185 Problem Four judges (parole board members) rank eight convicts on “parole readiness.” By using the coefficient of concordance, indicate the degree of consistency of the judges. Convict 1 2 3 4 5 6 7 8 1 Judge 2 3 4 1 2 3 4 5 6 7 8 1 4 3 2 6 5 7 8 1 3 2 4 5 6 8 7 1 2 4 3 5 7 6 8 Solve the problem by getting the following requirements: 1). Specific Problem 2). Null Hypothesis 3). Alternative Hypothesis 4). Statistical Method Used 5). Level of Significance 6). Region of Rejection 7). Computation of the value of 8). Decision 9). Conclusion POST TEST-Answers Kendall’s Coefficient of Concordance (W) I - Jegie 186 Specific Problem Is there any consistency in the ranking of the four judges in the parole readiness of the eight convicts? II - Null Hypothesis There is no consistency in the ranking of the four judges in the parole readiness of the eight convicts. III - Alternative Hypothesis There is consistency in the ranking of the four judges in the parole readiness of the eight convicts. IV - Level of Significance: Assume = 0.01 V - Statistical Method Used: VI - Rejection Region Use Kendall Coefficient of Concordance Degrees of freedom (df) = n – 1 = 8 – 1 = 7 The critical value at 0.01 level of significance X2 0.01 = 18.475. Appendix D VII- Computation = 0.92 VIII- Please refer to X2 obs = m(n–1)(w) = 4 ( 8 – 1 ) ( 0.92 ) = 25.76 Decision Since the observed absolute value of chi-square x 2 obs = 25.76 is greater than the critical value at 0.01 level of significance x2 0.01 = 18.475, therefore accept the alternative hypothesis. IX – Conclusion There is a significant relationship on consistency in the ranking of the four judges in the parole readiness of the eight convicts. PRACTICE Problem Kendall’s Coefficient of Concordance (W) Jegie 187 Three judges rated eight essays with the following results. Test if there is a significant relationship in judging the eight essays by the three judges. Level of significance is 0.01. Essay 1 2 3 4 5 6 7 8 JUDGE A B C 8 6 4 1 3 2 5 7 7 5 6 2 3 1 4 8 8 6 5 1 2 3 4 7 Please be guided by the nine steps in solving the problem. 1). Specific Problem 2). Null Hypothesis 3). Alternative Hypothesis 4). Statistical Method Used 5). Level of Significance 6). Region of Rejection 7). Computation of the Value of w 8). Decision 9). Conclusion PRACTICE Problem - Answers Kendall’s Coefficient of Concordance (W) I - Jegie 188 Specific Problem Is there an agreement among these three judges in the ranking of the eight essays? II - Null Hypothesis There is no significant agreement among the three judges in the ranking of the eight essays. III - Alternative Hypothesis There is an agreement among the three judges in the ranking of the eight essays. IV - Level of Significance: assume = 0.05 V - Statistical Method Used Use Kendall Coefficient of Concordance VI - Rejection Region Degrees of freedom (df) = n - 1 = 8 –1 =7 The critical value at 5% level of significance X2 VII- = 3.355 for two-tailed test Computation = 0.94 VIII- 0.05 x2 = 19.74 Decision Since the computed value of x2 = 19.74 is greater than x20.05 = 14.067, therefore reject the null hypothesis and accept the alternative hypothesis. IX – Conclusion There is an agreement among the three judges in the ranking of the eight essays.