What are concordant and discordant pairs?
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Use concordant and discordant pairs to describe the relationship between pairs of observations. To
calculate the concordant and discordant pairs, the data are treated as ordinal, so ordinal data should be
appropriate for your application. The number of concordant and discordant pairs are used in
calculations for Kendall's tau, which measures the association between two ordinal variables.
The procedure compares the classifications for two variables (for example, X and Y) on the same two
items. If the direction of classifications is the same, the pairs are concordant. For example, both X and
Y rate item 1 higher than item 2. If the direction of the classification is not the same, the pair is
discordant. For example, X rates item 1 higher than item 2 but Y rates item 1 lower than item 2.
Specifically, there are two sets of paired observations – (Xi, Yi) and (Xj, Yj):
The pairs are concordant if Xi > Xj and Yi > Yj or Xi < Xj and Yi < Yj
The pairs are discordant if Xi > Xj and YI < Yj or Xi < Xj and Yi > Yj
For example, judges at a singing competition rate 5 contestants on a scale from 1 to 5, where 1 is the
best.
Judge Robert Juan Sophia Helena Marie
Judge X 3 1 2 4 5
Judge Y 1 2 3 5 4
Arranging the ratings in order helps to determine which pairs are concordant and which are discordant.
Judge Juan Sophia Robert Helena Marie
Judge X 1 2 3 4 5
Judge Y 2 3 1 5 4
Compare the ratings for Juan and Sophia to determine if this pair of the judges' ratings are concordant
or discordant. Judge X's rating for Juan (1) is less than their rating for Sophia (2); Judge Y's rating for
Juan (2) is less than their rating for Sophia (3). Because both judges' ratings for Juan are less than their
ratings for Sophia, this pair of ratings is concordant. The following shows the comparison of all pairs
of ratings that include Juan:
Juan vs Sophia: XJ, XS = 1, 2 and YJ, YS= 2, 3; 1 < 2 and 2 < 3 so the pair is concordant.
Juan vs Robert: XJ, XR = 1, 3 and YJ, YR = 2, 1; 1 < 3 and 2 > 1 so the pair is discordant.
Juan vs Helena: XJ, XH = 1, 4 and YJ, YH= 2, 5; 1 < 4 and 2 < 5 so the pair is concordant.
Juan vs Marie : XJ, XM = 1, 5 and YJ, YM = 2, 4; 1 < 5 and 2 < 4 so the pair is concordant.
Then, compare the remaining pairs:
Sophia vs Robert: XS, XR = 2, 3 and YS, YR = 3, 1; 2 < 3 and 3 > 1 so the pair is discordant.
Sophia vs Helena: XS, XH = 2, 4 and YS, YH= 3, 5; 2 < 4 and 3 < 5 so the pair is concordant.
Sophia vs Marie: XS, XM = 2, 5 and YS, YM= 3, 4; 2 < 5 and 3 < 4 so the pair is concordant.
Robert vs Helena: XR, XH = 3, 4 and YR, YH= 1, 5; 3 < 4 and 1 < 5 so the pair is concordant.
Robert vs Marie: XR, XM = 3, 5 and YR, YM= 1, 4; 3 < 5 and 1 < 4 so the pair is concordant.
Helena vs Marie: XH, XM = 4, 5 and YH, YM= 5, 4; 4 < 5 and 5 > 4 so the pair is discordant.
7 of the pairs are concordant. 3 of the pairs are discordant.
Concordant = Having the same characteristics
KENDALL’S COEFFECIENT OF CONCORDANCE
Sample scenario:
“A survey was conducted about a popularity rating of possible senatorial candidates.”
The survey was administered by 3 persons
Senatorial Candidate Ranking from respondent 1 Ranking from respondent 2
A 5 4
B 4 5
C 3 1
D 6 3
E 1 2
F 2 6
G 7 7
Is there an agreement in the ranking of senatorial candidates from two respondents?
What is the significance of knowing if there is an agreement or not?
It is used to determine the degree of agreement among the raters (Judges) in the ranking system.
The Friedman Test is a version of the Repeated-Measures ANOVA that can be performed on
ordinal(ranked) data.
Ordinal data is displayed in the table below. Is there a difference between Weeks 1, 2, and 3 using
alpha = 0.05?
Figure 1.
Let's test to see if there are any differences with a hypothesis test.
Steps for Friedman Test
1. Define Null and Alternative Hypotheses
2. State Alpha
3. Calculate Degrees of Freedom
4. State Decision Rule
5. Calculate Test Statistic
6. State Results
7. State Conclusion
1. Define Null and Alternative Hypotheses
Figure 2.
2. State Alpha
alpha = 0.05
3. Calculate Degrees of Freedom
df = k – 1, where k = number of groups
df = 3 – 1 = 2
4. State Decision Rule
We look up our critical value in the Chi-Square Table and find a critical value of plus/minus 5.99.
Figure 3.
5. Calculate Test Statistic
First, we must rank the scores of every subject, as shown below in red:
Figure 4.
We then replace our original values with the rankings we've just found:
Figure 5.
A Chi-Square value is then calculated using the sums of the ranks of each group:
Figure 6.
6. State Results
Figure 7.
Do not reject the null hypothesis.
7. State Conclusion
Figure 8.
