Consider the formula for rank correlation coefficient.
Correction term for ρ when ranks are tied
Consider
the formula for rank correlation coefficient. We have
If there is a tie involving m items, we have to add
to the term D2 in ρ. We
have to add as many terms like (m3 – m) / 12 as there are ties.
Let
us calculate the correction terms for certain values of m. These are provided
in the following table.
Illustrative examples: If there
is a tie involving 2 items, then the correction term is 0.5 If there are 2 ties involving 2 items each, then the correction term is
0.5 + 0.5 = 1 If there are 3 ties with 2 items each, then the correction term is 0.5 +
0.5 + 0.5 = 1.5 If there
is a tie involving 3 items, then the correction term is 2 If there are 2 ties involving 3 items each, then the correction term is
2 + 2 = 4 If there is a tie with 2 items and another tie with 3 items, then the
correction term is 0.5 + 2 = 2.5 If there are 2 ties with 2 items each and another tie with 3 items, then
the correction term is 0.5 + 0.5 + 2 = 3
Problem 9 : Resolving ties in ranks The
following are the details of ratings scored by two popular insurance schemes.
Determine the rank correlation coefficient between them.Solution: From the given values, we have to
determine the ranks. Step 1. Arrange
the scores for Insurance Scheme I in descending order and rank them as
1,2,3,…,8.
The score 87 appears twice. The
corresponding ranks are 3, 4. Their average is (3 + 4) / 2 = 3.5. Assign this
rank to the two equal scores in Scheme I. The score 80 appears twice. The
corresponding ranks are 7, 8. Their average is (7 + 8) / 2 = 7.5. Assign this
rank to the two equal scores in Scheme I. The
revised ranks for Insurance Scheme I are as follows:Step 2. Arrange
the scores for Insurance Scheme II in descending order and rank them as
1,2,3,…,8.The score
57 appears thrice. The corresponding ranks are 4, 5, 6. Their average is (4 + 5 + 6) / 3 = 15 / 3 = 5. Assign this rank to the
three equal scores in Scheme II. The
revised ranks for Insurance Scheme II are as follows:Step 3. Calculation of D2: Assign
the revised ranks to the given pairs of values
and calculate D2 as follows:Inference: It
is inferred that the two insurance schemes are highly, positively correlated.
Tags : Research Methodology - Correlation And Regression Analysis
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