Rank Correlation Using Repeated Ranks

The process of arranging individuals and items regarding proficiency and merit in specific characteristics is known as ranking. In contrast, the number indicating the position of the individuals and items is the assigned rank. Evaluating the correlation between the ranks of individuals and items for two different constraints is known as the process of assessing rank correlation.  

With the help of rank correlation, you can find an association between two distinguishing traits. It is to be noted that Spearman’s rank-order correlation can be considered as the nonparametric version of the Pearson product-moment correlation. Spearman’s correlation coefficient measures the strengths and the direction of the association identified between multiple ranked variables.

The rank correlation coefficient assesses the significance of the relationship between two rankings by measuring the similarities between them.

With the help of rank correlation, you can find an association between two distinguishing traits. It is to be noted that Spearman’s rank-order correlation can be considered as the nonparametric version of the Pearson product-moment correlation. Spearman’s correlation coefficient measures the strengths and the direction of the association identified between multiple ranked variables.

Merits of Rank Correlation Coefficient

• Spearman’s rank correlation coefficient and Karl Pearson’s correlation coefficient can be interpreted similarly
•  The rank correlation coefficient is easy to understand and easier to calculate
• The rank correlation coefficient is the only formula for evaluating the association between qualitative characteristics
• The rank correlation coefficient is considered the nonparametric kind of Karl Pearson’s product-moment correlation coefficient
• Rank correlation coefficient does not require the assumption of normality of the observed sample population

Demerits of Rank Correlation Coefficient

• It is not appropriate for group data
• It is not designed for large-scale observations
• It takes a long time

Spearman’s Rank Correlation

Spearman’s rank correlation coefficient is applied when the data is on an ordinal scale. The Greek letter is used to represent it (rho). Subjectivity data, such as competition scores, can also be used to determine Spearman’s correlation. By assigning ranks to the data, it can be ranked from low to high or vice versa.

The formula for Spearman’s rank correlation coefficient is as follows:

where Di is equal to R1i – R2i

R1i denotes the position of I in the first set of data.

R2i = in the second set of data, the rank of I and

n = number of observation pairs

The strength of a monotonic, increasing or decreasing, the relation between paired data is measured by Spearman’s rank correlation coefficient, which is a statistical measure. It follows Pearson’s interpretation. The monotonic relationship is stronger the closer it is to 1.

Assumptions made in Spearman’s Rank correlation-

• You must use ordinal, interval, or ratio data
• Your data must also be monotonically related because Spearman gauges a monotonic association’s strength
• Essentially, this indicates that if one variable rise (or falls), the other rises with it (or decreases)

Rank Correlation using Repeated Ranks

It is challenging to assign rankings to two or more items with the same value (i.e., a tie). In these circumstances, the objects are assigned an average of the ranks they would have obtained. For example, if two people are ranked equal in the seventh place, they are given the rank [7+8] / 2 = 7.5 each, which is a common rank to be assigned, and the next rank will be 9. If three people are ranked equal in the seventh place, they are given the rank [7+ 8 +9] /3 = 8 each, which is a common rank to be assigned, and the next rank will be 10.

Another example of this can be:

Common ranks are assigned to the repeated elements if two or more persons have the same value. This is the average of the ranks they would have gotten if there had been no repetition. For example, if we have a sequence of 50, 70, 80, 80, 85, 90, the first rank is awarded to 90 as it is the largest value, the second to 85, and so on.

Because both values are the same, the same rank will be assigned, which is the average of the ranks that would have been assigned if there had been no repetition. As a result, both 80 will get an average of 3 and 4, i.e. (Average of 3 and 4 i.e. (3 + 4) / 2= 3.5). 70 receive the 5th position, and 50 receive the 6th rank.

When there are multiple items with the same value, a separate formula is applied in this scenario.

Where mi is the number of times, the ith rank has been repeated.

Conclusion

Correlation is a statistical term that shows how closely two variables vary together. A positive correlation suggests that the two variables increase or decrease in lockstep. The degree to which one variable grows while the other declines are called a negative correlation. The Spearman’s Rank Correlation measures how closely two ranked (ordered) variables are related. When each of their quantities rates two sets of data, this method determines the relationship’s strength and direction.

This method is effective in discovering relationships and the sensitivity of measured outcomes to influencing factors and determining the degree and direction of the correlation between two sets of data when ranked by each of their quantities.