“Tell Me About Statistics!” vol.55 How to calculate Spearman’s rank correlation coefficient?

on 1月 25, 2024

We explain the calculation the Spearman’s rank correlation coefficient, which we introduced in Vol.54.

First, we find the “tie length,” which is the number of items in the same rank. This implies the same ranking. When we conducted a patient satisfaction survey (5-point evaluation) at a clinic, we obtained the results shown in Table 1. We calculate the “tie length” for “somewhat satisfied” for level of satisfaction with the reception staff’s response.

[Table 1] List of patient satisfaction survey results

“Tie length” is expressed as “t” and can be determined using the following steps:

[1] Sort data on reception staff satisfaction levels in descending or ascending order.

[2] Count the number of instances with the same rank.

For example, there are three instances for “somewhat satisfied (4)” in reception staff satisfaction level, so “t” is 3. (Table 2)

[Table 2] Summary of reception staff satisfaction level

How to calculate Spearman’s rank correlation coefficient

[Formula]
The Spearman’s rank correlation coefficient is expressed as “r.”

* x and y denote the sum of ties for each item.

Tx=(n³-n-x) ÷ 12

Ty=(n³-n-y) ÷ 12

Following [1] and [2], proceed with the calculation as follows.

[3] Determine t³−t and find the total Σ(t³−t) (called the tie total).
In this case, Σ(t³−t) = 90.
[4] Determine the ranking. If there are ties, the average is used as the rank. For example, “somewhat satisfied (4)” is ranked 7th to 9th, so the average rank is 8.

[Table 3] List of reception staff satisfaction level

[5] The ranking of reception staff satisfaction level is “rank ①,” and the ranking of clinic overall satisfaction level is “rank ②.” Let the difference between rank ① and rank ② be “d”.

[6] Find the square of d and find Σd², which will be 103. (Table 4)

[Table 4] List of satisfaction levels for two items