Analyze data using Spearman’s rank correlation coefficient (rho)
- It measures the relationship between two logically related variables. Like the conventional correlation coefficient (r), Spearman rho can have any value between -1 to +1. A Spearman correlation of 1 results when the two variables being compared are monotonically related, even if their relationship is not linear. In a monotonic relationship, the variables tend to move in the same relative direction, but not necessarily at a constant rate. Monotonic relationship can be nonlinear. In a linear relationship, the variables move in the same direction at a constant rate.
CLICK HERE to refer to the diagrams of Linear, Nonlinear and monotonic relationships.
In Statistics, Correlation is largely a measure of an association between variables. In logically correlated data, the change in the magnitude of 1 variable is related to a corresponding change in the magnitude of another variable, either in the same (positive correlation: High-High, Low-Low) or in the opposite (negative correlation: Low-High or High-Low) direction. Very often, the term correlation is used in the context of a linear relationship between 2 continuous variables and expressed as Pearson product-moment correlation. For monotonically (nonlinear) distributed continuous ordinal data(indicate the order or rank of things) or for data with relevant outliers, Spearman rank correlation can be used as a measure. Spearman rho refers to the ranked values rather than the original measurements.
Variable A (arranged in an ascending order-smallest to largest) | Rank (R1) (assigned after arranging in order) | Variable B | Rank (R2) | Difference in rank (R1-R2)=d | d² |
---|---|---|---|---|---|
200 | 10 | 6.2 | 1 | +9 | 81 |
1100 | 9 | 5.1 | 2 | +7 | 49 |
1900 | 8 | 3.1 | 4 | +4 | 16 |
2400 | 7 | 4.6 | 3 | +4 | 16 |
3100 | 6 | 2.5 | 5 | +1 | 1 |
3900 | 5 | 1.5 | 7 | -2 | 4 |
4200 | 4 | 1.8 | 6 | -2 | 4 |
5300 | 3 | 0.4 | 10 | -7 | 49 |
5900 | 2 | 1.3 | 8 | -6 | 36 |
6500 | 1 | 0.5 | 9 | -8 | 64 |
Total d² = 320 |
Spearman rank
where n = number of paired observations
There are (n) degrees freedom (10)
Significance Level
Compare your result to the critical values for Spearman's Rank correlation (ignoring + / -). If rho is greater than or equal to the critical value, then there is a significant correlation and the null hypothesis can be rejected.Number of pairs of measurement (n) | p = 0.05 (95%) (+ or -) | p = 0.01 (99%) (+ or -) |
---|---|---|
5 | 1 | |
6 | 0.886 | 1.000 |
7 | 0.786 | 0.929 |
8 | 0.738 | 0.881 |
9 | 0.683 | 0.833 |
10 | 0.648 | 0.818 |
11 | 0.623 | 0.794 |
12 | 0.591 | 0.780 |
13 | 0.566 | 0.745 |
14 | 0.545 | 0.716 |
15 | 0.525 | 0.689 |
16 | 0.507 | 0.666 |
17 | 0.490 | 0.645 |
18 | 0.476 | 0.625 |
19 | 0.462 | 0.608 |
20 | 0.450 | 0.591 |
25 | 0.400 | 0.526 |
30 | 0.364 | 0.478 |
35 | 0.336 | 0.442 |
40 | 0.314 | 0.413 |