Rank Randomization for Association (Spearman's ρ )

David M. Lane

Prerequisites

Pearson's r, Randomization Test for Pearson's r

Learning Objectives
1. Compute Spearman's ρ
2. Test Spearman's ρ for significance

The rank randomization test for association is equivalent to the randomization test for Pearson's r except that the numbers are converted to ranks before the analysis is done. Table 1 shows 5 values of X and Y. Table 2 shows these same data converted to ranks (separately for X and Y).

Table 1. Example data.

X Y
1.0 1.0
2.4 2.0
3.8 2.3
4.0 3.7
11.0 2.5

Table 2. Ranked data.

X Y
1 1
2 2
3 3
4 5
5 4

The approach is to consider the X variable fixed and compare the correlation obtained in the actual ranked data to the correlations that could be obtained by rearranging the Y variable ranks. For the ranked data shown in Table 2, the correlation between X and Y is 0.90. The correlation of ranks is called "Spearman's ρ."

Table 3. Ranked data with correlation of 1.0.

X Y
1 1
2 2
3 3
4 4
5 5

There is only one arrangement of Y that produces a higher correlation than 0.90: A correlation of 1.0 results if the fourth and fifth observations' Y values are switched (see Table 3). There are also three other arrangements that produce an r of 0.90 (see Tables 4, 5, and 6). Therefore, there are five arrangements of Y that lead to correlations as high or higher than the actual ranked data (Tables 2 through 6).

Table 4. Ranked data with correlation of 0.90.

X Y
1 1
2 2
3 4
4 3
5 5

Table 5. Ranked data with correlation of 0.90.

X Y
1 1
2 3
3 2
4 4
5 5

Table 6. Ranked data with correlation of 0.90.

X Y
1 2
2 1
3 3
4 4
5 5

The next step is to calculate the number of possible arrangements of Y. The number is simply N!, where N is the number of pairs of scores. Here, the number of arrangements is 5! = 120. Therefore, the probability value is 5/120 = 0.042. Note that this is a one-tailed probability since it is the proportion of arrangements that give a correlation as large or larger. The two-tailed probability is 0.084.

Since it is hard to count up all the possibilities when the sample size is even moderately large, it is convenient to have a table of critical values.

From the table linked to above, you can see that the critical value for a one-tailed test with 5 observations at the 0.05 level is 0.90. Since the correlation for the sample data is 0.90, the association is significant at the 0.05 level (one-tailed). As shown above, the probability value is 0.042. Since the critical value for a two-tailed test is 1.0, Spearman's ρ is not significant in a two-tailed test.