RANK CORRELATION COEFFICIENT

Is there a significant relationship between ones’ education level and their public speaking skill? Is there a correlation between TOEFL scores and the ability to speak in English? Is there any relatedness between the alumni’s GPAs and their linguistic intelligence? Such problems can be addressed by the use of the Spearman’s Rank Correlation Coefficient. We apply it when we are to test whether two variables of ordinal level are related/associated/correlated.

To test whether there is association between an ordinal-level variable and another ordinal-level variable, we apply the 5-step procedure of hypothesis testing discussed earlier. The test statistic to use is , where d_i is the difference between the ranks for each pair of observations. However, if the sample size is large, i.e. more than 50, the null hypothesis also may be tested by the statistic .
The statistic r_s stated above is called the sample Spearman’s correlation coefficient. The parameter to be tested is ρ_s, the population Spearman’s correlation coefficient. If there is no association/relationship between the variables, ρ_s = 0. Thus, to test whether the association exists, the null hypothesis is ρ_s = 0, meaning “there is no correlation between the variables”. The alternate hypothesis in this case is ρ_s ≠ 0, meaning “there is a correlation between the variables”. (In general, there are three possible expressions for alternate hypotheses, that is ρ_s ≠ 0, ρ_s > 0 and ρ_s < 0. The last two are called directional associations. If we hypothesize that the variables X and Y move in the same direction, i.e. X increases as Y increases and X decreases as Y decreases, the alternate hypothesis is ρ_s > 0. On the other hand, if X and Y move in the opposite direction, i.e. X increases as Y decreases and X decreases as Y increases, then the alternate hypothesis is ρ_s < 0.)

Before we continue, please note that the Spearman’s correlation coefficient r_s or ρ_s ranges from -1 to 1, that is, -1 ≤ r_s ≤ 1 and -1 ≤ ρ_s ≤ 1. A coefficient correlation of 1 or -1 indicates a perfect correlation between the variables. More specifically, 1 indicates a perfect positive correlation while -1 indicates a perfect negative correlation. Zero correlation means that there is no correlation between the variables.

Example

A researcher hypothesizes that there is a relationship between the students’ TOEFL scores and their grades on English. To test the hypothesis, he has collected 12 sample data on the TOEFL scores and the grades on English achieved by the students under study. The results are summarized as follows.

At .05 level of significance, is it reasonable to conclude that there is a relationship between the variables?

Let’s follow the 5-step procedure of hypothesis testing.

Step 1
H₀: There is no relationship between TOEFL scores and the grades on English.
H₁: There is a relationship between TOEFL scores and the grades on English.

Step 2
The level of significance is α = .05.
Step 3
The test statistic is .
Step 4
Referring to The Table of Critical Values of the Spearman’s Ranked Correlation Coefficient with n = 12 and α = .05 we obtain the critical value .587. Thus, the decision rule is: reject the null hypothesis if r_s < – .587 or r_s > .587.

Step 5
The sampling result is as given above. What we have to do next is to calculate the test statistic r_s.

(Note: in the table above, d_i = rank(X_i) – rank(Y_i).)

As the table shows, . This results in .

In Step 4 it is stated that we have to reject the null hypothesis on condition that r_s < – .587 or r_s > .587. Since .827 > .587, we reject the null hypothesis and conclude that there is a significant relationship between TOEFL scores and the grades on English.