13.12
Spearman's Rank Correlation Test
Spearman's rank correlation test, also known as Spearman's rho, is a nonparametric method for assessing the strength and direction of association betw…
Spearman's rank correlation is a nonparametric test used to find the association between two variables in paired data.
Consider an example of finding a correlation between the eggshell thickness and the hatching order for 35 turtle eggs.
Here, the hatching order cannot be measured parametrically. Instead, it can only be ranked based on its order. Similarly, the eggshell thickness can be ranked in increasing order of thickness measured after hatching.
To conduct the test, we begin with the null hypothesis that there is no correlation between the two variables and the alternative hypothesis that a correlation exists between them.
The quantity Rho-sub-S or Spearman's rank correlation coefficient is a population parameter estimated from the sample statistic R-sub-S using the following equation.
The sample size is more than 30, so the critical value is calculated using the following equation.
Notice the sample statistic is beyond the threshold of the critical value, suggesting that there is a correlation between eggshell thickness and hatching order.
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Q1: When should you use Spearman's rank correlation instead of Pearson's correlation?
Use Spearman's rank correlation when data violates parametric assumptions, such as non-normal distributions or unknown data distributions. It's also ideal for ordinal data or when relationships are non-linear but monotonic. Spearman's test is more robust and flexible than Pearson's, making it suitable for a wider range of scenarios in introduction to nonparametric statistics.
Q2: How does Spearman's rank correlation handle data that cannot be measured parametrically?
Spearman's rank correlation converts raw data into ranks instead of using actual values. For example, hatching order cannot be measured parametrically but can be ranked sequentially. This ranking approach makes the test applicable to ordinal data and situations where precise measurements are impossible or meaningless.
Q3: What does the Spearman's rho coefficient value tell you about the relationship between two variables?
Spearman's rho ranges from -1 to +1, indicating both strength and direction of association. A positive value shows direct correlation, while negative indicates inverse correlation. The magnitude reflects how closely the ranked variables move together, with values closer to ±1 indicating stronger monotonic relationships.
Q4: How is the critical value determined in Spearman's rank correlation test?
For sample sizes larger than 30, the critical value is calculated using a specific equation rather than consulting a standard table. For smaller samples (n ≤ 30), critical values are obtained from standard reference tables. The sample statistic is then compared against this critical value to determine statistical significance.
Q5: What is the efficiency rating of Spearman's rho compared to Pearson's correlation?
Spearman's rho has an efficiency rating of approximately 0.91 when compared to Pearson's correlation under ideal parametric conditions. This means Spearman's test with 100 data pairs yields results comparable to Pearson's with 91 pairs. This rating reflects relative effectiveness in capturing correlation strength, not accuracy or correctness.
Q6: What are the null and alternative hypotheses in a Spearman's rank correlation test?
The null hypothesis states there is no correlation between the two variables being tested. The alternative hypothesis asserts that a correlation exists between them. The test determines whether sample evidence is strong enough to reject the null hypothesis and conclude a significant association.
Q7: Can Spearman's rank correlation detect non-linear relationships between variables?
Yes, Spearman's rank correlation can detect both linear and non-linear monotonic associations, unlike Pearson's correlation which only measures linear relationships. This flexibility makes Spearman's test valuable for identifying associations that increase or decrease consistently but not necessarily in a straight-line pattern.