Hypothesis tests using Spearman's coefficient

Hypothesis tests using Spearman’s coefficient

Understanding Spearman’s Coefficient

Spearman’s rank correlation coefficient, denoted as ρ, is a non-parametric measure of statistical dependence between two variables.
It measures the strength and direction of association between two ranked variables, providing a value between -1 and +1.
A Spearman correlation of +1 indicates a perfect positive association of ranks, -1 a perfect negative association, and 0 indicates that there is no association between the ranks.

Hypothesis Testing with Spearman’s Coefficient

In a hypothesis test using Spearman’s rank correlation, the null hypothesis (H0) typically states that there is no correlation between the two variables.
The alternative hypothesis (H1) can be two-sided (the variables are not uncorrelated) or one-sided (the variables are positively/negatively correlated).

Calculating Spearman’s Rank Correlation

To calculate Spearman’s coefficient, first rank the data for each variable separately.
Then calculate the difference d between the ranks for each observation, square these differences and sum them (Σd²).
The coefficient is then calculated using the formula 1 - ( (6Σd²) / (n(n²-1)) ), where n is the number of observations.

Making a Decision: Critical Values

After calculating the Spearman’s rank coefficient, it’s necessary to compare it with a critical value which depends on the sample size and the chosen level of significance.
If the absolute value of the rank correlation coefficient is greater than the critical value, then the null hypothesis is rejected.

Assumptions for Spearman’s Test

The assumptions for conducting a Spearman’s test are that data is ordinal (i.e., can be ranked), the variables are related monotonically, and the data values are paired and come from the same population.
Violations of these assumptions may lead to erroneous conclusions about the correlation.

Advantages of Spearman’s Coefficient

An advantage of using Spearman’s rank correlation coefficient is that it is less sensitive to extreme values or outliers than the Pearson correlation coefficient.
It is also suitable for use with both continuous and discrete ordinal variables.

Limitations of Spearman’s Coefficient

As a non-parametric test, while Spearman’s coefficient makes fewer assumptions, it may be less powerfully discriminatory than parametric methods like Pearson’s correlation.
It may not correctly estimate the correlation when the relationship between variables is not monotonic.

Spearman’s Coefficient in Practice

The application of Spearman’s rank correlation test can be seen in various fields where one needs to analyse the relationship between variables that do not necessarily require a linear relationship.
Examples include psychology, sociology, epidemiology, and market research.
It’s a powerful tool for identifying non-linear relationships that other correlation coefficients may fail to detect.