Hypothesis Tests using Spearman’s Coefficient

Introduction

A hypothesis test utilising Spearman’s coefficient is a statistical method to determine the probability that a correlation exists between two sets of data.
This method leverages the Spearman’s Rank Correlation Coefficient to reach a statistical conclusion.
Applying this test can validate or reject a research assumption about the relationship between variables.

A null hypothesis (H0) typically states that there is no correlation between the two sets of data points, i.e. ρ = 0.
The alternative hypothesis (H1) commonly suggests that there is a correlation between the data sets, i.e. ρ ≠ 0. Although, based on the research question, one may also set the alternative hypothesis to be more specific predicting a positive or negative correlation.
It is crucial to set up the hypotheses carefully as these form the premise of the test results.

After the hypotheses are framed, observe the data set and assign ranks to each element. Use these ranks to calculate differences and squared differences.
Using Spearman’s formula, calculate the coefficient.
Use the obtained Spearman coefficient along with the sample size to find the corresponding t-value.
Compare this t-value against the critical t-value for the desired significance level (commonly 5% or 1%) and degrees of freedom (n-2).

If the computed t-value is greater in magnitude than the critical t-value, reject the null hypothesis in favour of the alternative. It indicates that there is significant evidence to believe that the variables are correlated.
If the computed value is less than the critical value, we do not have enough evidence to reject the null hypothesis. The potential correlation between the variables is considered not statistically significant.
Always state the conclusion in context of the original research question or problem.

It’s crucial to note Spearman’s test assumes data sets are ordinal, and each variable pair is monotonically related.
It’s unable to accurately determine correlations in data that contains tied ranks.
It also may not perform well when the data set lacks robustness, i.e. when the sample size is too small.
Assumptions violation or incorrect application can lead to misinterpretation of results. Always check assumptions before proceeding with the test.

Hypothesis tests with Spearman’s Coefficient are widely used in the field of social sciences, research, and psychology among others.
For example, you might use a hypothesis test with Spearman’s coefficient to confirm or reject the assumption that there’s a correlation between people’s subjective ratings of a film’s quality and the film’s box office earnings.