# 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.

• 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.