What particular measure of correlation would be most appropriate for use with two variables measured at an ordinal level?
Spearman's Rho
Ordinal data comes in two general types. The first type of ordinal data is often called
continuous ordinal data. This type of data is found in situations where the researcher has a
relatively detailed set of rankings where the cases have been ranked in a relatively broad range of
categories and where the number or tied rankings is relatively small. Researchers working with
this type of ordinal data use Spearman's Rho to determine the level of correlation that exists
between variables.
When dealing with variables that are measured at an ordinal level, the most appropriate measure of correlation is the Spearman's rank correlation coefficient (or Spearman's rho). Spearman's rho assesses the strength and direction of the monotonic relationship between two ordinal variables. Unlike other correlation measures, the Spearman's rho does not assume a linear relationship between the variables. Instead, it looks at the ranks of the observations, taking into account the order in which they appear.
When dealing with variables measured at an ordinal level, the most appropriate measure of correlation to use is Spearman's rank correlation coefficient (rho or rs). Here's how you can calculate it:
1. Assign ranks to each value in both variables. For example, if you have two variables X and Y, rank the values of X from lowest to highest and rank the values of Y from lowest to highest. If there are ties, assign the average rank to the tied values.
2. Calculate the difference between the ranks of each corresponding pair of X and Y values.
3. Square the differences calculated in step 2.
4. Calculate the sum of squared differences.
5. Use the following formula to calculate Spearman's rank correlation coefficient (rho):
rho = 1 - (6 * sum of squared differences) / (n * (n^2 - 1))
In this formula, n represents the number of pairs of values.
Spearman's rank correlation coefficient ranges from -1 to +1. A positive value indicates a positive monotonic relationship (as one variable increases, the other tends to increase), while a negative value indicates a negative monotonic relationship (as one variable increases, the other tends to decrease). A value of zero indicates no monotonic relationship.
Please note that Spearman's rank correlation coefficient measures the strength and direction of monotonic relationships between variables, but it does not provide information about the linearity or causality of the relationship.