Spearman Rho vs ANOVA

Question

Assume that we have an ordinal scale IV (e.g. EDUCATION LVL) and an interval scale DV (or for Anova, response variable) (e.g. JOB_INVOLVEMENT)

As i know, Spearman rho gives us (a) whether there is a sig. association b/w IV & DV, and (b) magnitude of the relationship.

On the other hand, Anova (& post-hoc tests) results give us whether there are any sig. differences among the means of DV for each treatments (categories of IV). So it doesnt only explain the association by determining which levels of IV (PRIMARY&SECONDARY/BACHELOR/MASTER&PHD+) have sig effect on DV (JOB INVOLVEMENT) , but also the comparisons of their effects on DV.

It seems that Anova provides what Spearman rho does, even beyond that. The question is: Do we need to conduct Spearman rho rather than Anova in any particular case?

Answer 1

First, Spearman's rho, like other correlation measures, does not posit a dependent and independent variable. ANOVA does. So, they answer different questions.

Second, Spearman's rho (and other correlation measures) may be more easily understood than the results of ANOVA.

Answer 2

With Anova, you are assuming that residuals have a normal distribution. If deviation from Normality is a serious issue (see for example a discussion here: ANOVA assumption normality/normal distribution of residuals), you can't draw valide inference (i.e. your confidence intervals and p-values are not valid).

With Spearman's correlation, you are measuring the direction of association between the two variables, but just using rankings. While this makes your results robust to outliers, your inference becomes less efficient (in case the normality assumption of residuals is not violated). The big issue I see with using Spearman's correlation is that you have an ordinal variables, thus few distinct values. While a correction for ties have been proposed for Spearman's correlation, if you have too few distinct values you have "jumps" from a value to another (i.e. transformed values are not equally spaced anymore), thus the usual advantage of robustness to outliers of Spearman's correlation does not hold anymore (or is at least questionable; see here: http://sci.stat.consult.narkive.com/GN2zNUes/which-correlation-coefficient-in-case-of-many-ties for a discussion on this issue).

To sum up, I would either use ANOVA (possibly after transforming your data if there are issues with the distribution of residuals) or, as a non-parametric alternative, use a Kruskal-Wallis test. I would consider the Spearman's correlation coefficient only if education is measured in years (so that you have a reasonable amount of distinct values).

Please notice that, with Spearman's correlation, you are treating (the ranked transformation of) education as a continuous variables, so you are answering the question of whether more/less educated people are more/less involved; on the contrary, with ANOVA and Kruskal-Wallis you are comparing groups of educational level regardless of their ranking.

Finally, I add that an alternative to Spearman's correlation (Kendall's Tau) is nowadays considered better if I well understand (Kendall Tau or Spearman's rho?) even if I think Spearman's correlation is still rather popular (maybe because its interpretation seems easier).