Is homoscedasticity an assumption for Pearson's correlation?

I'm running correlation analysis in SPSS between my variables and I'm starting by checking the assumptions to run Pearson's correlation (r). I'm confused as to whether or not homoscedasticity is one of the assumptions for Pearson's correlation. In some places online it says that homoscedasticity isn't an assumption for Pearson's correlation, but in other places it says that it is an assumption.

Also, in the case that it is an assumption, is there an easier way to test for homoscedasticity in SPSS than having to observe every single scatterplot between all my variables (I have a ton)? Is there some sort of test that tests for homoscedasticity? For example, when testing for normality there is the Shapiro-wilk test. Is there something like that for homoscedasticity?

Thanks,

FBH

1 Answer

Please forgive me for being a tad pedantic here at the start of this answer. But, Pearson's correlation is a defined quantity. As such, it is what it is, and the value itself does not have any assumptions. In particular, this is just the scaled covariance of any 2 variables.

However, if you wish to test if the correlation (of a given population) is some value based on a sample, then there are assumptions that this test would require. Such a test would assume a bivariate normal distribution among the variables.

So, homoscedasticity is indirectly an assumption for such distributions. But, the real reason people might argue that this is an assumption is because the correlation is often used as a measure of the linear relationship. And, borrowing from regression, it would be an assumption for regression analyses (and then by extension, might be assumed for correlation analyses).

• +1 One can of course construct a test for a Pearson correlation under other assumptions ... or indeed conduct a nonparametric test. Even with a bivariate Gaussian assumption this is only strictly required under the null (about which a sample might not be particularly relevant); unless we're trying to compute power, we might well consider a broader class of alternatives. Apr 8, 2022 at 2:36
• Thanks for ask your comments... But I'm still a little bit confused... If I check for normality, linearity, outliers, and make sure my data are on interval or ratio scale, and see that my variables are from normal distributions, have no outliers, are all measured on an interval or ratio scale, and all relationships between them are linear.... then is it fine to calculate Pearson's correlation to measure correlations between the variables? Or do I need to also manually check to make sure that all relationships between variables are homoscedastic first before calculating Pearson's r? Thanks Apr 11, 2022 at 21:49
• Please clarify what you mean by "all relationships between variables"...as this suggests you are talking about 3 or more variables...and correlation is defined between 2 (and only 2) variables. Apr 12, 2022 at 12:57
• Sorry, my wording was incorrect. I have two variables, each with several levels, and I'm correlating each of the levels of the first variable with each of the levels of the second variable. Apr 15, 2022 at 22:33
• @GreggH , if I transform the data (e.g. log) and now follows a line between x and y, i should apply Pearson to the transformed values, right? Dec 11, 2023 at 22:04