Solved – Mantel test vs. Pearson’s correlation coefficient

correlationpearson-r

Why is it correct not to measure the correlation between two distance matrices by calculating the Pearson's correlation coefficient between all distance pairs but to use the Mantel test instead?

Edit:
Wikipedia says that because "distances are not independent of each other – since changing the "position" of one object would change n-1 of these distances (the distance from that object to each of the others) – we can not assess the relationship between the two matrices by simply evaluating the correlation coefficient between the two sets of distances and testing its statistical significance. The Mantel test deals with this problem."

This argument is not clear to me. Why is dependence between the distances a problem?

Best Answer

If you have $n$ objects you will have $\frac{n(n-1)}2$ distances. So, unless $n<4$, you always have more distances than objects. This artificially inflates the number of independent samples when using Pearson's correlation.

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Solved – Reporting coefficient of determination using Spearman’s rho

Pearson's r and Spearman's rho are both already effect size measures. Spearman's rho, for example, represents the degree of correlation of the data after data has been converted to ranks. Thus, it already captures the strength of relationship.

People often square a correlation coefficient because it has a nice verbal interpretation as the proportion of shared variance. That said, there's nothing stopping you from interpreting the size of relationship in the metric of a straight correlation.

It does not seem to be customary to square Spearman's rho. That said, you could square it if you wanted to. It would then represent the proportion of shared variance in the two ranked variables.

I wouldn't worry so much about normality and absolute precision on p-values. Think about whether Pearson or Spearman better captures the association of interest. As you already mentioned, see the discussion here on the implication of non-normality for the choice between Pearson's r and Spearman's rho.

Solved – Why does the Phi coefficient approximates the Pearson’s correlation

By default, chisq.test() applies a continuity correction when computing the test statistic for 2x2 tables. If you switch off this behavior, then:

x = c(1,  1,  0,  0,  1,  0,  1,  1,  1)
y = c(1,  1,  0,  0,  0,  0,  1,  1,  1)
cor(x,y)
sqrt(chisq.test(table(x,y), correct=FALSE)$statistic/length(x)) # phi

will give you exactly the same answer. And this essentially also answers why $\sqrt{\chi^2/n}$ with the continuity correction approximates cor(x,y) -- as $n$ increases, the continuity correction has less and less influence on the result.

The continuity correction is described here: Yates's correction for continuity

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Related Solutions

Solved – Reporting coefficient of determination using Spearman’s rho

Solved – Why does the Phi coefficient approximates the Pearson’s correlation

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