Get overlap between two normally distributed variables

Question

I have three variables ($X_G$,$X_B$, and $X_R$), each of which is normally distributed. Is there a way to show the overlap between each of the pairs using the mean and the standard deviation?

An independent t-test shows that the three groups are very much significantly different, which is good, but I want a way to show that one pair is most different (while two other pairs are equally different)

To show what I mean graphically I've plotted the distributions below;

i.e. red and green are very different, while blue and green are less different. Is there a good way to quantify this difference in overlap? The t-statistic for red and blue is 16.6789, while the t-statistic for green and red is 38.9686. I do find this kind of odd, given how much overlap red and blue seem to have though...

What I'd like is to be able to directly calculate the information presented in the nomogram here. However, I guess I'd thought there would be standard implementations in matlab/R to do this, but couldn't find any.

Answer 1

Maybe this thread answer your question. It includes some R code to compute the percentage of overlap between 2 normal densities. Since densities add up to one, their proportions (% of overlap) might be directly compared.

Percentage of overlapping regions of two normal distributions

Is this what you were looking for?

Answer 2

You are basically asking about the size of the effect. For the difference in means between a pair of variables, Cohen's d is a (standardized) measure of such an effect size and Cohen presented an interpretation in terms of normal distributions' overlap in his book Statistical Power Analysis for the Behavioral Sciences. Note that this is not necessarily the only relevant notion of what “most different” could mean in this context.

Regarding the comment in the last paragraph, one important point is that the t statistic and other tests are based on the sampling distribution for the mean. They tell you how likely it would have been to observe a given difference if the null hypothesis were true, not how big this difference is. As the size of the sample increases, the variance of the sample mean decreases, as reflected in the lower standard error and narrower confidence intervals. Consequently, t can in principle be arbitrarily high and the p value arbitrarily small even if there is a lot of overlap and the difference is tiny in practical terms.

Unlike the standard error of the mean, the variance of the data themselves and the overlap between their distributions obviously does not tend toward 0. Intuitively, adding observations does not make them less different from each other (but it does make their distribution more precisely known, hence the smaller standard error of the mean).