Calculating the Overlap of Two Normal Distributions Using Monte Carlo Integration

I read this post over at the blog Cartesian Faith about Probability and Monte Carlo methods. The post describe how to numerically intregate using Monte Carlo methods. I thought the results looked cool so I used the method to calculate the overlap of two normal distributions that are separated by a Cohen’s d of 0.8. You should head over to the original post if you want a more detailed explanation of the method. And I should add that this is not the most efficient way to calculate the overlap of two gaussian distributions, but it is a fun and pretty intuitive way, plus you can make a cool plot of the result. However, I also show how to get the overlap using the cumulative distribution function and using R’s built-in integration function.

Overlapping proportions of two normal distributions

So two gaussian distributions that are separated by a standardized mean difference (Cohen’s d) of 0.8 look like this

Overlap of two gaussian distributions using monte carlo integration. By Kristoffer Magnusson

To calculate the overlap we just divide the number of points in the overlap region with the total numbers of points in one of the distributions. To get more stable results I calculate the mean overlap using both distributions. What we’re calculating is sometimes called the overlapping coefficient (OVL).

The faster but less cool way

If we just want to convert from Cohen’s d to OVL, we can use the cumulative distribution function pnorm().

This result is very close to our monte carlo estimate. Another easy way is to use R’s built-in integrate() function, which will work with unequal variances as well.

Written by Kristoffer Magnusson, a researcher in clinical psychology. You should follow him on Twitter and come hang out on the open science discord Git Gud Science.


Published January 08, 2014 (View on GitHub)

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Archived Comments (3)

Gavin 2020-03-28

Excellent coding, thank you very much. Have you perhaps been able to adapt this to a multivariate normal case? i.e. when mu1 = c(0,1,2) and Sigma1=1*diag(3) as an example?

FRI END 2018-04-20


This is really helpful and exactly the code I need. I cannot thank you enough for this!
I was wondering if there was a way to get this to work for multiple Ms & Sds?
So I would like estimates of OVL for a table of 20+ Ms and SDs (for some review work), but I cannot get this code to work.

So I've got the code the OVL stat so that:
int_f <- function(x, mu1, mu2, sd1, sd2) {
f1 <- dnorm(x, mean=mu1, sd=sd1)
f2 <- dnorm(x, mean=mu2, sd=sd2)
Overlap <- function(mu1, mu2, sd1, sd2) {
integrate(int_f, -Inf, Inf, mu1, mu2, sd1, sd2)$value


This correctly returns:
[1] 0.9203441

Which is what I'd want. But when I have a dataset that has columns Mean1, Mean2, Sd1, Sd2 with N=20 rows of data and I run:
Mydata$Overlap <-Overlap(Mydata$Mean1, Mydata$Mean2, Mydata$Sd1, Mydata$Sd2)

I get this error:
Error in integrate(int_f, -Inf, Inf, mu1, mu2, sd1, sd2) : evaluation of function gave a result of wrong length

Do you know why this might be happening? Sorry if this is a bit beyond the remit of this post. Any help is greatly appreciated!

PS: Why does this not work when other custom functions don't seem to have the same problem, such as the below which does work:
ColSumTest <- function(x, y) {
Mydata$Sum <- ColSumTest(Mydata$Mean1,Mydata$Mean2)

Matti Meyer 2015-11-22

Hi Kristoffer, could you tell me please if there is some literature to get deeper into this topic? Or which literature did you used to get into it?