Pitman-Morgan Test: Test the Difference Between Correlated Variances

Step 1: If you need it, watch this video to show you how to use the spreadsheet below
Step 2: Enter your 'variance 1' (larger), 'variance 2' (smaller), 'sample size' and 'correlation'
Step 3: Enjoy results

When studying the independent groups t-test, one very often reads about testing the assumption of homogeneity of variance. However, when studying the dependent groups t-test (a.k.a., paired t-test; t-test for correlated samples), it is rare to see this assumption specified. To my knowledge, the assumption of homogeneity of variance applies to both t-tests. However, I do not know of any stats packages that provide a test of homogeneity of variance when performing a paired t-test.

Based on my google scholar research, the Pitman-Morgan Test appears to be the most frequently cited approach to testing the assumption of homogeneity of variance, when the variances are correlated (i.e., pretty much every paired t-test case, in practice). However, it is rare to see the Pitman-Morgan Test available in software packages, hence the spreadsheet above.

In addition to testing the assumption of homogeneity, I believe this type of analysis is interesting in its own right. That is, sometimes one's hypothesis is relevant to determining whether one group of data is more variable than another (matching) group of data.

In psychology, an example would include if you want to know if a condition (e.g., anxiety or depression) is associated with greater variance within your sample, after a treatment of some sort. This will usually suggest that people in your sample are responding to the treatment differentially (some very well, others not at all).

In the hard sciences, you may be sourcing an element from two different suppliers. You want to know if the concentration of the element you are receiving is more variable from one supplier, in comparison to another.

In finance, this test may be appropriate to determine if the investment returns from one security are more variable than another security. Typically, people perform a CAPM analysis (linear regression) and estimate beta to answer this question. Although I would have to think about it more, I believe the Pitman-Morgan test may be a more appropriate, direct, and robust approach to answering this question.

To use the interactive spreadsheet below, click 'click to edit' to activate the spreadsheet. Then enter the four required data points: Variance 1, Variance 2, sample size, and correlation between variable 1 and variable 2.

The spreadsheet below provides results in two ways, both of which are based on Pitman and Morgan's papers (Morgan, 1939; Pitman, 1939). One is based on a t-test, which I found in Gardner (2001, p.57). I prefer this method, myself, as it seems more conventional to report a t (or F) value associated with a homogeneity of variance test. The second option is the rDS method that Snedecor and Cochran (1989, p. 193). It is relatively common to see the rDS approach reported in the literature, but I think most people would be surprised to see a correlation reported when testing homogeneity of variance. Both methods yield identical p values. They are essentially the same test.

If you find any errors, please feel free to contact me.


Gardner, R.C. (2001). Psychological Statistics Using SPSS for Windows. New Jersey: Prentice Hall

Morgan, W.A. (1939). A test for the significance of the difference between the two variances in a sample from a normal bivariate population. Biometrika 31, 13-19

Pitman, E. J. G. 1939. A note on normal correlation. Biometrika 31, 9-12.

Snedecor, G.W., & Cochran, W.G. (1989). Statistical Methods. Iowa: Blackwell