Inter-rater Reliability, Cohen's Kappa and Surprises with R

Introduction

This is part three in adventures with statsmodels.stats, after power and multicomparison.

This time it is about Cohen's Kappa, a measure of inter-rater agreement or reliability. Suppose we have two raters that each assigns the same subjects or objects to one of a fixed number of categories. The question then is: How well do the raters agree in their assignments? Kappa provides a measure of association, the largest possible value is one, the smallest is minus 1, and it has a corresponding statistical test for the hypothesis that agreement is only by chance. Cohen's kappa and similar measures have widespread use, among other fields, in medical or biostatistic. In one class of applications, raters are doctors, subjects are patients and the categories are medical diagnosis. Cohen's Kappa provides a measure how well the two sets of diagnoses agree, and a hypothesis test whether the agreement is purely random.

For more background see this Wikipedia page which was the starting point for my code.

Most of the following focuses on weighted kappa, and the interpretation of different weighting schemes. In the last part, I add some comments about R, which provided me with several hours of debugging, since I'm essentially an R newbie and have not yet figured out some of it's "funny" behavior.

Multiple Comparison and Tukey HSD or why statsmodels is awful

Introduction

Statistical tests are often grouped into one-sample, two-sample and k-sample tests, depending on how many samples are involved in the test. In k-sample tests the usual Null hypothesis is that a statistic, for example the mean as in a one-way ANOVA, or the distribution in goodness-of-fit tests, is the same in all groups or samples. The common test is the joint test that all samples have the same value, against the alternative that at least one sample or group has a different value.

However, often we are not just interested in the joint hypothesis if all samples are the same, but we would also like to know for which pairs of samples the hypothesis of equal values is rejected. In this case we conduct several tests at the same time, one test for each pair of samples.

This results, as a consequence, in a multiple testing problem and we should correct our test distribution or p-values to account for this.

I mentioned some of the one- and two sample test in statsmodels before. Today, I just want to look at pairwise comparison of means. We have k samples and we want to test for each pair whether the mean is the same, or not.

Instead of adding more explanations here, I just want to point to R tutorial and also the brief description on Wikipedia. A search for "Tukey HSD" or multiple comparison on the internet will find many tutorials and explanations.

The following are examples in statsmodels and R interspersed with a few explanatory comments.

Statistical Power in Statsmodels

I merged last week a branch of mine into statsmodels that contains large parts of basic power calculations and some effect size calculations. The documentation is in this section . Some parts are still missing but I thought I have worked enough on this for a while.

(Adding the power calculation for a new test now takes approximately: 3 lines of real code, 200 lines of wrapping it with mostly boiler plate and docstrings, and 30 to 100 lines of tests.)

The first part contains some information on the implementation. In the second part, I compare the calls to the function in the R pwr package to the calls in my (statsmodels') version.

I am comparing it to the pwr package because I ended up writing almost all unit tests against it. The initial development was based on the SAS manual, I used the explanations on the G-Power website for F-tests, and some parts were initially written based on articles that I read. However, during testing I adjusted the options (and fixed bugs), so I was able to match the results to pwr, and I think pwr has just the right level of abstraction and easiness of use, that I ended up with code that is pretty close to it.

If you just want to see the examples, skip to the second part.