Statistics - Cohen's kappa coefficient

Cohen's kappa coefficient is a statistic which measures inter-rater agreement for qualitative (categorical) items. It is generally thought to be a more robust measure than simple percent agreement calculation, since k takes into account the agreement occurring by chance. Cohen's kappa measures the agreement between two raters who each classify N items into C mutually exclusive categories.

Cohen's kappa coefficient is defined and given by the following function −


${k = \frac{p_0 - p_e}{1-p_e} = 1 - \frac{1-p_o}{1-p_e}}$

Where −

  • ${p_0}$ = relative observed agreement among raters.

  • ${p_e}$ = the hypothetical probability of chance agreement.

${p_0}$ and ${p_e}$ are computed using the observed data to calculate the probabilities of each observer randomly saying each category. If the raters are in complete agreement then ${k}$ = 1. If there is no agreement among the raters other than what would be expected by chance (as given by ${p_e}$), ${k}$ ≤ 0.


Problem Statement

Suppose that you were analyzing data related to a group of 50 people applying for a grant. Each grant proposal was read by two readers and each reader either said "Yes" or "No" to the proposal. Suppose the disagreement count data were as follows, where A and B are readers, data on the diagonal slanting left shows the count of agreements and the data on the diagonal slanting right, disagreements −

Yes No
A Yes 20 5
No 10 15

Calculate Cohen's kappa coefficient.


Note that there were 20 proposals that were granted by both reader A and reader B and 15 proposals that were rejected by both readers. Thus, the observed proportionate agreement is

${p_0 = \frac{20+15}{50} = 0.70}$

To calculate ${p_e}$ (the probability of random agreement) we note that −

  • Reader A said "Yes" to 25 applicants and "No" to 25 applicants. Thus reader A said "Yes" 50% of the time.

  • Reader B said "Yes" to 30 applicants and "No" to 20 applicants. Thus reader B said "Yes" 60% of the time.

Using formula P(A and B) = P(A) x P(B) where P is probability of event occuring.

The probability that both of them would say "Yes" randomly is 0.50 x 0.60 = 0.30 and the probability that both of them would say "No" is 0.50 x 0.40 = 0.20. Thus the overall probability of random agreement is ${p_e}$ = 0.3 + 0.2 = 0.5.

So now applying our formula for Cohen's Kappa we get:

${k = \frac{p_0 - p_e}{1-p_e} = \frac{0.70 - 0.50}{1-0.50} = 0.40}$