Cohen's kappa

Cohen's kappa coefficient is a statistical measure of inter-rater agreement or inter-annotator agreement^[1] for qualitative (categorical) items. It is generally thought to be a more robust measure than simple percent agreement calculation since κ takes into account the agreement occurring by chance. Some researchers (e.g. Strijbos, Martens, Prins, & Jochems, 2006) have expressed concern over κ's tendency to take the observed categories' frequencies as givens, which can have the effect of underestimating agreement for a category that is also commonly used; for this reason, κ is considered an overly conservative measure of agreement.

Others (e.g., Uebersax, 1987) contest the assertion that kappa "takes into account" chance agreement. To do this effectively would require an explicit model of how chance affects rater decisions. The so-called chance adjustment of kappa statistics supposes that, when not completely certain, raters simply guess—a very unrealistic scenario.

Nevertheless, and despite potentially better alternatives, Cohen's kappa enjoys continued popularity. A possible reason for this is that kappa is, under certain conditions, equivalent to the intraclass correlation coefficient.

Calculation

Cohen's kappa measures the agreement between two raters who each classify N items into C mutually exclusive categories. The first mention of a kappa-like statistic is attributed to Galton (1892), see Smeeton (1985).

The equation for κ is:

$\kappa ={\frac {\Pr(a)-\Pr(e)}{1-\Pr(e)}},\!$

where Pr(a) is the relative observed agreement among raters, and Pr(e) is the hypothetical probability of chance agreement, using the observed data to calculate the probabilities of each observer randomly saying each category. If the raters are in complete agreement then κ = 1. If there is no agreement among the raters (other than what would be expected by chance) then κ ≤ 0.

The seminal paper introducing kappa as a new technique was published by Jacob Cohen in the journal Educational and Psychological Measurement in 1960.

A similar statistic, called pi, was proposed by Scott (1955). Cohen's kappa and Scott's pi differ in terms of how Pr(e) is calculated.

Note that Cohen's kappa measures agreement between two raters only. For a similar measure of agreement (Fleiss' kappa) used when there are more than two raters, see Fleiss (1971). The Fleiss kappa, however, is a multi-rater generalization of Scott's pi statistic, not Cohen's kappa.

Example

Suppose that you were analyzing data related to people applying for a grant. Each grant proposal was read by two people and each reader either said "Yes" or "No" to the proposal. Suppose the data were as follows, where rows are reader A and columns are reader B:

	Yes	No
Yes	20	5
No	10	15

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 percentage agreement is Pr(a)=(20+15)/50 = 0.70.

To calculate Pr(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.

Therefore the probability that both of them would say "Yes" randomly is 0.50*0.60=0.30 and the probability that both of them would say "No" is 0.50*0.40=0.20. Thus the overall probability of random agreement is Pr("e") = 0.3+0.2 = 0.5.

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

$\kappa ={\frac {\Pr(a)-\Pr(e)}{1-\Pr(e)}}={\frac {0.70-0.50}{1-0.50}}=0.40\!$

Inconsistent results

One of the problems with Cohen's Kappa is that it does not always produce the expected answer. For instance, in the following two cases there is equal agreement between A and B (60 out of 100 in both cases) so we would expect the relative values of Cohen's Kappa to reflect this. However, calculating Cohen's Kappa for each:

	Yes	No
Yes	45	15
No	25	15

$\kappa ={\frac {0.60-0.54}{1-0.54}}=0.1304$

	Yes	No
Yes	25	35
No	5	35

$\kappa ={\frac {0.60-0.46}{1-0.46}}=0.2593$

we find that it shows greater similarity between A and B in the second case, compared to the first.

Significance and Magnitude

Statistical significance for kappa is rarely reported, probably because even relatively low values of kappa can nonetheless be significantly different form zero but not of sufficient magnitude to satisfy investigators. Still, its standard error has been described and is computed by various computer programs.

If statistical significance is not a useful guide, what magnitude of kappa reflects adequate agreement? Guidelines would be helpful, but factors other than agreement can influence its magnitude, which makes interpretation of a given magnitude problematic. As Sim and Wright noted, two important factors are prevalence (are the codes equiprobable or do their probabilities vary) and bias (are the marginal probabilities for the two observers similar or different). Other things being equal, kappas are higher when codes are equiprobable and distributed similarly by the two observers.

Another factor is the number of codes. As number of codes increases, kappas become higher. Based on a simulation study, Bakeman and colleagues concluded that for fallible observers, values for kappa were lower when codes were fewer. And, in agreement with Sim & Wrights’s statement concerning prevalence, kappas were higher when codes were roughly equiprobable. Thus Bakeman et al. concluded that “no one value of kappa can be regarded as universally acceptable.” They also provide a computer program that lets users compute values for kappa specifying number of codes, their probability, and observer accuracy. For example, given equiprobable codes and observers who are 85% accurate, value of kappa are .49, .60, .66, and .69 when number of codes is 2, 3, 5, and 10, respectively.

Nonetheless, magnitude guidelines have appeared in the literature. Perhaps the first was Landis and Koch, who characterized values < 0 as indicating no agreement and 0–.20 as slight, .21–.40 as fair, .41–.60 as moderate, .61–.80 as substantial, and .81–1 as almost perfect agreement. This set of guidelines is however by no means universally accepted; Landis and Koch supplied no evidence to support it, basing it instead on personal opinion. It has been noted that these guidelines may be more harmful than helpful.^[2] Fleiss's equally arbitrary guidelines characterize kappas over .75 as excellent, .40 to .75 as fair to good, and below .40 as poor.

Weighted Kappa

Weighted kappa lets you count disagreements differently and is especially useful when codes are ordered. Three matrices are involved, the matrix of observed scores, the matrix of expected scores based on chance agreement, and the weight matrix. Weight matrix cells located on the diagonal (upper-left to bottom-right) represent agreement and thus contain zeros. Off-diagonal cells contain weights indicating the seriousness of that disagreement. Often, cells one off the diagonal are weighted 1, those two off 2, etc.

The equation for weighted κ is:

$\kappa =1-{\frac {\sum _{i=1}^{k}\sum _{j=1}^{k}w_{ij}x_{ij}}{\sum _{i=1}^{k}\sum _{j=1}^{k}w_{ij}m_{ij}}}$

where k=number of codes and $w_{ij}$, $x_{ij}$, and $m_{ij}$ are elements in the weight, observed, and expected matrices, respectively. When diagonal cells contain weights of 0 and all off-diagonal cells weights of 1, this formula produces the same value of kappa as the calculation given above.

Kappa Maximum

Kappa assumes its theoretical maximum value of 1 only when both observers distribute codes the same, that is, when corresponding row and column sums are identical. Anything less is less than perfect agreement. Still, the maximum value kappa could achieve given unequal distributions helps interpret the value of kappa actually obtained. The equation for κ maximum is:

$\kappa _{max}={\frac {P_{max}-P_{exp}}{1-P_{exp}}}$

where $P_{exp}=\sum _{i=1}^{k}P_{i+}P_{+i}$, as usual, $P_{max}=\sum _{i=1}^{k}MIN(P_{i+},P_{+i})$,

k=number of codes, $P_{i+}$ are the row probabilities, and $P_{+i}$ are the column probabilities.

See also

• Fleiss' kappa
• Intraclass correlation

Notes

1. ^ Carletta, Jean. (1996) Assessing agreement on classification tasks: The kappa statistic. Computational Linguistics, 22(2), pp. 249–254.

References

• Kilem Gwet (May 2002). "Inter-Rater Reliability: Dependency on Trait Prevalence and Marginal Homogeneity". Statistical Methods For Inter-Rater Reliability Assessment 2:  ???.
• Banerjee, M. et al. (1999). "Beyond Kappa: A Review of Interrater Agreement Measures" The Canadian Journal of Statistics / La Revue Canadienne de Statistique, Vol. 27, No. 1, pp. 3-23 <http://www.jstor.org/stable/3315487>
• Brennan, R. L. and Prediger, D. J. (1981) "Coefficient λ: Some Uses, Misuses, and Alternatives" Educational and Psychological Measurement, 41, 687-699.
• Cohen, Jacob. (1960) A coefficient of agreement for nominal scales, Educational and Psychological Measurement Vol.20, No.1, pp. 37–46.
• Cohen, J. (1968). Weighed kappa: Nominal scale agreement with provision for scaled disagreement or partial credit. Psychological Bulletin, 70, 213-220.
• Fleiss, J.L. (1971) "Measuring nominal scale agreement among many raters." Psychological Bulletin, Vol. 76, No. 5 pp. 378—382
• Fleiss, J. L. (1981) Statistical methods for rates and proportions. 2nd ed. (New York: John Wiley) pp. 38—46
• Fleiss, J.L. and Cohen, J. (1973) "The equivalence of weighted kappa and the intraclass correlation coefficient as measures of reliability" in Educational and Psychological Measurement, Vol. 33 pp. 613—619
• Galton, F. (1892). Finger Prints Macmillan, London.
• Gwet, K. (2008). "Computing inter-rater reliability and its variance in the presence of high agreement." British Journal of Mathematical and Statistical Psychology, Vol. 61, pp. 29-48
• Gwet, K. (2008). "Variance Estimation of Nominal-Scale Inter-Rater Reliability with Random Selection of Raters." Psychometrika, Vol. 73, No. 3, pp. 407-430
• Gwet, K. (2008). "Intrarater Reliability." Wiley Encyclopedia of Clinical Trials, Copyright 2008 John Wiley & Sons, Inc.
• Scott, W. (1955). "Reliability of content analysis: The case of nominal scale coding." Public Opinion Quarterly, 17, 321-325.
• Sim, J. and Wright, C. C. (2005) "The Kappa Statistic in Reliability Studies: Use, Interpretation, and Sample Size Requirements" in Physical Therapy. Vol. 85, pp. 257—268
• Smeeton, N.C. (1985) "Early History of the Kappa Statistic" in Biometrics. Vol. 41, p.795.
• Strijbos, J., Martens, R., Prins, F., & Jochems, W. (2006). Content analysis: What are they talking about? Computers & Education, 46, 29-48.
• Uebersax JS. Diversity of decision-making models and the measurement of interrater agreement. Psychological Bulletin, 1987, 101, 140-146.

External links

• Kappa, its meaning, problems, and several alternatives
• Kappa Statistics: Pros and Cons
• Windows program for kappa, weighted kappa, and kappa maximum

Online calculators

• Cohen's Kappa for Maps
• Online Kappa Calculator (multiple raters and classes)
• Vassar College's Kappa Calculator
• AgreeStat: A user-friendly point-and-click Excel VBA program for the statistical analysis of inter-rater reliability data