Fleiss' kappa

Fleiss' kappa is a generalisation of Scott's pi statistic, a statistical measure of inter-rater reliability. It is also related to Cohen's kappa statistic and Youden's J statistic which may be more appropriate in certain instances. Whereas Scott's pi and Cohen's kappa work for only two raters, Fleiss' kappa works for any number of raters giving categorical ratings, to a fixed number of items, at the condition that for each item raters are randomly sampled. It can be interpreted as expressing the extent to which the observed amount of agreement among raters exceeds what would be expected if all raters made their ratings completely randomly. It is important to note that whereas Cohen's kappa assumes the same two raters have rated a set of items, Fleiss' kappa specifically allows that although there are a fixed number of raters (e.g., three), different items may be rated by different individuals (Fleiss, 1971, p. 378). That is, Item 1 is rated by Raters A, B, and C; but Item 2 could be rated by Raters D, E, and F. The condition of random sampling among raters makes Fleiss' kappa not suited for cases where all raters rate all patients.

Agreement can be thought of as follows, if a fixed number of people assign numerical ratings to a number of items then the kappa will give a measure for how consistent the ratings are. The kappa, ${\displaystyle \kappa \,}$, can be defined as,

(1)

${\displaystyle \kappa ={\frac {{\bar {P}}-{\bar {P_{e}}}}{1-{\bar {P_{e}}}}}}$

The factor ${\displaystyle 1-{\bar {P_{e}}}}$ gives the degree of agreement that is attainable above chance, and, ${\displaystyle {\bar {P}}-{\bar {P_{e}}}}$ gives the degree of agreement actually achieved above chance. If the raters are in complete agreement then ${\displaystyle \kappa =1~}$. If there is no agreement among the raters (other than what would be expected by chance) then ${\displaystyle \kappa \leq 0}$.

An example of the use of Fleiss' kappa may be the following: Consider several psychiatrists are asked to look at ten patients. For each patient, 14 psychiatrists gives one of possibly five diagnoses. These are compiled into a matrix, and Fleiss' kappa can be computed from this matrix (see example below) to show the degree of agreement between the psychiatrists above the level of agreement expected by chance.

Let N be the total number of subjects, let n be the number of ratings per subject, and let k be the number of categories into which assignments are made. The subjects are indexed by i = 1, ... N and the categories are indexed by j = 1, ... k. Let n_ij represent the number of raters who assigned the i-th subject to the j-th category.

First calculate p_j, the proportion of all assignments which were to the j-th category:

(2)

${\displaystyle p_{j}={\frac {1}{Nn}}\sum _{i=1}^{N}n_{ij},\quad \quad 1=\sum _{j=1}^{k}p_{j}}$

Now calculate ${\displaystyle P_{i}\,}$, the extent to which raters agree for the i-th subject (i.e., compute how many rater-rater pairs are in agreement, relative to the number of all possible rater-rater pairs):

(3)

${\displaystyle P_{i}={\frac {1}{n(n-1)}}\sum _{j=1}^{k}n_{ij}(n_{ij}-1)}$

${\displaystyle ={\frac {1}{n(n-1)}}\sum _{j=1}^{k}(n_{ij}^{2}-n_{ij})}$

${\displaystyle ={\frac {1}{n(n-1)}}\left[\left(\sum _{j=1}^{k}n_{ij}^{2}\right)-(n)\right]}$

Note that ${\displaystyle P_{i}}$ is bound between 0, when ratings are assigned equally over all categories, and 1 when all ratings are assigned to a single category.

Now compute ${\displaystyle {\bar {P}}}$, the mean of the ${\displaystyle P_{i}}$'s, and ${\displaystyle {\bar {P_{e}}}}$, which go into the formula for ${\displaystyle \kappa }$:

(4)

${\displaystyle {\bar {P}}={\frac {1}{N}}\sum _{i=1}^{N}P_{i}}$

${\displaystyle ={\frac {1}{Nn(n-1)}}\left(\sum _{i=1}^{N}\sum _{j=1}^{k}n_{ij}^{2}-Nn\right)}$

(5)

${\displaystyle {\bar {P_{e}}}=\sum _{j=1}^{k}p_{j}^{2}}$

Table of values for computing the worked example

${\displaystyle n_{ij}}$	1	2	3	4	5	${\displaystyle P_{i}\,}$
1	0	0	0	0	14	1.000
2	0	2	6	4	2	0.253
3	0	0	3	5	6	0.308
4	0	3	9	2	0	0.440
5	2	2	8	1	1	0.330
6	7	7	0	0	0	0.462
7	3	2	6	3	0	0.242
8	2	5	3	2	2	0.176
9	6	5	2	1	0	0.286
10	0	2	2	3	7	0.286
Total	20	28	39	21	32
${\displaystyle p_{j}\,}$	0.143	0.200	0.279	0.150	0.229

In the following example, for each of ten "subjects" (${\displaystyle N}$) fourteen raters (${\displaystyle n}$), sampled from a larger group, assign a total of five categories (${\displaystyle k}$). The categories are presented in the columns, while the subjects are presented in the rows. Each cell lists the number of raters who assigned the indicated (row) subject to the indicated (column) category.

Landis and Koch (1977) gave the following table for interpreting ${\displaystyle \kappa }$ values for a 2-annotator 2-class example. This table is however by no means universally accepted. They 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, as the number of categories and subjects will affect the magnitude of the value. For example, the kappa is higher when there are fewer categories.

Condition	${\displaystyle \kappa }$	Interpretation
Subjective example: only for two annotators, on two classes. See Landis & Koch 1977	< 0	Poor agreement
	0.01 – 0.20	Slight agreement
	0.21 – 0.40	Fair agreement
	0.41 – 0.60	Moderate agreement
	0.61 – 0.80	Substantial agreement
	0.81 – 1.00	Almost perfect agreement

Statistical packages can calculate a standard score (Z-score) for Cohen's kappa or Fleiss's Kappa, which can be converted into a P-value. However, even when the P value reaches the threshold of statistical significance (typically less than 0.05), it only indicates that the agreement between raters is significantly better than would be expected by chance. The p value does not tell you, by itself, whether the agreement is good enough to have high predictive value.

• Pearson product-moment correlation coefficient
• Matthews correlation coefficient
• Krippendorff's alpha

1. ^ MiniTab Inc. Kappa statistics for Attribute Agreement Analysis. https://support.minitab.com/en-us/minitab/18/help-and-how-to/quality-and-process-improvement/measurement-system-analysis/how-to/attribute-agreement-analysis/attribute-agreement-analysis/interpret-the-results/all-statistics-and-graphs/kappa-statistics/ Accessed Jan 22 2019.
2. ^ Fleiss, J. L. (1971) "Measuring nominal scale agreement among many raters." Psychological Bulletin, Vol. 76, No. 5 pp. 378–382
3. ^ Scott, W. (1955). "Reliability of content analysis: The case of nominal scale coding." Public Opinion Quarterly, Vol. 19, No. 3, pp. 321–325.
4. ^ Powers, D. M. W. (2011). "Evaluation: From Precision, Recall and F-Measure to ROC, Informedness, Markedness & Correlation". Journal of Machine Learning Technologies 2 (1): 37–63
5. ^ Powers, David M. W. (2012). "The Problem with Kappa". Conference of the European Chapter of the Association for Computational Linguistics (EACL2012) Joint ROBUS-UNSUP Workshop.
6. ^ Landis, J. R. and Koch, G. G. (1977) "The measurement of observer agreement for categorical data" in Biometrics. Vol. 33, pp. 159–174
7. ^ Gwet, K. L. (2014) Handbook of Inter-Rater Reliability (4th Edition), Chapter 6. (Gaithersburg : Advanced Analytics, LLC) ISBN 978-0970806284. http://www.agreestat.com/book4/9780970806284_chap2.pdf
8. ^ Sim, J. and Wright, C. C. (2005) "The Kappa Statistic in Reliability Studies: Use, Interpretation, and Sample Size Requirements" in Physical Therapy. Vol. 85, No. 3, pp. 257–268
9. ^ Hallgren, Kevin A. (2012) “Computing Inter-Rater Reliability for Observational Data: An Overview and Tutorial” in Tutorials in quantitative methods for psychology, Vol. 8, No. 1 pp 23–34.

• 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
• Fleiss, J. L. (1981) Statistical methods for rates and proportions. 2nd ed. (New York: John Wiley) pp. 38–46
• Gwet, K. L. (2008) "Computing inter-rater reliability and its variance in the presence of high agreement Archived 2016-03-03 at the Wayback Machine", British Journal of Mathematical and Statistical Psychology, Vol. 61, pp29–48

The Wikibook Algorithm implementation has a page on the topic of: Fleiss' kappa

• Cloud-based inter-rater reliability analysis, Cohen's kappa, Gwet's AC1/AC2, Krippendorff's alpha, Brennan-Prediger, Fleiss generalized kappa, intraclass correlation coefficients
• Kappa: Pros and Cons – contains a good bibliography of articles about the coefficient
• Online Kappa Calculator Archived 2009-02-28 at the Wayback Machine – calculates a variation of Fleiss' kappa