Fleiss' kappa (named after
Joseph L. Fleiss
Joseph L. Fleiss (November 13, 1937 – June 12, 2003) was an American professor of biostatistics at the Columbia University Mailman School of Public Health, where he also served as head of the Division of Biostatistics from 1975 to 1992. He is ...
) is a
statistical measure
In statistics, as opposed to its general use in mathematics, a parameter is any measured quantity of a statistical population that summarises or describes an aspect of the population, such as a mean or a standard deviation. If a population exa ...
for assessing the
reliability of agreement between a fixed number of raters when assigning
categorical rating
Level of measurement or scale of measure is a classification that describes the nature of information within the values assigned to variables. Psychologist Stanley Smith Stevens developed the best-known classification with four levels, or scal ...
s to a number of items or classifying items. This contrasts with other kappas such as
Cohen's kappa
Cohen's kappa coefficient (''κ'', lowercase Greek kappa) is a statistic that is used to measure inter-rater reliability (and also intra-rater reliability) for qualitative (categorical) items. It is generally thought to be a more robust measure tha ...
, which only work when assessing the agreement between not more than two raters or the intra-rater reliability (for one appraiser versus themself). The measure calculates the degree of agreement in classification over that which would be expected by chance.
Fleiss' kappa can be used with binary or
nominal-scale. It can also be applied to
Ordinal data
Ordinal data is a categorical, statistical data type where the variables have natural, ordered categories and the distances between the categories are not known. These data exist on an ordinal scale, one of four levels of measurement described b ...
(ranked data): the MiniTab online documentation gives an example. However, this document notes: "When you have ordinal ratings, such as defect severity ratings on a scale of 1–5,
Kendall's coefficients, which account for ordering, are usually more appropriate statistics to determine association than kappa alone." Keep in mind however, that Kendall rank coefficients are only appropriate for rank data.
Introduction
Fleiss' kappa is a generalisation of
Scott's pi statistic, a
statistical
Statistics (from German: ''Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industria ...
measure of
inter-rater reliability
In statistics, inter-rater reliability (also called by various similar names, such as inter-rater agreement, inter-rater concordance, inter-observer reliability, inter-coder reliability, and so on) is the degree of agreement among independent obse ...
. It is also related to Cohen's kappa statistic and
Youden's J statistic
Youden's J statistic (also called Youden's index) is a single statistic that captures the performance of a dichotomous diagnostic test. Informedness is its generalization to the multiclass case and estimates the probability of an informed decision ...
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,
, can be defined as,
(1)
:
The factor
gives the degree of agreement that is attainable above chance, and,
gives the degree of agreement actually achieved above chance. If the raters are in complete agreement then
. If there is no agreement among the raters (other than what would be expected by chance) then
.
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
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
(see
example below) to show the degree of agreement between the psychiatrists above the level of agreement expected by chance.
Definition
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)
:
Now calculate
, 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)
:
::
::
Now compute
, the mean of the
's, and
which go into the formula for
:
(4)
:
::
(5)
:
Worked example
In the following example, for each of ten "subjects" (
) fourteen raters (
), sampled from a larger group, assign a total of five categories (
). 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.
Data
See table to the right.
''N'' = 10, ''n'' = 14, ''k'' = 5
Sum of all cells = 140
Sum of ''P''
''i'' = 3.780
Calculations
The value
is the proportion of all assignments (
, here
) that were made to the
th category. For example, taking the first column,
:
And taking the second row,
:
In order to calculate
, we need to know the sum of
,
:
Over the whole sheet,
:
:
:
Interpretation
Landis and Koch (1977) gave the following table for interpreting
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.
Tests of Significance
Statistical packages can calculate a
standard score
In statistics, the standard score is the number of standard deviations by which the value of a raw score (i.e., an observed value or data point) is above or below the mean value of what is being observed or measured. Raw scores above the mean ...
(Z-score) for
Cohen's kappa
Cohen's kappa coefficient (''κ'', lowercase Greek kappa) is a statistic that is used to measure inter-rater reliability (and also intra-rater reliability) for qualitative (categorical) items. It is generally thought to be a more robust measure tha ...
or Fleiss's Kappa, which can be converted into a
P-value
In null-hypothesis significance testing, the ''p''-value is the probability of obtaining test results at least as extreme as the result actually observed, under the assumption that the null hypothesis is correct. A very small ''p''-value means ...
. 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.
See also
*
Pearson product-moment correlation coefficient
In statistics, the Pearson correlation coefficient (PCC, pronounced ) ― also known as Pearson's ''r'', the Pearson product-moment correlation coefficient (PPMCC), the bivariate correlation, or colloquially simply as the correlation coefficient ...
*
Matthews correlation coefficient
In statistics, the phi coefficient (or mean square contingency coefficient and denoted by φ or rφ) is a measure of association for two binary variables. In machine learning, it is known as the Matthews correlation coefficient (MCC) and used as a ...
*
Krippendorff's alpha
Krippendorff's alpha coefficient, named after academic Klaus Krippendorff, is a statistical measure of the agreement achieved when coding a set of units of analysis. Since the 1970s, ''alpha'' has been used in content analysis where textual units a ...
References
# 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.
# Fleiss, J. L. (1971) "Measuring nominal scale agreement among many raters." ''Psychological Bulletin'', Vol. 76, No. 5 pp. 378–382
# Scott, W. (1955). "Reliability of content analysis: The case of nominal scale coding." ''Public Opinion Quarterly'', Vol. 19, No. 3, pp. 321–325.
# 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
# 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.
# Landis, J. R. and Koch, G. G. (1977) "The measurement of observer agreement for categorical data" in ''Biometrics''. Vol. 33, pp. 159–174
# Gwet, K. L. (2014) ''Handbook of Inter-Rater Reliability'' (4th Edition), Chapter 6. (Gaithersburg : Advanced Analytics, LLC) . http://www.agreestat.com/book4/9780970806284_chap2.pdf
# 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
# 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.
Further reading
* 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", ''British Journal of Mathematical and Statistical Psychology'', Vol. 61, pp29–48
External links
AgreeStat 360: cloud-based inter-rater reliability analysis, Cohen's kappa, Gwet's AC1/AC2, Krippendorff's alpha, Brennan-Prediger, Fleiss generalized kappa, intraclass correlation coefficientscontains a good bibliography of articles about the coefficient.
Online Kappa Calculator calculates a variation of Fleiss' kappa.
Categorical variable interactions
Inter-rater reliability
Summary statistics for contingency tables
{{good article
de:Cohens Kappa#Fleiss' Kappa