Author: Kilem L. Gwet, Ph.D.
Date: 2026-04-26

# Calculating Chance-corrected Agreement Coefficients (CAC)

#### Kilem L. Gwet, Ph.D.

#### 2026-04-26

    library(irrCAC)

# Abstract

The irrCAC is an R package that provides several functions for calculating various chance-corrected agreement coefficients. This package closely follows the general framework of inter-rater reliability assessment presented by Gwet (2014). A similar package was developed for STATA users by Klein (2018).

The functions included in this package can handle 3 types of input data: (1) the contingency table, (2) the distribution of raters by subject and by category, (3) the raw data, which is essentially a plain dataset where each row represents a subject and each column, the ratings associated with one rater. The list of all datasets containined in this package can be listed as follows:

      data(package="irrCAC")

# Computing Agreement Coefficients

## Computing agreement Coefficients from Contingency tables

cont3x3abstractors is one of 2 datasets included in this package and that contain rating data from 2 raters organized in the form of a contingency table. The following r script shows how to compute Cohen’s kappa, Scott’s Pi, Gwet’s \(\mbox{AC}_1\), Brennan-Prediger, Krippendorff’s alpha, Bangdiwala’s B, and the percent agreement coefficients from this dataset.

      cont3x3abstractors
    #>         Ectopic AIU NIU
    #> Ectopic      13   0   0
    #> AIU           0  20   7
    #> NIU           0   4  56
      kappa2.table(cont3x3abstractors)
    #>      coeff.name coeff.val   coeff.se     coeff.ci coeff.pval
    #> 1 Cohen's Kappa 0.7964094 0.05891072 (0.68,0.913)      0e+00
      scott2.table(cont3x3abstractors)
    #>   coeff.name coeff.val   coeff.se      coeff.ci coeff.pval
    #> 1 Scott's Pi 0.7962397 0.05905473 (0.679,0.913)      0e+00
      gwet.ac1.table(cont3x3abstractors)
    #>   coeff.name coeff.val   coeff.se      coeff.ci coeff.pval
    #> 1 Gwet's AC1 0.8493305 0.04321747 (0.764,0.935)      0e+00
      bp2.table(cont3x3abstractors)
    #>         coeff.name coeff.val   coeff.se      coeff.ci coeff.pval
    #> 1 Brennan-Prediger     0.835 0.04693346 (0.742,0.928)      0e+00
      krippen2.table(cont3x3abstractors)
    #>             coeff.name coeff.val   coeff.se     coeff.ci coeff.pval
    #> 1 Krippendorff's Alpha 0.7972585 0.05905473 (0.68,0.914)      0e+00
      bangdiwala.table(cont3x3abstractors)
    #>       coeff.name coeff.val   coeff.se      coeff.ci coeff.pval
    #> 1 Bangdiwala's B 0.8059604 0.05449465 (0.698,0.914)      0e+00
      pa2.table(cont3x3abstractors)
    #>          coeff.name coeff.val   coeff.se      coeff.ci coeff.pval
    #> 1 Percent Agreement      0.89 0.03128898 (0.828,0.952)      0e+00

Suppose that you only want to obtain Gwet’s \(\mbox{AC}_1\) coefficient, but don’t care about the associated precision measures such as the standard error, confidence intervals or p-values. You can accomplish this as follows:

      ac1 <- gwet.ac1.table(cont3x3abstractors)$coeff.val

Then use the variable ac1 to obtain \(\mbox{AC}_1=0.849\).

Another contingency table included in this package is named cont4x4diagnosis. You may use it to experiment with the r functions listed above. If you are interested in the weighted agreement coefficients, you can find more information in weighting.html. Note that Bangdiwala’s weighted coefficient is not implemented in this package.

## Computing agreement coefficients from the distribution of raters by subject & category

Included in this package is a small dataset named distrib.6raters, which contains the distribution of 6 raters by subject and category. Each row represents a subject (i.e. a psychiatric patient) and the number of raters (i.e. psychiatrists) who classified it into each category used in the inter-rater reliability study. Here is the dataset and how it can be used to compute the various agreement coefficients:

    distrib.6raters
    #>    Depression Personality.Disorder Schizophrenia Neurosis Other
    #> 1           0                    0             0        6     0
    #> 2           0                    3             0        0     3
    #> 3           0                    1             4        0     1
    #> 4           0                    0             0        0     6
    #> 5           0                    3             0        3     0
    #> 6           2                    0             4        0     0
    #> 7           0                    0             4        0     2
    #> 8           2                    0             3        1     0
    #> 9           2                    0             0        4     0
    #> 10          0                    0             0        0     6
    #> 11          1                    0             0        5     0
    #> 12          1                    1             0        4     0
    #> 13          0                    3             3        0     0
    #> 14          1                    0             0        5     0
    #> 15          0                    2             0        3     1
    gwet.ac1.dist(distrib.6raters)
    #>   coeff.name     coeff     stderr      conf.int      p.value        pa
    #> 1 Gwet's AC1 0.4448007 0.08418757 (0.264,0.625) 5.779637e-05 0.5511111
    #>          pe  n q r.min r.max
    #> 1 0.1914815 15 5     6     6
    fleiss.kappa.dist(distrib.6raters)
    #>      coeff.name     coeff     stderr     conf.int      p.value        pa
    #> 1 Fleiss' Kappa 0.4139265 0.08119291 (0.24,0.588) 8.113618e-05 0.5511111
    #>          pe  n q r.min r.max
    #> 1 0.2340741 15 5     6     6
    krippen.alpha.dist(distrib.6raters)
    #>             coeff.name     coeff     stderr      conf.int      p.value
    #> 1 Krippendorff's Alpha 0.4204384 0.08243228 (0.244,0.597) 8.078607e-05
    #>          pa        pe  n q r.min r.max
    #> 1 0.5560988 0.2340741 15 5     6     6
    bp.coeff.dist(distrib.6raters)
    #>         coeff.name     coeff     stderr      conf.int      p.value        pa
    #> 1 Brennan-Prediger 0.4388889 0.08312142 (0.261,0.617) 5.814998e-05 0.5511111
    #>    pe  n q r.min r.max
    #> 1 0.2 15 5     6     6

Once again, you can request a single value from these functions. To get only Krippendorff’s alpha coefficient without it’s precission measures, you may proceed as follows:

      alpha <- krippen.alpha.dist(distrib.6raters)$coeff

The newly-created alpha variable gives the coefficient \(\alpha = 0.4204384\).

Two additional datasets that represent ratings in the form of a distribution of raters by subject and by category, are included in this package. These datasets are cac.dist4cat and cac.dist4cat. Note that these 2 datasets contain more columns than needed to run the 4 functions presented in this section. Therefore, the columns associated with the response categories must be extracted from the original datasets before running the functions. For example, computing Gwet’s \(\mbox{AC}_1\) coefficient using the cac.dist4cat dataset should be done as follows:

      ac1 <- gwet.ac1.dist(cac.dist4cat[,2:4])$coeff

Note that the input dataset supplied to the function is cac.dist4cat[,2:4]. That is, only columns 2, 3, and 4 are extracted from the original datset and used as input data. We know from the value of the newly created variable that \(\mbox{AC}_1=0.3518903\).

## Computing agreement coefficients from raw ratings

One example dataset of raw ratings included in this package is cac.raw4raters and looks like this:

      cac.raw4raters
    #>    Rater1 Rater2 Rater3 Rater4
    #> 1       1      1     NA      1
    #> 2       2      2      3      2
    #> 3       3      3      3      3
    #> 4       3      3      3      3
    #> 5       2      2      2      2
    #> 6       1      2      3      4
    #> 7       4      4      4      4
    #> 8       1      1      2      1
    #> 9       2      2      2      2
    #> 10     NA      5      5      5
    #> 11     NA     NA      1      1
    #> 12     NA     NA      3     NA

As you can see, a dataset of raw ratings is merely a listing of ratings that the raters assigned to the subjects. Each row is associated with a single subject.Typically, the same subject would be rated by all or some of the raters. The dataset cac.raw4raters contains some missing ratings represented by the symbol NA, suggesting that some raters did not rate all subjects. As a matter of fact, in this particular case, no rater rated all subjects.

Here is you can compute the various agreement coefficients using the raw ratings:

    pa.coeff.raw(cac.raw4raters)
    #> $est
    #>          coeff.name        pa pe coeff.val coeff.se  conf.int      p.value
    #> 1 Percent Agreement 0.8181818  0 0.8181818  0.12561 (0.542,1) 2.172686e-05
    #>   tot.obs     w.name
    #> 1      41 unweighted
    #>
    #> $weights
    #>      [,1] [,2] [,3] [,4] [,5]
    #> [1,]    1    0    0    0    0
    #> [2,]    0    1    0    0    0
    #> [3,]    0    0    1    0    0
    #> [4,]    0    0    0    1    0
    #> [5,]    0    0    0    0    1
    #>
    #> $categories
    #> [1] 1 2 3 4 5
    #>
    #> $obs
    #>   Rater1 Rater2 Rater3 Rater4
    #> 1      9     10     11     11
    gwet.ac1.raw(cac.raw4raters)
    #> $est
    #>   coeff.name        pa        pe coeff.val coeff.se  conf.int      p.value
    #> 1        AC1 0.8181818 0.1903212   0.77544  0.14295 (0.461,1) 0.0001043605
    #>   tot.obs     w.name
    #> 1      41 unweighted
    #>
    #> $weights
    #>      [,1] [,2] [,3] [,4] [,5]
    #> [1,]    1    0    0    0    0
    #> [2,]    0    1    0    0    0
    #> [3,]    0    0    1    0    0
    #> [4,]    0    0    0    1    0
    #> [5,]    0    0    0    0    1
    #>
    #> $categories
    #> [1] 1 2 3 4 5
    #>
    #> $obs
    #>   Rater1 Rater2 Rater3 Rater4
    #> 1      9     10     11     11
    fleiss.kappa.raw(cac.raw4raters)
    #> $est
    #>      coeff.name        pa        pe coeff.val coeff.se  conf.int      p.value
    #> 1 Fleiss' Kappa 0.8181818 0.2387153   0.76117  0.15302 (0.424,1) 0.0002095865
    #>   tot.obs     w.name
    #> 1      41 unweighted
    #>
    #> $weights
    #>      [,1] [,2] [,3] [,4] [,5]
    #> [1,]    1    0    0    0    0
    #> [2,]    0    1    0    0    0
    #> [3,]    0    0    1    0    0
    #> [4,]    0    0    0    1    0
    #> [5,]    0    0    0    0    1
    #>
    #> $categories
    #> [1] 1 2 3 4 5
    #>
    #> $obs
    #>   Rater1 Rater2 Rater3 Rater4
    #> 1      9     10     11     11
    krippen.alpha.raw(cac.raw4raters)
    #> $est
    #>             coeff.name    pa   pe coeff.val coeff.se  conf.int      p.value
    #> 1 Krippendorff's Alpha 0.805 0.24   0.74342  0.14548 (0.423,1) 0.0001693123
    #>   tot.obs     w.name
    #> 1      41 unweighted
    #>
    #> $weights
    #>      [,1] [,2] [,3] [,4] [,5]
    #> [1,]    1    0    0    0    0
    #> [2,]    0    1    0    0    0
    #> [3,]    0    0    1    0    0
    #> [4,]    0    0    0    1    0
    #> [5,]    0    0    0    0    1
    #>
    #> $categories
    #> [1] 1 2 3 4 5
    #>
    #> $obs
    #>   Rater1 Rater2 Rater3 Rater4
    #> 1      9     10     11     11
    conger.kappa.raw(cac.raw4raters)
    #> $est
    #>       coeff.name        pa        pe coeff.val coeff.se  conf.int      p.value
    #> 1 Conger's Kappa 0.8181818 0.2334252   0.76282  0.14917 (0.435,1) 0.0001683533
    #>   tot.obs     w.name
    #> 1      41 unweighted
    #>
    #> $weights
    #>      [,1] [,2] [,3] [,4] [,5]
    #> [1,]    1    0    0    0    0
    #> [2,]    0    1    0    0    0
    #> [3,]    0    0    1    0    0
    #> [4,]    0    0    0    1    0
    #> [5,]    0    0    0    0    1
    #>
    #> $categories
    #> [1] 1 2 3 4 5
    #>
    #> $obs
    #>   Rater1 Rater2 Rater3 Rater4
    #> 1      9     10     11     11
    bp.coeff.raw(cac.raw4raters)
    #> $est
    #>         coeff.name        pa  pe coeff.val coeff.se  conf.int      p.value
    #> 1 Brennan-Prediger 0.8181818 0.2   0.77273  0.14472 (0.454,1) 0.0001187804
    #>   tot.obs     w.name
    #> 1      41 unweighted
    #>
    #> $weights
    #>      [,1] [,2] [,3] [,4] [,5]
    #> [1,]    1    0    0    0    0
    #> [2,]    0    1    0    0    0
    #> [3,]    0    0    1    0    0
    #> [4,]    0    0    0    1    0
    #> [5,]    0    0    0    0    1
    #>
    #> $categories
    #> [1] 1 2 3 4 5
    #>
    #> $obs
    #>   Rater1 Rater2 Rater3 Rater4
    #> 1      9     10     11     11

Most users of this package will only be interessted in the agreement coefficients and possibly in the related statistics such as the standard error and p-values. In this case, you should run these functions as follows (\(\mbox{AC}_1\) is used here as an example. Feel free to experiment with the other coefficients):

    ac1 <- gwet.ac1.raw(cac.raw4raters)$est
    ac1
    #>   coeff.name        pa        pe coeff.val coeff.se  conf.int      p.value
    #> 1        AC1 0.8181818 0.1903212   0.77544  0.14295 (0.461,1) 0.0001043605
    #>   tot.obs     w.name
    #> 1      41 unweighted

You can even request only the \(\mbox{AC}_1\) coefficient estimate 0.77544. You will then proceed as follows:

    ac1 <- gwet.ac1.raw(cac.raw4raters)$est
    ac1$coeff.val
    #> [1] 0.77544

# References:

1. Gwet, K.L. (2014). “Handbook of Inter-Rater Reliability,” 4th Edition. Advanced Analytics, LLC
2. Klein, D. (2018) doi:https://doi.org/10.1177/1536867X1801800408. “Implementing a general framework for assessing interrater agreement in Stata,” The Stata Journal Volume 18, Number 4, pp. 871-901.
3. Shankar, V. and Bangdiwala, S.I. (2014). Observer agreement paradoxes in \(2\times2\) tables: Comparison of agreement measures. BMC Medical Research Methodology, 14(1). https://doi.org/10.1186/1471-2288-14-100.