Last updated: 2025-10-08
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| File | Version | Author | Date | Message |
|---|---|---|---|---|
| Rmd | 466f85f | Dave Tang | 2025-10-08 | Cohen’s Kappa with random results |
| html | 9a6ead6 | Dave Tang | 2025-10-06 | Build site. |
| Rmd | 47f2160 | Dave Tang | 2025-10-06 | Manually calculate Fleiss’ Kappa |
| html | a8d6f16 | Dave Tang | 2025-10-06 | Build site. |
| Rmd | 59c9392 | Dave Tang | 2025-10-06 | Inter-rater reliability |
Measures of inter-rater reliability (IRR) provide an index indicating how much agreement there is between raters/observers, correcting for agreement that would happen just by chance.
Install {irr}.
install.packages("irr")Measures agreement between two raters who classify items into categories.
\[ \kappa = \frac{P_o - P_e}{1 - P_e} \]
where
cohen_kappa <- function(x, y) {
stopifnot(length(x) == length(y))
confusion_matrix <- table(x, y)
n <- sum(confusion_matrix)
P_o <- sum(diag(confusion_matrix)) / n
row_marginals <- rowSums(confusion_matrix) / n
col_marginals <- colSums(confusion_matrix) / n
P_e <- sum(row_marginals * col_marginals)
kappa <- (P_o - P_e) / (1 - P_e)
return(list(
kappa = kappa,
observed = P_o,
expected = P_e,
confusion_matrix = confusion_matrix
))
}Agreement between two doctors on 50 patients.
| Doctor 2: Disease | Doctor 2: No Disease | Row Total | |
|---|---|---|---|
| Doctor 1: Disease | 15 | 5 | 20 |
| Doctor 1: No Disease | 10 | 20 | 30 |
| Column Total | 25 | 25 | 50 |
Total agreement = 15 + 20 = 35.
\[ P_o = \frac{35}{50} = 0.70 \quad \text{(70% agreement observed)} \]
How much agreement would we expect just by chance, given how often each doctor says “Disease” versus “No Disease”?
Now multiply matching probabilities:
Expected agreement = 0.20 + 0.30 = 0.50 (50%)
Now calculate Cohen’s Kappa manually:
\[ \kappa = \frac{Po - Pe}{1 - Pe} = \frac{0.70 - 0.50}{1 - 0.50} = \frac{0.20}{0.50} = 0.40 \]
Using our function and irr::kappa2().
doc1 <- c(rep('D', 15), rep('N', 20), rep('N', 10), rep('D', 5))
doc2 <- c(rep('D', 15), rep('N', 20), rep('D', 10), rep('N', 5))
cohen_kappa(doc1, doc2)$kappa[1] 0.4irr::kappa2(data.frame(x = doc1, y = doc2)) Cohen's Kappa for 2 Raters (Weights: unweighted)
Subjects = 50
Raters = 2
Kappa = 0.4
z = 2.89
p-value = 0.00389 Generalises Cohen’s Kappa to more than two raters. Each item is rated by k raters (not necessarily the same raters for every item). Compute the agreement per item, then average across items, correcting for chance.
\[ \kappa = \frac{\bar{P} - \bar{P_e}}{1 - \bar{P_e}} \]
where
Data:
Psychiatric diagnoses of n=30 patients provided by different sets of m=6 raters. Data were used by Fleiss (1971) to illustrate the computation of Kappa for m raters.
data(diagnoses)
dim(diagnoses)[1] 30 6Fleiss’ Kappa.
kappam.fleiss(diagnoses) Fleiss' Kappa for m Raters
Subjects = 30
Raters = 6
Kappa = 0.43
z = 17.7
p-value = 0 Manually calculate.
lapply(diagnoses, \(x) as.integer(sub("\\. .*", "", x))) |>
as.data.frame() |>
as.matrix() -> ratings
# patients
N <- nrow(ratings)
# doctors
n <- ncol(ratings)
cats <- sort(unique(as.numeric(ratings)))
# build item × category counts
counts <- t(apply(ratings, 1, function(row) {
tab <- table(factor(row, levels = cats))
as.integer(tab)
}))
colnames(counts) <- cats
# category proportions across all items
p_j <- colSums(counts) / (N * n)
# agreement per item
P_i <- (rowSums(counts^2) - n) / (n * (n - 1))
# observed and expected agreement
P_bar <- mean(P_i)
P_e <- sum(p_j^2)
fkappa <- (P_bar - P_e) / (1 - P_e)
fkappa[1] 0.4302445Since the expected number of agreements is taken into consideration, random guesses should result in a Kappa close to zero.
set.seed(1984)
replicate(
100,
{
a <- rbinom(n = 100, size = 1, prob = 0.5)
b <- rbinom(n = 100, size = 1, prob = 0.5)
irr::kappa2(data.frame(a, b))$value
}
) |>
mean()[1] 0.003047436
sessionInfo()R version 4.5.0 (2025-04-11)
Platform: x86_64-pc-linux-gnu
Running under: Ubuntu 24.04.2 LTS
Matrix products: default
BLAS: /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3
LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.26.so; LAPACK version 3.12.0
locale:
[1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C
[3] LC_TIME=en_US.UTF-8 LC_COLLATE=en_US.UTF-8
[5] LC_MONETARY=en_US.UTF-8 LC_MESSAGES=en_US.UTF-8
[7] LC_PAPER=en_US.UTF-8 LC_NAME=C
[9] LC_ADDRESS=C LC_TELEPHONE=C
[11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
time zone: Etc/UTC
tzcode source: system (glibc)
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] irr_0.84.1 lpSolve_5.6.23 lubridate_1.9.4 forcats_1.0.0
[5] stringr_1.5.1 dplyr_1.1.4 purrr_1.0.4 readr_2.1.5
[9] tidyr_1.3.1 tibble_3.2.1 ggplot2_3.5.2 tidyverse_2.0.0
[13] workflowr_1.7.1
loaded via a namespace (and not attached):
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[5] digest_0.6.37 magrittr_2.0.3 timechange_0.3.0 evaluate_1.0.3
[9] grid_4.5.0 RColorBrewer_1.1-3 fastmap_1.2.0 rprojroot_2.0.4
[13] jsonlite_2.0.0 processx_3.8.6 whisker_0.4.1 ps_1.9.1
[17] promises_1.3.2 httr_1.4.7 scales_1.4.0 jquerylib_0.1.4
[21] cli_3.6.5 rlang_1.1.6 withr_3.0.2 cachem_1.1.0
[25] yaml_2.3.10 tools_4.5.0 tzdb_0.5.0 httpuv_1.6.16
[29] vctrs_0.6.5 R6_2.6.1 lifecycle_1.0.4 git2r_0.36.2
[33] fs_1.6.6 pkgconfig_2.0.3 callr_3.7.6 pillar_1.10.2
[37] bslib_0.9.0 later_1.4.2 gtable_0.3.6 glue_1.8.0
[41] Rcpp_1.0.14 xfun_0.52 tidyselect_1.2.1 rstudioapi_0.17.1
[45] knitr_1.50 farver_2.1.2 htmltools_0.5.8.1 rmarkdown_2.29
[49] compiler_4.5.0 getPass_0.2-4