Randolph's Free-Marginal Multirater Kappa

Computes Randolph's free-marginal multirater kappa for $m$ raters over $N$ subjects. This agreement coefficient does not assume fixed marginal distributions (i.e., it is free-marginal).

Usage

randolphKappa(x, levels = NULL, ...)

Arguments

x: a matrix of size $N \times m$ with subjects in rows and raters in columns; cells contain the assigned categories
levels: the categories a rater could have chosen from, or a single number giving how many there were. Defaults to NULL, i.e. the categories actually observed in x. See Details.
...: reserved for future bootstrap options. No confidence interval is implemented yet, so any argument given here raises an error instead of being ignored.

Value

a numeric scalar containing Randolph's kappa

Details

Let $k$ be the number of categories, $m$ the number of raters, and $N$ the number of subjects. Randolph's kappa is $$\kappa = \frac{P_o - 1/k}{1 - 1/k},$$ where the observed agreement $P_o$ is the proportion of agreeing rater pairs, $$P_o = \frac{1}{N} \sum_{i=1}^{N} \frac{\sum_{j} n_{ij}(n_{ij}-1)}{m(m-1)}.$$ Here, $n_{ij}$ denotes the number of raters who assigned subject $i$ to category $j$. This is the same observed agreement as in Fleiss' kappa (and as computed by percAgreement()); Randolph's coefficient differs from Fleiss' only in the chance agreement $P_e$, which is fixed at $1/k$ instead of being estimated from the marginals.

$P_e = 1/k$ refers to the categories a rater could have chosen, not to those that happen to occur in the data. If a category was available but never used, the default levels = NULL understates $k$ and thus overstates chance agreement, which biases $\kappa$ downwards. Supply levels whenever the coding scheme is known.

Long-format ratings can first be reshaped with raterFrame().

References

Randolph, J. J. (2005). Free-Marginal Multirater Kappa (multirater $\kappa_{\mathrm{free}}$): An Alternative to Fleiss’ Fixed-Marginal Multirater Kappa. Online submission.

Examples

## Matrix (subjects x raters), 5 subjects, 3 raters
x <- matrix(c(
  1,1,1,
  2,2,2,
  1,2,1,
  3,3,3,
  2,2,1
), ncol = 3, byrow = TRUE)
randolphKappa(x)
#> [1] 0.6

# the raters could have chosen from five categories, not just the three
# they used
randolphKappa(x, levels = 5)
#> [1] 0.6666667

## Long format with a formula
df <- data.frame(
  subject = rep(1:5, each = 3),
  rater   = rep(paste0("r", 1:3), times = 5),
  rating  = c(1,1,1, 2,2,2, 1,2,1, 3,3,3, 2,2,1)
)
randolphKappa(raterFrame(rating ~ subject | rater, 
                         data = df, dropSubj=TRUE))
#> [1] 0.6