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import numpy as np
from numpy.typing import NDArray
def fleiss_kappa(ratings: NDArray[np.int_]) -> float:
n, k = ratings.shape # n is number of items, k is number of raters
N = np.sum(
ratings[0]
) # Total number of ratings per item (assumed to be the same for all items)
# Calculating p_i
p_i = np.sum(ratings * (ratings - 1), axis=1) / (N * (N - 1))
P_o = np.mean(p_i) # Mean of p_i
# Calculating p_e
p = np.sum(ratings, axis=0) / (
n * N
) # Sum over items, divide by total number of ratings
P_e = np.sum(p**2)
# Calculating kappa
kappa = float((P_o - P_e) / (1 - P_e))
return kappa
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