ENH: inter_rater.fleiss_kappa p-values and confidence interval

[josef-pkt]

https://stackoverflow.com/questions/78323943/statistic-values-of-fleiss-kappa-using-statsmodels-stats-inter-rater/78324041#78324041

Note our fleiss_kappa includes also randolph's kappa, i.e. we would need p-values also for those.

(needs reference, I have not looked at this in a long time)

copy from answer

import numpy as np
import pandas as pd
from statsmodels.stats.inter_rater import fleiss_kappa
from scipy.stats import norm

np.random.seed(42)

data = {
    f'Item{i+1}': np.random.choice([0, 1, 2], size=30, p=[0.33, 0.33, 0.34]) for i in range(15)
}
df = pd.DataFrame(data)

formatted_data = {
    f"Category {cat}": [(df[item] == cat).sum() for item in df] for cat in range(3)
}
formatted_df = pd.DataFrame(formatted_data)

kappa = fleiss_kappa(formatted_df.values)

category_totals = formatted_df.sum(axis=1) 
p = np.sum((category_totals / (30 * 15))**2)  

n = 15  
k = 3   
N = n * 30  

variance = (1 / (N * (n - 1))) * (N * p * (1 - p) + (n * (k - 1) * (p - (1 / k)**2)))
if variance > 0:
    z_value = kappa / np.sqrt(variance)
    p_value = 2 * (1 - norm.cdf(np.abs(z_value)))
    z_critical = norm.ppf(0.975)
    margin_of_error = z_critical * np.sqrt(variance)
    lower_bound = kappa - margin_of_error
    upper_bound = kappa + margin_of_error

    print("Fleiss' kappa:", kappa)
    print("Z-value:", z_value)
    print("P-value:", p_value)
    print("Confidence interval (95%):", (lower_bound, upper_bound))
else:
    print("Variance calculation error: Non-positive variance", variance)

Fleiss' kappa: -0.008536683290635389
Z-value: -0.1312124600755962
P-value: 0.8956072394628303
Confidence interval (95%): (-0.13605194965657783, 0.11897858307530704)