# Source code for statsmodels.stats.inter_rater

```
# -*- coding: utf-8 -*-
"""Inter Rater Agreement
contains
--------
fleiss_kappa
cohens_kappa
aggregate_raters:
helper function to get data into fleiss_kappa format
to_table:
helper function to create contingency table, can be used for cohens_kappa
Created on Thu Dec 06 22:57:56 2012
Author: Josef Perktold
License: BSD-3
References
----------
Wikipedia: kappa's initially based on these two pages
https://en.wikipedia.org/wiki/Fleiss%27_kappa
https://en.wikipedia.org/wiki/Cohen's_kappa
SAS-Manual : formulas for cohens_kappa, especially variances
see also R package irr
TODO
----
standard errors and hypothesis tests for fleiss_kappa
other statistics and tests,
in R package irr, SAS has more
inconsistent internal naming, changed variable names as I added more
functionality
convenience functions to create required data format from raw data
DONE
"""
import numpy as np
from scipy import stats #get rid of this? need only norm.sf
class ResultsBunch(dict):
template = '%r'
def __init__(self, **kwds):
dict.__init__(self, kwds)
self.__dict__ = self
self._initialize()
def _initialize(self):
pass
def __str__(self):
return self.template % self
def _int_ifclose(x, dec=1, width=4):
'''helper function for creating result string for int or float
only dec=1 and width=4 is implemented
Parameters
----------
x : int or float
value to format
dec : 1
number of decimals to print if x is not an integer
width : 4
width of string
Returns
-------
xint : int or float
x is converted to int if it is within 1e-14 of an integer
x_string : str
x formatted as string, either '%4d' or '%4.1f'
'''
xint = int(round(x))
if np.max(np.abs(xint - x)) < 1e-14:
return xint, '%4d' % xint
else:
return x, '%4.1f' % x
[docs]def aggregate_raters(data, n_cat=None):
'''convert raw data with shape (subject, rater) to (subject, cat_counts)
brings data into correct format for fleiss_kappa
bincount will raise exception if data cannot be converted to integer.
Parameters
----------
data : array_like, 2-Dim
data containing category assignment with subjects in rows and raters
in columns.
n_cat : None or int
If None, then the data is converted to integer categories,
0,1,2,...,n_cat-1. Because of the relabeling only category levels
with non-zero counts are included.
If this is an integer, then the category levels in the data are already
assumed to be in integers, 0,1,2,...,n_cat-1. In this case, the
returned array may contain columns with zero count, if no subject
has been categorized with this level.
Returns
-------
arr : nd_array, (n_rows, n_cat)
Contains counts of raters that assigned a category level to individuals.
Subjects are in rows, category levels in columns.
'''
data = np.asarray(data)
n_rows = data.shape[0]
if n_cat is None:
#I could add int conversion (reverse_index) to np.unique
cat_uni, cat_int = np.unique(data.ravel(), return_inverse=True)
n_cat = len(cat_uni)
data_ = cat_int.reshape(data.shape)
else:
cat_uni = np.arange(n_cat) #for return only, assumed cat levels
data_ = data
tt = np.zeros((n_rows, n_cat), int)
for idx, row in enumerate(data_):
ro = np.bincount(row)
tt[idx, :len(ro)] = ro
return tt, cat_uni
[docs]def to_table(data, bins=None):
'''convert raw data with shape (subject, rater) to (rater1, rater2)
brings data into correct format for cohens_kappa
Parameters
----------
data : array_like, 2-Dim
data containing category assignment with subjects in rows and raters
in columns.
bins : None, int or tuple of array_like
If None, then the data is converted to integer categories,
0,1,2,...,n_cat-1. Because of the relabeling only category levels
with non-zero counts are included.
If this is an integer, then the category levels in the data are already
assumed to be in integers, 0,1,2,...,n_cat-1. In this case, the
returned array may contain columns with zero count, if no subject
has been categorized with this level.
If bins are a tuple of two array_like, then the bins are directly used
by ``numpy.histogramdd``. This is useful if we want to merge categories.
Returns
-------
arr : nd_array, (n_cat, n_cat)
Contingency table that contains counts of category level with rater1
in rows and rater2 in columns.
Notes
-----
no NaN handling, delete rows with missing values
This works also for more than two raters. In that case the dimension of
the resulting contingency table is the same as the number of raters
instead of 2-dimensional.
'''
data = np.asarray(data)
n_rows, n_cols = data.shape
if bins is None:
#I could add int conversion (reverse_index) to np.unique
cat_uni, cat_int = np.unique(data.ravel(), return_inverse=True)
n_cat = len(cat_uni)
data_ = cat_int.reshape(data.shape)
bins_ = np.arange(n_cat+1) - 0.5
#alternative implementation with double loop
#tt = np.asarray([[(x == [i,j]).all(1).sum() for j in cat_uni]
# for i in cat_uni] )
#other altervative: unique rows and bincount
elif np.isscalar(bins):
bins_ = np.arange(bins+1) - 0.5
data_ = data
else:
bins_ = bins
data_ = data
tt = np.histogramdd(data_, (bins_,)*n_cols)
return tt[0], bins_
[docs]def fleiss_kappa(table, method='fleiss'):
"""Fleiss' and Randolph's kappa multi-rater agreement measure
Parameters
----------
table : array_like, 2-D
assumes subjects in rows, and categories in columns
method : str
Method 'fleiss' returns Fleiss' kappa which uses the sample margin
to define the chance outcome.
Method 'randolph' or 'uniform' (only first 4 letters are needed)
returns Randolph's (2005) multirater kappa which assumes a uniform
distribution of the categories to define the chance outcome.
Returns
-------
kappa : float
Fleiss's or Randolph's kappa statistic for inter rater agreement
Notes
-----
no variance or hypothesis tests yet
Interrater agreement measures like Fleiss's kappa measure agreement relative
to chance agreement. Different authors have proposed ways of defining
these chance agreements. Fleiss' is based on the marginal sample distribution
of categories, while Randolph uses a uniform distribution of categories as
benchmark. Warrens (2010) showed that Randolph's kappa is always larger or
equal to Fleiss' kappa. Under some commonly observed condition, Fleiss' and
Randolph's kappa provide lower and upper bounds for two similar kappa_like
measures by Light (1971) and Hubert (1977).
References
----------
Wikipedia https://en.wikipedia.org/wiki/Fleiss%27_kappa
Fleiss, Joseph L. 1971. "Measuring Nominal Scale Agreement among Many
Raters." Psychological Bulletin 76 (5): 378-82.
https://doi.org/10.1037/h0031619.
Randolph, Justus J. 2005 "Free-Marginal Multirater Kappa (multirater
K [free]): An Alternative to Fleiss' Fixed-Marginal Multirater Kappa."
Presented at the Joensuu Learning and Instruction Symposium, vol. 2005
https://eric.ed.gov/?id=ED490661
Warrens, Matthijs J. 2010. "Inequalities between Multi-Rater Kappas."
Advances in Data Analysis and Classification 4 (4): 271-86.
https://doi.org/10.1007/s11634-010-0073-4.
"""
table = 1.0 * np.asarray(table) #avoid integer division
n_sub, n_cat = table.shape
n_total = table.sum()
n_rater = table.sum(1)
n_rat = n_rater.max()
#assume fully ranked
assert n_total == n_sub * n_rat
#marginal frequency of categories
p_cat = table.sum(0) / n_total
table2 = table * table
p_rat = (table2.sum(1) - n_rat) / (n_rat * (n_rat - 1.))
p_mean = p_rat.mean()
if method == 'fleiss':
p_mean_exp = (p_cat*p_cat).sum()
elif method.startswith('rand') or method.startswith('unif'):
p_mean_exp = 1 / n_cat
kappa = (p_mean - p_mean_exp) / (1- p_mean_exp)
return kappa
[docs]def cohens_kappa(table, weights=None, return_results=True, wt=None):
'''Compute Cohen's kappa with variance and equal-zero test
Parameters
----------
table : array_like, 2-Dim
square array with results of two raters, one rater in rows, second
rater in columns
weights : array_like
The interpretation of weights depends on the wt argument.
If both are None, then the simple kappa is computed.
see wt for the case when wt is not None
If weights is two dimensional, then it is directly used as a weight
matrix. For computing the variance of kappa, the maximum of the
weights is assumed to be smaller or equal to one.
TODO: fix conflicting definitions in the 2-Dim case for
wt : {None, str}
If wt and weights are None, then the simple kappa is computed.
If wt is given, but weights is None, then the weights are set to
be [0, 1, 2, ..., k].
If weights is a one-dimensional array, then it is used to construct
the weight matrix given the following options.
wt in ['linear', 'ca' or None] : use linear weights, Cicchetti-Allison
actual weights are linear in the score "weights" difference
wt in ['quadratic', 'fc'] : use linear weights, Fleiss-Cohen
actual weights are squared in the score "weights" difference
wt = 'toeplitz' : weight matrix is constructed as a toeplitz matrix
from the one dimensional weights.
return_results : bool
If True (default), then an instance of KappaResults is returned.
If False, then only kappa is computed and returned.
Returns
-------
results or kappa
If return_results is True (default), then a results instance with all
statistics is returned
If return_results is False, then only kappa is calculated and returned.
Notes
-----
There are two conflicting definitions of the weight matrix, Wikipedia
versus SAS manual. However, the computation are invariant to rescaling
of the weights matrix, so there is no difference in the results.
Weights for 'linear' and 'quadratic' are interpreted as scores for the
categories, the weights in the computation are based on the pairwise
difference between the scores.
Weights for 'toeplitz' are a interpreted as weighted distance. The distance
only depends on how many levels apart two entries in the table are but
not on the levels themselves.
example:
weights = '0, 1, 2, 3' and wt is either linear or toeplitz means that the
weighting only depends on the simple distance of levels.
weights = '0, 0, 1, 1' and wt = 'linear' means that the first two levels
are zero distance apart and the same for the last two levels. This is
the sample as forming two aggregated levels by merging the first two and
the last two levels, respectively.
weights = [0, 1, 2, 3] and wt = 'quadratic' is the same as squaring these
weights and using wt = 'toeplitz'.
References
----------
Wikipedia
SAS Manual
'''
table = np.asarray(table, float) #avoid integer division
agree = np.diag(table).sum()
nobs = table.sum()
probs = table / nobs
freqs = probs #TODO: rename to use freqs instead of probs for observed
probs_diag = np.diag(probs)
freq_row = table.sum(1) / nobs
freq_col = table.sum(0) / nobs
prob_exp = freq_col * freq_row[:, None]
assert np.allclose(prob_exp.sum(), 1)
#print prob_exp.sum()
agree_exp = np.diag(prob_exp).sum() #need for kappa_max
if weights is None and wt is None:
kind = 'Simple'
kappa = (agree / nobs - agree_exp) / (1 - agree_exp)
if return_results:
#variance
term_a = probs_diag * (1 - (freq_row + freq_col) * (1 - kappa))**2
term_a = term_a.sum()
term_b = probs * (freq_col[:, None] + freq_row)**2
d_idx = np.arange(table.shape[0])
term_b[d_idx, d_idx] = 0 #set diagonal to zero
term_b = (1 - kappa)**2 * term_b.sum()
term_c = (kappa - agree_exp * (1-kappa))**2
var_kappa = (term_a + term_b - term_c) / (1 - agree_exp)**2 / nobs
#term_c = freq_col * freq_row[:, None] * (freq_col + freq_row[:,None])
term_c = freq_col * freq_row * (freq_col + freq_row)
var_kappa0 = (agree_exp + agree_exp**2 - term_c.sum())
var_kappa0 /= (1 - agree_exp)**2 * nobs
else:
if weights is None:
weights = np.arange(table.shape[0])
#weights follows the Wikipedia definition, not the SAS, which is 1 -
kind = 'Weighted'
weights = np.asarray(weights, float)
if weights.ndim == 1:
if wt in ['ca', 'linear', None]:
weights = np.abs(weights[:, None] - weights) / \
(weights[-1] - weights[0])
elif wt in ['fc', 'quadratic']:
weights = (weights[:, None] - weights)**2 / \
(weights[-1] - weights[0])**2
elif wt == 'toeplitz':
#assume toeplitz structure
from scipy.linalg import toeplitz
#weights = toeplitz(np.arange(table.shape[0]))
weights = toeplitz(weights)
else:
raise ValueError('wt option is not known')
else:
rows, cols = table.shape
if (table.shape != weights.shape):
raise ValueError('weights are not square')
#this is formula from Wikipedia
kappa = 1 - (weights * table).sum() / nobs / (weights * prob_exp).sum()
#TODO: add var_kappa for weighted version
if return_results:
var_kappa = np.nan
var_kappa0 = np.nan
#switch to SAS manual weights, problem if user specifies weights
#w is negative in some examples,
#but weights is scale invariant in examples and rough check of source
w = 1. - weights
w_row = (freq_col * w).sum(1)
w_col = (freq_row[:, None] * w).sum(0)
agree_wexp = (w * freq_col * freq_row[:, None]).sum()
term_a = freqs * (w - (w_col + w_row[:, None]) * (1 - kappa))**2
fac = 1. / ((1 - agree_wexp)**2 * nobs)
var_kappa = term_a.sum() - (kappa - agree_wexp * (1 - kappa))**2
var_kappa *= fac
freqse = freq_col * freq_row[:, None]
var_kappa0 = (freqse * (w - (w_col + w_row[:, None]))**2).sum()
var_kappa0 -= agree_wexp**2
var_kappa0 *= fac
kappa_max = (np.minimum(freq_row, freq_col).sum() - agree_exp) / \
(1 - agree_exp)
if return_results:
res = KappaResults( kind=kind,
kappa=kappa,
kappa_max=kappa_max,
weights=weights,
var_kappa=var_kappa,
var_kappa0=var_kappa0)
return res
else:
return kappa
_kappa_template = '''\
%(kind)s Kappa Coefficient
--------------------------------
Kappa %(kappa)6.4f
ASE %(std_kappa)6.4f
%(alpha_ci)s%% Lower Conf Limit %(kappa_low)6.4f
%(alpha_ci)s%% Upper Conf Limit %(kappa_upp)6.4f
Test of H0: %(kind)s Kappa = 0
ASE under H0 %(std_kappa0)6.4f
Z %(z_value)6.4f
One-sided Pr > Z %(pvalue_one_sided)6.4f
Two-sided Pr > |Z| %(pvalue_two_sided)6.4f
'''
'''
Weighted Kappa Coefficient
--------------------------------
Weighted Kappa 0.4701
ASE 0.1457
95% Lower Conf Limit 0.1845
95% Upper Conf Limit 0.7558
Test of H0: Weighted Kappa = 0
ASE under H0 0.1426
Z 3.2971
One-sided Pr > Z 0.0005
Two-sided Pr > |Z| 0.0010
'''
class KappaResults(ResultsBunch):
'''Results for Cohen's kappa
Attributes
----------
kappa : cohen's kappa
var_kappa : variance of kappa
std_kappa : standard deviation of kappa
alpha : one-sided probability for confidence interval
kappa_low : lower (1-alpha) confidence limit
kappa_upp : upper (1-alpha) confidence limit
var_kappa0 : variance of kappa under H0: kappa=0
std_kappa0 : standard deviation of kappa under H0: kappa=0
z_value : test statistic for H0: kappa=0, is standard normal distributed
pvalue_one_sided : one sided p-value for H0: kappa=0 and H1: kappa>0
pvalue_two_sided : two sided p-value for H0: kappa=0 and H1: kappa!=0
distribution_kappa : asymptotic normal distribution of kappa
distribution_zero_null : asymptotic normal distribution of kappa under
H0: kappa=0
The confidence interval for kappa and the statistics for the test of
H0: kappa=0 are based on the asymptotic normal distribution of kappa.
'''
template = _kappa_template
def _initialize(self):
if 'alpha' not in self:
self['alpha'] = 0.025
self['alpha_ci'] = _int_ifclose(100 - 0.025 * 200)[1]
self['std_kappa'] = np.sqrt(self['var_kappa'])
self['std_kappa0'] = np.sqrt(self['var_kappa0'])
self['z_value'] = self['kappa'] / self['std_kappa0']
self['pvalue_one_sided'] = stats.norm.sf(self['z_value'])
self['pvalue_two_sided'] = stats.norm.sf(np.abs(self['z_value'])) * 2
delta = stats.norm.isf(self['alpha']) * self['std_kappa']
self['kappa_low'] = self['kappa'] - delta
self['kappa_upp'] = self['kappa'] + delta
self['distribution_kappa'] = stats.norm(loc=self['kappa'],
scale=self['std_kappa'])
self['distribution_zero_null'] = stats.norm(loc=0,
scale=self['std_kappa0'])
def __str__(self):
return self.template % self
```