ICC，全称为intraclass correlation coefficient，即组内相关系数。它是衡量和评价观察者间信度（inter-observer reliability）和复测信度（test-retest reliability）的信度系数（reliability coefficient）指标之一。它最先由Bartko于1966年用于测量和评价信度的大小。ICC等于个体的变异度除以总的变异度，故其值介于0～1之间。0表示不可信，1表示完全可信。一般认为信度系数低于0.4表示信度较差，大于0.75表示信度良好，对于定量资料常常需要更高的ICC值。
统计学中，组内相关系数常用于评价具有某种特定亲属关系（如双胞胎，兄弟姐妹等）的个体间某种定量属性（如遗传力）的相似程度，另一方面主要用于评价不同测定方法或评定者对同一定量测量结果的一致性或可靠性。有很多不同的ICC统计量，这些统计量并不估计相同的总体参数。由于对同一组数据不同ICC计算结果不同，有关这些ICC统计量的恰当应用一直是争论的焦点。1979年，Columbia University生物统计系的Shrout和Fleiss教授提出了研究者评价评定者测量结果的可靠性时选择恰当系数的准则。
下面是集中不用的实现ICC的方法：
第一种：这些可以通过rpy2包从 Python 中使用。
from rpy2.robjects import DataFrame, FloatVector, IntVectorfrom rpy2.robjects.packages import importrfrom math import isclose
groups = [1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4,
4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8]
values = [1, 2, 0, 1, 1, 3, 3, 2, 3, 8, 1, 4, 6, 4, 3,
3, 6, 5, 5, 6, 7, 5, 6, 2, 8, 7, 7, 9, 9, 9, 9, 8]
r_icc = importr("ICC")
df = DataFrame({"groups": IntVector(groups),
"values": FloatVector(values)})
icc_res = r_icc.ICCbare("groups", "values", data=df)
icc_val = icc_res[0] # icc_val now holds the icc value
# check whether icc value equals reference valueprint(isclose(icc_val, 0.728, abs_tol=0.001))
第二种：pengouin库以 6 种不同的方式计算 ICC，以及相关的置信水平和 p 值。
您可以使用pip install pingouin或安装它conda install -c conda-forge pingouin
import pingouin as pg
data = pg.read_dataset('icc')
icc = pg.intraclass_corr(data=data, targets='Wine', raters='Judge' , ratings='Scores')
第三种：R包psych具有类内相关 (ICC) 的实现，可计算多种类型的变体，包括 ICC(1,1)、ICC(1,k)、ICC(2,1)、ICC(2,k)、ICC (3,1) 和 ICC(3,k) 以及其他指标。
1. 首先安装psych并lme4在 R 中：
install.packages("psych")
install.packages("lme4")
2. 使用 rpy2 在 Python 中计算 ICC 系数：
import rpy2from rpy2.robjects import IntVector, pandas2rifrom rpy2.robjects.packages import importr
psych = importr("psych")
values = rpy2.robjects.r.matrix(
IntVector(
[9, 2, 5, 8,
6, 1, 3, 2,
8, 4, 6, 8,
7, 1, 2, 6,
10, 5, 6, 9,
6, 2, 4, 7]),
ncol=4, byrow=True
)
icc = psych.ICC(values)# Convert to Pandas DataFrame
icc_df = pandas2ri.rpy2py(icc[0])
第四种：import osimport numpy as npfrom numpy import ones, kron, mean, eye, hstack, dot, tilefrom numpy.linalg import pinvdef icc(Y, icc_type='ICC(2,1)'):
''' Calculate intraclass correlation coefficient
ICC Formulas are based on:
Shrout, P. E., & Fleiss, J. L. (1979). Intraclass correlations: uses in
assessing rater reliability. Psychological bulletin, 86(2), 420.
icc1: x_ij = mu + beta_j + w_ij
icc2/3: x_ij = mu + alpha_i + beta_j + (ab)_ij + epsilon_ij
Code modifed from nipype algorithms.icc
https://github.com/nipy/nipype/blob/master/nipype/algorithms/icc.py
Args:
Y: The data Y are entered as a 'table' ie. subjects are in rows and repeated
measures in columns
icc_type: type of ICC to calculate. (ICC(2,1), ICC(2,k), ICC(3,1), ICC(3,k))
Returns:
ICC: (np.array) intraclass correlation coefficient
'''
[n, k] = Y.shape
# Degrees of Freedom
dfc = k - 1
dfe = (n - 1) * (k-1)
dfr = n - 1
# Sum Square Total
mean_Y = np.mean(Y)
SST = ((Y - mean_Y) ** 2).sum()
# create the design matrix for the different levels
x = np.kron(np.eye(k), np.ones((n, 1))) # sessions
x0 = np.tile(np.eye(n), (k, 1)) # subjects
X = np.hstack([x, x0])
# Sum Square Error
predicted_Y = np.dot(np.dot(np.dot(X, np.linalg.pinv(np.dot(X.T, X))),
X.T), Y.flatten('F'))
residuals = Y.flatten('F') - predicted_Y
SSE = (residuals ** 2).sum()
MSE = SSE / dfe
# Sum square column effect - between colums
SSC = ((np.mean(Y, 0) - mean_Y) ** 2).sum() * n
MSC = SSC / dfc # / n (without n in SPSS results)
# Sum Square subject effect - between rows/subjects
SSR = SST - SSC - SSE
MSR = SSR / dfr
if icc_type == 'icc1':
# ICC(2,1) = (mean square subject - mean square error) /
# (mean square subject + (k-1)*mean square error +
# k*(mean square columns - mean square error)/n)
# ICC = (MSR - MSRW) / (MSR + (k-1) * MSRW)
NotImplementedError("This method isn't implemented yet.")
elif icc_type == 'ICC(2,1)' or icc_type == 'ICC(2,k)':
# ICC(2,1) = (mean square subject - mean square error) /
# (mean square subject + (k-1)*mean square error +
# k*(mean square columns - mean square error)/n)
if icc_type == 'ICC(2,k)':
k = 1
ICC = (MSR - MSE) / (MSR + (k-1) * MSE + k * (MSC - MSE) / n)
elif icc_type == 'ICC(3,1)' or icc_type == 'ICC(3,k)':
# ICC(3,1) = (mean square subject - mean square error) /
# (mean square subject + (k-1)*mean square error)
if icc_type == 'ICC(3,k)':
k = 1
ICC = (MSR - MSE) / (MSR + (k-1) * MSE)
return ICC