ICC

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Basic

ICC: intra-class correlation

Interpretation

  • Intraclass Correlation Coefficient: Definition + Example we’d like to measure the absolute agreement among judges. ... An ICC of 0.782 indicates that the exams can be rated with “good” reliability by different raters.
  • From the wikipedia page
    • If ICC is high, values from the same group tend to be similar.
    • If ICC is small (~0), there is no tendency for values from the same group to be similar.
  • If ICC is high (consider the subjects/students and raters/teachers example),
    • It indicates a high similarity between values from the same group/subject (the set of measurements made by different raters on the same subject).
    • It indicates high reliability/agreement among raters/teachers about ...

ICC(1,1), ICC(2,k), ICC(3,k)

Intraclass Correlation Coefficient in R

How to choose the correct ICC forms:

  • ICC(1)/One-Way Random Effects Model: Rarely used in clinical reliability analysis. Suppose a new medical device has been developed to measure blood pressure, and you want to assess its reliability. You randomly select 10 patients and measure their blood pressure using this new device. To get a measure of reliability, you repeat this process three times on different days.
  • Two-Way Random Effects Model. This model is appropriate for evaluating rater-based clinical assessment methods that are designed for routine clinical use. This model is used when all subjects (essays in this case) are rated by all raters (teachers in this case), and both subjects and raters are considered to be random effects sampled from larger populations. The ICC calculated under this model measures the reliability of the ratings, assuming that if we were to repeat the competition with new essays and new teachers, the ratings would be similarly reliable.
    • ICC(2,1) or ICC(A,1): This measures the absolute agreement among the raters. It takes into account both variance due to the raters and variance due to the interaction between raters and subjects.
    • ICC(2,k) or ICC(A,k): This measures the consistency among the raters. It does not take the interaction term into account.
    • The choice between absolute agreement and consistency depends on whether it’s important that the raters give the exact same scores (absolute agreement), or whether it’s sufficient that they rank the subjects in the same order (consistency).
  • Two-Way Mixed Effects Model. The two-way mixed-effects model is less commonly used in inter-rater reliability analysis. In this model, subjects (e.g., essays in the classroom example) are considered as a random sample from a larger population, while raters (e.g., teachers) are considered as the only raters of interest. This model is used when you want to generalize the findings to the larger population of subjects, but you’re only interested in the specific set of raters used in the study.
    • ICC(3,1) or ICC(C,1): This measures the absolute agreement among the raters. It takes into account both variance due to the raters and variance due to the interaction between raters and subjects.
    • ICC(3,k) or ICC(C,k): This measures the consistency among the raters. It does not take the interaction term into account.
    • The choice between absolute agreement and consistency depends on whether it’s important that the raters give the exact same scores (absolute agreement), or whether it’s sufficient that they rank the subjects in the same order (consistency).

Derive from Correlation

Random intercept model

R packages

The main input is a matrix of n subjects x p raters. Each rater is a class/group.

  • psych: ICC()
  • irr: icc() for one-way or two-way model. This works on my data 30k by 58. The default option gives ICC(1). It can also compute ICC(A,1)/agreement and ICC(C,1)/consistency.
  • psy: icc(). No options are provided. I got an error: vector memory exhausted (limit reached?) when the data is 30k by 58.
  • rptR:

Negative ICC

Examples

psych package data

It shows ICC1 = ICC(1,1) 

R> library(psych)
R> (o <- ICC(anxiety, lmer=FALSE) )
Call: ICC(x = anxiety, lmer = FALSE)

Intraclass correlation coefficients 
                         type  ICC   F df1 df2     p lower bound upper bound
Single_raters_absolute   ICC1 0.18 1.6  19  40 0.094     -0.0405        0.44
Single_random_raters     ICC2 0.20 1.8  19  38 0.056     -0.0045        0.45
Single_fixed_raters      ICC3 0.22 1.8  19  38 0.056     -0.0073        0.48
Average_raters_absolute ICC1k 0.39 1.6  19  40 0.094     -0.1323        0.70
Average_random_raters   ICC2k 0.43 1.8  19  38 0.056     -0.0136        0.71
Average_fixed_raters    ICC3k 0.45 1.8  19  38 0.056     -0.0222        0.73

 Number of subjects = 20     Number of Judges =  3

R> library(irr)
R> (o2 <- icc(anxiety, model="oneway")) # subjects be considered as random effects
 Single Score Intraclass Correlation

   Model: oneway 
   Type : consistency 

   Subjects = 20 
     Raters = 3 
     ICC(1) = 0.175

 F-Test, H0: r0 = 0 ; H1: r0 > 0 
   F(19,40) = 1.64 , p = 0.0939 

 95%-Confidence Interval for ICC Population Values:
  -0.077 < ICC < 0.484

R> o$results["Single_raters_absolute", "ICC"]
[1] 0.1750224
R> o2$value
[1] 0.1750224

R> icc(anxiety, model="twoway", type = "consistency")
 Single Score Intraclass Correlation

   Model: twoway 
   Type : consistency 

   Subjects = 20 
     Raters = 3 
   ICC(C,1) = 0.216

 F-Test, H0: r0 = 0 ; H1: r0 > 0 
   F(19,38) = 1.83 , p = 0.0562 

 95%-Confidence Interval for ICC Population Values:
  -0.046 < ICC < 0.522
R> icc(anxiety, model="twoway", type = "agreement")
 Single Score Intraclass Correlation

   Model: twoway 
   Type : agreement 

   Subjects = 20 
     Raters = 3 
   ICC(A,1) = 0.198

 F-Test, H0: r0 = 0 ; H1: r0 > 0 
 F(19,39.7) = 1.83 , p = 0.0543 

 95%-Confidence Interval for ICC Population Values:
  -0.039 < ICC < 0.494

library(magrittr)
library(tidyr)
library(ggplot2)

set.seed(1)
r1 <- round(rnorm(20, 10, 4))
r2 <- round(r1 + 10 + rnorm(20, 0, 2))
r3 <- round(r1 + 20 + rnorm(20, 0, 2))
df <- data.frame(r1, r2, r3) %>% pivot_longer(cols=1:3)
df %>% ggplot(aes(x=name, y=value)) + geom_point()

df0 <- cbind(r1, r2, r3)
icc(df0, model="oneway")  #  ICC(1) = -0.262  --> Negative. 
                          #  Shift can mess up the ICC. See wikipedia.
icc(df0, model="twoway", type = "consistency")  # ICC(C,1) = 0.846 --> Make sense
icc(df0, model="twoway", type = "agreement")    # ICC(A,1) = 0.106 --> Why?

ICC(df0)
Call: ICC(x = df0, lmer = T)

Intraclass correlation coefficients 
                         type   ICC     F df1 df2       p lower bound upper bound
Single_raters_absolute   ICC1 -0.26  0.38  19  40 9.9e-01     -0.3613      -0.085
Single_random_raters     ICC2  0.11 17.43  19  38 2.9e-13      0.0020       0.293
Single_fixed_raters      ICC3  0.85 17.43  19  38 2.9e-13      0.7353       0.920
Average_raters_absolute ICC1k -1.65  0.38  19  40 9.9e-01     -3.9076      -0.307
Average_random_raters   ICC2k  0.26 17.43  19  38 2.9e-13      0.0061       0.555
Average_fixed_raters    ICC3k  0.94 17.43  19  38 2.9e-13      0.8929       0.972

 Number of subjects = 20     Number of Judges =  3

Wine rating

Intraclass Correlation: Multiple Approaches from David C. Howell. The data appeared on the paper by Shrout and Fleiss 1979.

Winerating.png 

library(magrittr)
library(psych); library(lme4)
rating <- matrix(c(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)
(o <- ICC(rating))
o$results[, 1:2]
#                         type       ICC
#Single_raters_absolute   ICC1 0.1657423  # match with icc(, "oneway")
#Single_random_raters     ICC2 0.2897642  # match with icc(, "twoway", "agreement")
#Single_fixed_raters      ICC3 0.7148415  # match with icc(, "twoway", "consistency")
#Average_raters_absolute ICC1k 0.4427981
#Average_random_raters   ICC2k 0.6200510
#Average_fixed_raters    ICC3k 0.9093159

# Plot
rating2 <- data.frame(rating) %>% 
  dplyr::bind_cols(data.frame(subj = paste0("s", 1:nrow(rating)))) %>% 
  tidyr::pivot_longer(1:4, names_to="group", values_to="y")

rating2 %>% ggplot(aes(x=group, y=y)) + geom_point()

library(irr)
icc(rating, "oneway")
# Single Score Intraclass Correlation
#
#   Model: oneway 
#   Type : consistency 
#
#   Subjects = 6 
#     Raters = 4 
#     ICC(1) = 0.166
#
# F-Test, H0: r0 = 0 ; H1: r0 > 0 
#    F(5,18) = 1.79 , p = 0.165 
#
# 95%-Confidence Interval for ICC Population Values:
#  -0.133 < ICC < 0.723

icc(rating, "twoway", "agreement")
# Single Score Intraclass Correlation
#
#   Model: twoway 
#   Type : agreement 
#
#   Subjects = 6 
#     Raters = 4 
#   ICC(A,1) = 0.29
#
# F-Test, H0: r0 = 0 ; H1: r0 > 0 
#  F(5,4.79) = 11 , p = 0.0113 
#
# 95%-Confidence Interval for ICC Population Values:
#  0.019 < ICC < 0.761

icc(rating, "twoway", "consistency")
# Single Score Intraclass Correlation
#
#   Model: twoway 
#   Type : consistency 
#
#   Subjects = 6 
#     Raters = 4 
#   ICC(C,1) = 0.715
#
# F-Test, H0: r0 = 0 ; H1: r0 > 0 
#    F(5,15) = 11 , p = 0.000135 
#
# 95%-Confidence Interval for ICC Population Values:
#  0.342 < ICC < 0.946

anova(aov(y ~ subj + group, rating2))
# Analysis of Variance Table
#
# Response: y
#           Df Sum Sq Mean Sq F value    Pr(>F)    
# subj       5 56.208  11.242  11.027 0.0001346 ***
# group      3 97.458  32.486  31.866 9.454e-07 ***
# Residuals 15 15.292   1.019                      
# ---
# Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(11.242 - (97.458+15.292)/18) / (11.242 + 3*(97.458+15.292)/18)
# [1] 0.165751   # ICC(1) = (BMS - WMS) / (BMS + (k-1)WMS)
                 # k = number of raters

(11.242 - 1.019) / (11.242 + 3*1.019 + 4*(32.486-1.019)/6)
# [1] 0.2897922  # ICC(2,1) = (BMS - EMS) / (BMS + (k-1)EMS + k(JMS-EMS)/n)
                 # n = number of subjects/targets

(11.242 - 1.019) / (11.242 + 3*1.019)
# [1] 0.7149451  # ICC(3,1)

Wine rating2

Intraclass correlation (from Real Statistics Using Excel) with a simple example. 

R> wine <- cbind(c(1,1,3,6,6,7,8,9), c(2,3,8,4,5,5,7,9), 
                 c(0,3,1,3,5,6,7,9), c(1,2,4,3,6,2,9,8))
R> icc(wine, model="oneway")
 Single Score Intraclass Correlation

   Model: oneway 
   Type : consistency 

   Subjects = 8 
     Raters = 4 
     ICC(1) = 0.728

 F-Test, H0: r0 = 0 ; H1: r0 > 0 
    F(7,24) = 11.7 , p = 2.18e-06 

 95%-Confidence Interval for ICC Population Values:
  0.434 < ICC < 0.927

# For one-way random model, the order of raters is not important
R> wine2 <- wine
R> for(j in 1:8) wine2[j, ] <- sample(wine[j,])
R> icc(wine2, model="oneway")
 Single Score Intraclass Correlation

   Model: oneway 
   Type : consistency 

   Subjects = 8 
     Raters = 4 
     ICC(1) = 0.728

 F-Test, H0: r0 = 0 ; H1: r0 > 0 
    F(7,24) = 11.7 , p = 2.18e-06 

 95%-Confidence Interval for ICC Population Values:
  0.434 < ICC < 0.927

R> icc(wine, model="twoway", type="agreement")
 Single Score Intraclass Correlation

   Model: twoway 
   Type : agreement 

   Subjects = 8 
     Raters = 4 
   ICC(A,1) = 0.728

 F-Test, H0: r0 = 0 ; H1: r0 > 0 
    F(7,24) = 11.8 , p = 2.02e-06 

 95%-Confidence Interval for ICC Population Values:
  0.434 < ICC < 0.927

R> icc(wine, model="twoway", type="consistency")
 Single Score Intraclass Correlation

   Model: twoway 
   Type : consistency 

   Subjects = 8 
     Raters = 4 
   ICC(C,1) = 0.729

 F-Test, H0: r0 = 0 ; H1: r0 > 0 
    F(7,21) = 11.8 , p = 5.03e-06 

 95%-Confidence Interval for ICC Population Values:
  0.426 < ICC < 0.928

Two-way fixed effects model 

R> wine3 <- data.frame(wine) %>% 
            dplyr::bind_cols(data.frame(subj = paste0("s", 1:8))) %>% 
            tidyr::pivot_longer(1:4, names_to="group", values_to="y")
R> wine3 %>% ggplot(aes(x=group, y=y)) + geom_point()

R> anova(aov(y ~ subj + group, data = wine3))
Analysis of Variance Table

Response: y
          Df  Sum Sq Mean Sq F value    Pr(>F)    
subj       7 188.219 26.8884 11.7867 5.026e-06 ***
group      3   7.344  2.4479  1.0731    0.3818    
Residuals 21  47.906  2.2813                      
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

R> anova(aov(y ~ group + subj, data = wine3))
Analysis of Variance Table

Response: y
          Df  Sum Sq Mean Sq F value    Pr(>F)    
group      3   7.344  2.4479  1.0731    0.3818    
subj       7 188.219 26.8884 11.7867 5.026e-06 ***
Residuals 21  47.906  2.2812                      
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

R> library(car)
R> Anova(aov(y ~ subj + group, data = wine3))
Anova Table (Type II tests)

Response: y
           Sum Sq Df F value    Pr(>F)    
subj      188.219  7 11.7867 5.026e-06 ***
group       7.344  3  1.0731    0.3818    
Residuals  47.906 21                      
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Comparing intraclass correlation to Pearson correlation coefficient

Comparing intraclass correlation to Pearson correlation coefficient (with R)

In the second example (two raters, slope != 1), Pearson correlation is 1 but ICC is not large.

Is there an example where intra-class correlation ICC is large but Pearson correlation is small from two raters? One example where this can occur is when the raters are measuring the same quantity, but with different units or scales. In this case, the ICC can indicate a high level of agreement between the raters, while the Pearson correlation may not show a strong relationship between the two sets of measurements.

Cluster randomized trials

Fleiss Kappa statistic

Cohen's Kappa statistic

ICC and Kappa totally disagree

Concordance correlation coefficient/CCC

  • https://en.wikipedia.org/wiki/Concordance_correlation_coefficient
    [math]\displaystyle{ \rho_c = \frac{2\rho\sigma_x\sigma_y}{\sigma_x^2 + \sigma_y^2 + (\mu_x - \mu_y)^2}, }[/math]
    where [math]\displaystyle{ \mu_x }[/math] and [math]\displaystyle{ \mu_y }[/math] are the means for the two variables and [math]\displaystyle{ \sigma^2_x }[/math] and [math]\displaystyle{ \sigma^2_y }[/math] are the corresponding variances. [math]\displaystyle{ \rho }[/math] is the correlation coefficient between the two variables.
  • The CCC is used to evaluate the degree of agreement between two sets of measurements or observations, and it provides a value between -1 and 1.
  • The CCC is an extension of the Pearson correlation coefficient, but it takes into account not only the magnitude of the correlation between two variables but also the difference in their means. This makes it a more robust measure of agreement, especially when dealing with data that have a wide range of values or when there is a bias between the two sets of measurements.
  • The CCC ranges from -1 to 1, where values close to 1 indicate strong agreement between the two sets of measurements, values close to 0 indicate no agreement, and values close to -1 indicate strong disagreement or inverse agreement. A CCC of 0.8 or higher is generally considered to indicate good agreement, while a value of less than 0.5 indicates poor agreement.
  • Concordance Correlation Coefficient R-blogger
  • How to compare concordance correlation coefficient to Pearson's r?
  • R functions: DescTools::CCC(), yardstick::ccc(), epiR::epi.ccc()

ICC vs CCC

  • Methods for evaluating the agreement between diagnostic tests Morgan 2015
    • At the heart of this issue is quantifying the agreement between the results of two (or more) tests. That is, the two tests should yield similar results when applied to the same subject.
    • While the paired t test could then be used to test whether the mean difference significantly differs from zero, this test cannot provide evidence that there is agreement. That is, rejecting the null hypothesis of no difference between the two sets of test results would only allow us to say that the tests do not agree; failing to reject this hypothesis would not constitute proof that the tests agree.
    • M vs A plot in gene expression is an application of Bland-Altman plot. M = log2(R/G), the intensity log ratio and A is mean log intensity. The M value provides a measure of the fold change in expression between the two samples. By taking the log of the ratio of expression values, the M value is transformed into a scale that is symmetrical around zero. The log transformation of the ratio of expression values also helps to stabilize the variance of the data, making it easier to identify differentially expressed genes or transcripts between the two samples. The MA plot uses the average log expression (A value) in the x-axis because it provides a measure of the overall expression level of each gene or transcript, which is useful for identifying systematic biases in the data. See figures on SciencDirect. At each point on MA plot, M value is adjusted by subtracting the estimated bias (the height of the loess curve) at the same A value, that is, log2(R/G)=log2(R/G)−c(A), where c(A) is the lowess fit to MA plot.
  • How to Create a Bland-Altman Plot in R (Step-by-Step) y=A-B, x=(A+B)/2, BlandAltmanLeh R package, y=A-B=difference, x=(A+B)/2=average.
  • A simple example (Y=X+3 from wikipedia) 
    df <- 1:5; df <- cbind(df, df+3)

cor(df[, 1], df[, 2])
# 1

library(irr)
icc(df, model="twoway", type = "agreement")  
#  Single Score Intraclass Correlation
#
#    Model: twoway 
#    Type : agreement 
# 
#    Subjects = 5 
#      Raters = 2 
#    ICC(A,1) = 0.357

library(DescTools)
CCC(df[,1], df[, 2])
# $rho.c
#         est     lwr.ci    upr.ci
# 1 0.3076923 -0.0584809 0.6008884
# 
# $s.shift
# [1] 1
# 
# $l.shift
# [1] 2.12132
# 
# $C.b
# [1] 0.3076923

# Note that the sample variance/covariance estimators, we use "n" as denominator
2*cov(df[, 1], df[, 2]) / (sd(df[,1])^2 + sd(df[,2])^2 + (mean(df[,1] - mean(df[,2]))^2))
# [1] 0.3571429
2*cov(df[, 1], df[, 2])*4/5 / (var(df[,1])*4/5 + var(df[,2])*4/5 + (mean(df[,1] - mean(df[,2]))^2))
[1] 0.3076923

# And if we use one-way ANOVA
icc(df, model="oneway")  
#  Single Score Intraclass Correlation
# 
#    Model: oneway 
#    Type : consistency 
#
#   Subjects = 5 
#     Raters = 2 
#     ICC(1) = 0.0526

# Ronald Fisher
n <- 5
xbar <- mean(df)
s2 <- (sum((df[,1]-xbar)^2) + sum((df[,2]-xbar)^2))/ (2*n)
sum( (df[,1]-xbar)*(df[,2]-xbar) ) / (n*s2)
# [1] -0.05882353

Ordinal data

Estimating intracluster correlation for ordinal data 2023

Pitfalls

Common pitfalls in statistical analysis: Measures of agreement 2017

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