Revision as of 20:24, 11 March 2023

Basic

ICC: intra-class correlation

https://en.wikipedia.org/wiki/Intraclass_correlation (the random effect [math]\displaystyle{ \alpha_j }[/math] in the one-way random model should be subjects, not raters)
- For the 1st 2 dot plots on the wikipedia, y-axis is raters values, x-axis is groups. However in the R package such as like irr, rows = subjects, columns = raters (reversed).
- In the 3 scenario scatterplots on the wikipedia, x-axis represents rater 1 & y-axis represents rater 2. Different points represent different subjects.
- measure the reliability/consistency/agreement of ratings in studies where there are two or more raters (主角).
- It describes how strongly units in the same group resemble each other. (Cf ICC(1,1) model, the group has the same concept of subject. That is, ICC is used to measure to what extent are the ratings within a subject homogeneous.) So the dot plots shown on the wikipedia should be rotated in order to match with the row/column meaning in the real data matrix.
- (Early def of ICC) In the case of paired data (N groups, each group has 2 values), The key difference between this ICC and the interclass (Pearson) correlation is that the data are pooled to estimate the mean and variance. The reason for this is that in the setting where an intraclass correlation is desired, the pairs are considered to be unordered.
- Intraclass Correlation from Statistics How To
- Shrout, P.E., Fleiss, J.L. (1979), Intraclass correlation: uses in assessing rater reliability, Psychological Bulletin, 86, 420-428.
- ICC(1,1): each subject is measured by a different set of k randomly selected raters?;
- ICC(2,1): k raters are randomly selected, then, each subject is measured by the same set of k raters;
- ICC(3,1): similar to ICC(2,1) but k raters are fixed.
(Modern ICC definitions) [math]\displaystyle{ Y_{ij} = \mu + \alpha_i + \varepsilon_{ij}, }[/math] where [math]\displaystyle{ \alpha_i }[/math] is the random effect from subject i (or group/target but not rater/judge),

[math]\displaystyle{ ICC(1,1) = \frac{\sigma_\alpha^2}{\sigma_\alpha^2+\sigma_\varepsilon^2}. }[/math]

The intraclass correlation is the ratio of the variance attributable to subjects divided by the total of subject and error variances. See also the formula and estimation here. The intuition can be found in examples where the 1st case has large [math]\displaystyle{ \sigma_\alpha^2 }[/math] and the other does not.

The reason ICC(1,1) defined above is intraclass correlation [math]\displaystyle{ Corr(Y_{ij}, Y_{ik}) }[/math] which can be derived here: Calculating the Intraclass Correlation Coefficient (ICC) in SAS. Excellent!! To what extent are the ratings within a subject homogeneous? For continuous data, ICC often used to assess interrater reliability.
*Intraclass correlation – A discussion and demonstration of basic features Liljequist, PLOS One 2019. It gives a good comprehensive review with formulas (including a derivation of expected mean square relations) and some simulation studies. The table also shows the differences of agreement and consistency.
A Guideline of Selecting and Reporting Intraclass Correlation Coefficients for Reliability Research Koo 2016.
Intraclass correlation coefficient vs. F-test (one-way ANOVA)?
Good ICC, bad CV or vice-versa, how to interpret? Two examples are given show the difference of between-sample (think of using one value such as the average to represent a sample) variability and within-sample variability.
Intraclass Correlation Coefficient in R. icc() [irr package] and the function ICC() [psych package] are considered with a simple example.
Simulation. https://stats.stackexchange.com/q/135345 chapter 4 of this book: Headrick, T. C. (2010). Statistical simulation: Power method polynomials and other transformations. Boca Raton, FL: Chapman & Hall
https://bookdown.org/roback/bookdown-bysh/ch-corrdata.html
https://www.sas.com/content/dam/SAS/en_ca/User%20Group%20Presentations/Health-User-Groups/Maki-InterraterReliability-Apr2014.pdf

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.

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.
- Cookbook for R
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

Can ICC values be negative?
Negative Values of the Intraclass Correlation Coefficient Are Not Theoretically Possible
What to do with negative ICC values? Adjust the test or interpret it differently?
Intraclass correlation – A discussion and demonstration of basic features on PLOS One. A negative ICC(1) is simply a bad (unfortunate) estimate. This may occur by chance, especially if the sample size (number of subjects n) is small.

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.

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

Fleiss Kappa statistic

https://en.wikipedia.org/wiki/Fleiss%27_kappa
Fleiss’ Kappa in R: For Multiple Categorical Variables
It was used on the paper Comparison of pathway and gene-level models for cancer prognosis prediction Zheng 2020 to evaluate the stability of prediction models (subjects=genes/pathways, raters=replicates of CV prediction models).

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()

@@ Line 381: / Line 381: @@
 = Cohen's Kappa statistic =
 [https://stats.stackexchange.com/a/64997 ICC and Kappa totally disagree]
+= Concordance correlation coefficient/CCC =
+<ul>
+<li>https://en.wikipedia.org/wiki/Concordance_correlation_coefficient
+:<math>\rho_c = \frac{2\rho\sigma_x\sigma_y}{\sigma_x^2 + \sigma_y^2 + (\mu_x - \mu_y)^2},</math>
+where <math>\mu_x</math> and <math>\mu_y</math> are the means for the two variables and <math>\sigma^2_x</math> and <math>\sigma^2_y</math> are the corresponding variances. <math>\rho</math> is the correlation coefficient between the two variables.
+<li>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.
+<li>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'''.
+<li>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.
+<li>[https://www.r-bloggers.com/2020/01/concordance-correlation-coefficient/ Concordance Correlation Coefficient] R-blogger
+<li>[https://stats.stackexchange.com/a/395234 How to compare concordance correlation coefficient to Pearson's r?]
+<li>R functions: [https://www.rdocumentation.org/packages/DescTools/versions/0.99.48/topics/CCC DescTools::CCC()], [https://yardstick.tidymodels.org/reference/ccc.html yardstick::ccc()], [https://www.rdocumentation.org/packages/epiR/versions/2.0.43/topics/epi.ccc epiR::epi.ccc()]
+</ul>

ICC: Difference between revisions

Revision as of 20:24, 11 March 2023

Contents

Basic

Interpretation

R packages

Negative ICC

Examples

psych package data

Wine rating

Wine rating2

Fleiss Kappa statistic

Cohen's Kappa statistic

Concordance correlation coefficient/CCC

Navigation menu

ICC: Difference between revisions

Revision as of 20:24, 11 March 2023

Basic

Interpretation

R packages

Negative ICC

Examples

psych package data

Wine rating

Wine rating2

Fleiss Kappa statistic

Cohen's Kappa statistic

Concordance correlation coefficient/CCC

Navigation menu

Search