This page presents Intraclass Correlation Coefficient (ICC) in its "agreement" model, an algorithm to measure agreement (consensus, concordance) between 2 or more measurements that are normally distributed.

ICC has advantages over correlation coefficient, in that it is adjusted for the effects of the scale of measurements, and that it will represent agreements from more than two raters or measuring methods.

The calculation involves an initial Two Way Analysis of Variance, the components of which are then used to calculate the ICC

Example

120	115	125
130	140	125
100	98	105
150	156	145
90	90	95

Source of variance	df	ssq	msq	F	p
Between rows	4	6642.9	1660.7	50.7	<0.0001
Within rows	10	270	27
Between columns	2	8.1	4.9	0.1	0.8848
Residual error	8	261.9	32.7
Total overall	14	6912.9

The data in the example is artificially generated to demonstrate the procedure, It purports to be from an exercise to test 3 methods of measuring blood pressure, to check whether the results agreed with each other.

The table to the left shows the data, the columns represent the 3 methods of mesurements, the 5 rows the 5 subjects being measured.

The table to the right is the analysis of variance (truncated to 1 decimal point for brevity), showing components of variation.

The algorithm calculates the ICC and its 95% confidence interval using the variance components from the table. Six (6) methods of calculations are possible, depending on assumptions about the nature of the data

• Model 1 assumes that each method of measurement or rater is different, being subsets of a larger set of methods or raters, randomly chosen. This would be the case of having different research assistants measuring the height of children, with different sets of people doing the measurements at different sites, then combining the data together.
• Model 2 assumes the same methods or raters performs the evaluations in all cases, although these methods or raters may be a subset of a larger set of methods or raters. This is the model most frequently encountered in clinical research, when the same researchers carried out measurements on all the subjects.
• Model 3 makes no assumptions about the methods or raters.

The Intraclass Correlation also depends on whether calculations are based on individual measurements, or the mean of a cluster of measurement.

In most cases, single measurements are used. Unless the data has features that requires otherwise, the result of choice is single, model 2.

In this example, the ICC (truncated to 2 decimal point) is 0.95, 95% Confidence interval from 0.79 to 0.99

Interpretations

The results can be interpreted as follow

• In the older texts such as that from Portney and Watkins (see reference), ICC is used without further processing as an index of agreement. ICC of 0-0.2 indicates poor agreement, 0.3-0.4 indicates fair agreement; 0.5-0.6 indicates moderate agreement; 0.7-0.8 indicates strong agreement; and >0.8 indicates almost perfect agreement.
• More recently (as shown in this page), the approximate 95% confidence interval is estimated. An interval that does not traverse the null (0) value can be interpreted as statistically significant, that the agreement between measurements are greater than random chance.

The results from this example, ICC=0.95, 95% CI = 0.79 - 0.99, can therefore be interpreted as statistically significant, and near perfect agreement

References

Portney LG & Watkins MP (2000) Foundations of clinical research Applications to practice. Prentice Hall Inc. New Jersey ISBN 0-8385-2695-0 p 560-567

https://en.wikipedia.org/wiki/Intraclass_correlation Intraclass correlation on Wikipedia

https://rdrr.io/cran/irr/src/R/icc.R Source Code for ICC from package irr

Data Entry The data is a matrix of numbers with multiple columns     - Each row a subject     - Each column a particular instrument of measurement     - Each cell the value of that measurement (col) for that subject(row)

R provides an algorithm via its irr package. It produces Intraclass Correlation Coefficient, and the statistical significance of difference between the columns (measurements)

dat = ("
120	115	125
130	140	125
100	98	105
150	156	145
90	90	95
       ")

mx = read.table(textConnection(dat),header=FALSE)
#install.packages("irr")  # if not already installed
library(irr)
icc(mx, model = "twoway", type = "agreement", unit = "single")

The results are

> icc(mx, model = "twoway", type = "agreement", unit = "single")
 Single Score Intraclass Correlation

   Model: twoway 
   Type : agreement 

   Subjects = 5 
     Raters = 3 
   ICC(A,1) = 0.953

 F-Test, H0: r0 = 0 ; H1: r0 > 0 
  F(4,8.46) = 50.7 , p = 6.17e-06 

 95%-Confidence Interval for ICC Population Values:
  0.791 < ICC < 0.995