Intraclass correlation

In statistics, the intraclass correlation (or the "intraclass correlation coefficient" [Cite encyclopedia
last = Koch
first = Gary G.
authorlink = Gary G. Koch
title = Intraclass correlation coefficient
encyclopedia = Encyclopedia of Statistical Sciences
volume = 4
pages = 213–217
year = 1982
editor = Samuel Kotz and Norman L. Johnson
publisher = John Wiley & Sons
location = New York] ) is a measure of correlation, consistency or conformity for a data set when it has multiple groups.

The intra-class correlation is used to estimate the correlation of one variable between two members within a group, for instance between two children of one family. This is in contrast to Pearson's Correlation, where the variables of interest are modeled as two distinct traits, with the mean and variance of each being estimated separately. In the intraclass correlation, the trait's mean and variance are derived from pooled estimates across all members of all groups. Because of this, the intraclass correlation gives the proportion of variance attributable to between group differences. In the example of siblings nested in families, the intraclass correlation gives the proportion of variance accounted for by family membership, while the Pearson gives the proportion of shared variance between the two members of a pair without respect to group (family) membership. You might think of it as the equivalent of a matched-sample t-test.

There are several measures of ICC and they may yield different values for the same data set. [Cite journal
author = Reinhold Müller & Petra Büttner
title = A critical discussion of intraclass correlation coefficients
journal = Statistics in Medicine
year = 1994
month = December
volume = 13
issue = 23-24
pages = 2465–2476
pmid = 7701147
doi = 10.1002/sim.4780132310 See also comment:
* Cite journal
author = P. Vargha
title = Letter to the Editor
journal = Statistics in Medicine
volume = 16
pages = 821–823
year = 1997]

Early definition

Consider a data set with two groups represented as two columns of a data matrix $X(N\text{'}\; imes\; 2)$ then the intraclass correlation "r" is computed fromCite book
author = Ronald A. Fisher
title = Statistical Methods for Research Workers
publisher = Oliver and Boyd
address = Edinburgh
year = 1954
edition = Twelfth edition] : ,: ,: ,where "N" is the degrees of freedom(Note that the precise form of the formula differ between versions of Fisher's book: The 1954 edition uses $N\text{'}$ in places where the 1925 editionCite book
author = Ronald A. Fisher
title = Statistical Methods for Research Workers
publisher = Oliver and Boyd
address = Edinburgh
year = 1925
url = http://psychclassics.yorku.ca/Fisher/Methods/] uses $N$).This form is not the same as the int"er"class correlation.For the data set with two groups the intraclass correlation "r" will be confined to the interval [-1, +1] .

The intraclass correlation is also defined for data sets with more than two groups, e.g., for three groups it is computed as: ,: ,: .

(Also this form differs between editions of Fisher's book)

As the number of groups grow, the number of terms in the form will grow exponentially, but another form has been suggested that does not require so many computations: ,where "K" is the number of groups.This form is usually attributed to Harris. [Cite journal
author = J. Arthur Harris
title = On the Calculation of Intra-Class and Inter-Class Coefficinets of Correlation from Class Moments when the Number of Possible Combinations is Large
journl = Biometrika
volume = 9
issue = 3/4
pages = 446–472
month = October
year = 1913] The left term is non-negative, consequently the intraclass correlation must be : $r\; geq\; -1\; /(K-1)$.

"Modern" ICCs

Beginning with Ronald Fisher the intraclass correlation has been regarded within the framework of analysis of variance (ANOVA). Different ICCs arise with different ANOVA models, e.g., one-way analysis or two-way analysis, and they may produce marked different results.An article by McGraw and Wong lists these variations. [Cite journal
author = Kenneth O. McGraw & S. P. Wong
title = Forming inferences about some intraclass correlation coefficients
journal = Psychological Methods
volume = 1
pages = 30–46
year = 1996
url = http://www3.uta.edu/faculty/ricard/COED/McGraw%20(1996)%20Forming%20inferences%20about%20ICCs.pdf
doi = 10.1037/1082-989X.1.1.30 There are several errors in the article:
* Cite journal
author = Kenneth O. McGraw & S. P. Wong
title = Correction to McGraw and Wong (1996)
journal = Psychological Methods
volume = 1
pages = 390
year = 1996
url = http://psycnet.apa.org/index.cfm?fa=buy.optionToBuy&id=1996-06601-006]

Yet another measure that has been regarded as an intraclass correlation coefficient is the concordance correlation coefficient. [Cite journal
author = Carol A. E. Nickerson
title = A Note on 'A Concordance Correlation Coefficient to Evaluate Reproducibility'
journal = Biometrics
volume = 53
pages = 1503–1507
month = December
year = 1997
doi = 10.2307/2533516]

Output from the SPSS program may be used to compute an intraclass correlation. [Cite journal
author = Richard N. MacLennan
title = Interrater Reliability with SPSS for Windows 5.0
journal = The American Statistician
volume = 47
issue = 4
month = November
year = 1993
pages = 292–296
doi = 10.2307/2685289]

The open-source R-Project may also be used to compute the intraclass correlation (package 'psy').

References

There is an entire chapter that concerns the intraclass correlation in Ronald Fisher's classic book "Statistical Methods for Research Workers".

Notes

Other references

* Cite journal
author = P. E. Shrout & Joseph L. Fleiss
year = 1979
title = Intraclass Correlations: Uses in Assessing Rater Reliability
journal = Psychological Bulletin
volume = 86
issue = 2
pages = 420–428
doi = 10.1037//0033-2909.86.2.420

External links

* Michael T. Brannick, [http://luna.cas.usf.edu/~mbrannic/files/pmet/shrout1.htm "Shrout and Fleiss Computations for Intraclass correlations for interjudge reliability"]