Statistics Notes: Measurement error and correlation coefficients
BMJ 1996; 313 doi: https://doi.org/10.1136/bmj.313.7048.41 (Published 06 July 1996) Cite this as: BMJ 1996;313:41
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The formula for ICC is given as:
rI = m * SSB - SST / (m-1) * SST
This is algebraically incorrect. It should be:
rI = (m*SSB - SST) / ((m-1) * SST)
If algebraic precedence rules are followed, the former formula gives
very different results from the latter.
Regards,
Michael McStephen
Competing interests: No competing interests
I agree with Prof Bland and Prof Altman [1] that if one wants to use
an intraclass correlation coefficient (ICC) as index of measurement error
for test-retest reliability or reproducibility studies, the one-way ICC is
the measure of choice.
However, I'm not sure whether their formula for the one-way ICC is
correct. I reanalysed the corrected data for their table 1 from eBMJ (URL:
http://www.bmj.com/cgi/content/full/313/7048/41/DC1) with the Reliability
procedure of SPSS (SPSS Inc., Chicago). Beginning with release 8, users of
this computer program now can calculate ICCs directly. The result is 0.76
for the one-way ICC, single measure, which is the ICC Bland and Altman
obviously mean. This is slightly higher than the value given in their
statistics notes[1]: 0.75.
The calculation formula used in SPSS [2] for this ICC, adapted to the
notation used in Bland and Altman's statistics note, is: (MS(B) - MS(W)) /
[MS(B) + (m-1) MS(W)], where MS(B) and MS(W) are the mean squares for
between subjects und within subjects, respectively, and m is number of
observations per subject.
This is a formula, which most textbooks on this topic report. Actually, I
could not find any text citing a formula for this ICC as in Bland and
Altman.[1]
Perhaps they have derived their formula from the within-subject standard
deviation, which they prefer as an index of measurement error, and used
the sample sizes (m·n, where m is number of observations per subject, and
n is the number of subjects) instead of the degrees of freedom (m·n-1) as
denominator for the total sum of squares (SS(T)).
This could explain their formula. But I think this is not correct, and the
difference between the results of the two formulas may be greater for
other data sets.
References:
1 Bland JM, Altman DG. Measurement error and correlation coefficients. BMJ
1996;313:41-2.
2 McGraw KO, Wong SP. Forming inferences about some intraclass correlation
coefficients [published erratum appears in Psychol Methods 1996;1:390].
Psychol Methods 1996;1:30-46.
Peter Schuck, MD, PhD
FBK Research Institute,
Lindenstr. 5 ,
D-08645 Bad Elster, Germany.
Email: peter.schuck@kommunikaton.iz-plauen.de
Competing interests: No competing interests
Re: precedence rules in formula
Mea culpa! I am a rotten proof-reader. The original article as
submitted gave the formula in fraction form, with a numerator and
denominator. I did not notice when the article was type-set that a change
to the style of the formula had resulted in the mistake which Michael
McStephen has spotted. I thank him for putting me right.
My co-author was quite innocent, and would have noticed had he read
the proofs, being much more meticulous than myself!
Competing interests: No competing interests