CHOOSING AN INTRACLASS CORRELATION COEFFICIENT

David P. Nichols
Principal Support Statistician and
Manager of Statistical Support
SPSS Inc.
From SPSS Keywords, Number 67, 1998

Beginning with Release 8.0, the SPSS RELIABILITY procedure offers an
extensive set of options for estimation of intraclass correlation
coefficients (ICCs). Though ICCs have applications in multiple contexts,
their implementation in RELIABILITY is oriented toward the estimation of
guidance in choosing among the various available ICCs (which are all
discussed in McGraw & Wong, 1996). To request any of the available ICCs
via the dialog boxes, specify Statistics->Scale->Reliability, click on
the Statistics button, and check the Intraclass correlation coefficient
checkbox.

In all situations to be considered, the structure of the data is as N
cases or rows, which are the objects being measured, and k variables or
columns, which denote the different measurements of the cases or
objects. The cases or objects are assumed to be a random sample from a
larger population, and the ICC estimates are based on mean squares
obtained by applying analysis of variance (ANOVA) models to these data.

The first decision that must be made in order to select an appropriate
ICC is whether the data are to be treated via a one way or a two way
ANOVA model. In all situations, one systematic source of variance is
associated with differences among objects measured. This object (or
often, "person") factor is always treated as a random factor in the
ANOVA model. The interpretation of the ICCs is as the proportion of
relevant variance that is associated with differences among measured
objects or persons. What variance is considered relevant depends on the
particular model and definition of agreement used.

Suppose that the k ratings for each of the N persons have been produced
by a subset of j > k raters, so that there is no way to associate each
of the k variables with a particular rater. In this situation the one
way random effects model is used, with each person representing a level
of the random person factor. There is then no way to disentangle
variability due to specific raters, interactions of raters with persons,
and measurement error. All of these potential sources of variability are
combined in the within person variability, which is effectively treated
as error.

If there are exactly k raters who each rate all N persons, variability
among the raters is generally treated as a second source of systematic
variability. Raters or measures then becomes the second factor in a two
way ANOVA model. If the k raters are a random sample from a larger
population, the rater factor is considered random, and the two way
random effects model is used. Otherwise, the rater factor is treated as
a fixed factor, resulting in a two way mixed model. In the mixed model,
inferences are confined to the particular set of raters used in the
measurement process.

In the dialog boxes, when the Intraclass correlation coefficient
checkbox is checked, a dropdown list is enabled that allows you to
specify the appropriate model. If nothing further is specified, the
default is the two way mixed model. If either of the two way models is
selected, a second dropdown list is enabled, offering the option of
defining agreement in terms of consistency or in terms of absolute
agreement (if the one way model is selected, only measures of absolute
agreement are available, as consistency measures are not defined). The
default for two way models is to produce measures of consistency.

The difference between consistency and absolute agreement measures is
defined in terms of how the systematic variability due to raters or
measures is treated. If that variability is considered irrelevant, it is
not included in the denominator of the estimated ICCs, and measures of
consistency are produced. If systematic differences among levels of
ratings are considered relevant, rater variability contributes to the
denominators of the ICC estimates, and measures of absolute agreement
are produced.

The dialog boxes thus offer five different combinations of options: 1)
one way random model with measures of absolute agreement; 2) two way
random model with measures of consistency; 3) two way random model with
measures of absolute agreement; 4) two way mixed model with measures of
consistency; 5) two way mixed model with measures of absolute agreement.
In addition, you can specify a coverage level for confidence intervals
on the ICC estimates, and a test value for testing the null hypothesis
that the population ICC is a given value.

Each of the five possible sets of output includes two different ICC
estimates: one for the reliability of a single rating, and one for the
reliability for the mean or sum of k ratings. The appropriate measure to
use depends on whether you plan to rely on a single rating or a
combination of k ratings. Combining multiple ratings of course generally
produces more reliable measurements.

Note that the numerical values produced for the two way models are
identical for random and mixed models. However, the interpretations
under the two models are different, as are the assumptions. Since
treating the data matrix as a two way design leaves only one case per
cell, there is no way to disentangle potential interactions among raters
and persons from errors of measurement. The practical implications of
this are that when raters are treated as fixed in the mixed model, the
ICC estimates (for either consistency or absolute agreement) for the
combination of k ratings require the assumption of no rater by person
interactions. The estimates for the reliability of a single rating under
the mixed model and all estimates under the random model are the same
regardless of whether interactions are assumed. See McGraw & Wong for a
discussion of the assumptions and interpretations of the estimates under
the various models.

As a final note, though the ICCs are defined in terms of proportions of
variance, it is possible for empirical estimates to be negative (the
estimates all have upper bounds of 1, but no lower bounds). In the next
issue, we will discuss the problem of negative reliability estimates.

Reference:

McGraw, K. O., & Wong, S. P. (1996). Forming inferences about some
intraclass correlation coefficients. Psychological Methods, Vol. 1, No.
1, 30-46 (Correction, Vol. 1, No. 4, 390).


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