Intraclass correlation coefficients

Stata's icc can measure absolute agreement and consistency of agreement.

Consider a dataset containing 24 ratings of 6 targets by 4 judges. Assume that a different set of four judges is used to rate each target so that we have a one-way random-effects model.

We can estimate the correlation of ratings made on the same targets by typing

. webuse judges
(Ratings of targets by judges)


. icc rating target

Intraclass correlations
One-way random-effects model
Absolute agreement

Random effects: target           Number of targets =         6
                                 Number of raters  =         4



                rating 
 
        ICC       [95% conf. interval]
 
 
 
            Individual 
 
   .1657418      -.1329323    .7225601
               Average 
 
   .4427971      -.8844422    .9124154


F test that
  ICC=0.00: F(5.0, 18.0) = 1.79               Prob > F = 0.165

Note: ICCs estimate correlations between individual measurements
      and between average measurements made on the same target.

The correlation of measurements made on the same individual is 0.1657.

The correlation among mean ratings for each team of judges is 0.4428. The average ICC can be used when teams of different raters are used to rate a target. Teams of physicians are sometimes evaluated in this manner.

Now let's pretend the same team of judges rated each target. If the judges were drawn randomly from the population, then we would have a two-way random-effects model. We can estimate the correlations by typing

. icc rating target judge

Intraclass correlations
Two-way random-effects model
Absolute agreement

Random effects: target           Number of targets =         6
Random effects: judge            Number of raters  =         4



                rating 
 
        ICC       [95% conf. interval]
 
 
 
            Individual 
 
   .2897638       .0187865    .7610844
               Average 
 
   .6200505       .0711368     .927232


F test that
  ICC=0.00: F(5.0, 15.0) = 11.03              Prob > F = 0.000

Note: ICCs estimate correlations between individual measurements
      and between average measurements made on the same target.

The correlation of measurements made on the same individual is 0.2898.

The correlation among mean team ratings with those that might be produced by another team is 0.6201.

Finally, suppose the four judges are the only judges of interest. Judge is now a fixed effect in the model, and we have a two-way mixed-effects model. We can obtain the correlations by typing

. icc rating target judge, mixed

Intraclass correlations
Two-way mixed-effects model
Consistency of agreement

Random effects: target           Number of targets =         6
 Fixed effects: judge            Number of raters  =         4



                rating 
 
        ICC       [95% conf. interval]
 
 
 
            Individual 
 
   .7148407       .3424648    .9458583
               Average 
 
   .9093155       .6756747    .9858917


F test that
  ICC=0.00: F(5.0, 15.0) = 11.03              Prob > F = 0.000

Note: ICCs estimate correlations between individual measurements
      and between average measurements made on the same target.