Fischer RA. On the “probable error” of a correlation coefficient, derived from a small sample. Metron. 1921; 1:1–32. Three statistical models used in intraclass correlation theory are indicated: Model 1 (single-use model); Model 2 (two-way random model); and Model 3 (two-sided mixed model). The figure shows the relationship between these models and the three formulas ICC (1), ICC (A,1) and ICC (C,1). which can be interpreted as the fraction of the total variance due to the variation between groups. In his classic book, Ronald Fisher devotes an entire chapter to intraclassical correlation. [3] The system of equation Eq (10) is overdetermined and has five equations, but only three unknowns (σr, σc and σv); Two of the equations can be derived from the others. The solution eq (10) for the σ:s we receive (for example) the expressions (11) Eq (11) provides us with a statistical estimate of the variances σr2, σc2 and σv2. With Eq (11) in Eq (9), we obtain (12), i.e. the known ICC sampling formula icc (A,1) [6], which is therefore a statistical estimate of the intraclass population coefficient ρ2A. The ICC formula (A,1) gives an estimate of the reliability of the method if an absolute correspondence between the different measurements is desired [6].

Unlike the ICC formula (1), it also takes into account the additional variance due to random “bias” terms. We find that at the limit of the tiny small bias terms, that is to say when σc2 → 0, MSBM, MSWS and tend to be equal, since all three are then estimates of the same variance σv2. Model 2 is then reduced to Model 1 and the ICC formula (A,1) tends to provide the same result as the ICC formula (1). As a general rule, we should expect the 95% (A.1) confidence intervals obtained by group A and group B to overlap. In this overlap, as indicated in point 4.1, we can expect to find the model 2 population intraclassical coefficient ρ2A. The central zone of 95% CCI (1) is not the same as the confidence interval of ρ1, but the second can be derived from the first. In Figure 8, the curves show the upper and lower limits of the central zone of 95% of the simulated ICC distributions (1) for cases k = 3 and n = 10, 20, 50 and 100 as functions of the intraclass correlation coefficient assumed in the ρ1 simulation. To determine the confidence intervals, the diagram can be used as follows. ICC distributions (C,1) are insensitive to the presence of distortions; They remain the same, no matter how strong the bias is. With Eq (13) we find ρ2C = 102/(102 + 52) = 0.8 for the three ICC distributions (C,1).

They coincide with both the ICC (1) (Zero Bias) distribution and the ICC (A,1) distribution when distortions are very low. This indicates that the confidence limits of ρ2C (the ICC consistency population of Model 2) are the same as the confidence limits of ρ1 (ICC population of Model 1, i.e. in the absence of distortions. B). Thus, Figure 8 makes it possible to deduce graphically not only the confidence limits corresponding to an ICC value (1) calculated from an experimental matrix compatible with Model 1, but also the confidence limits corresponding to an ICC value (C,1) calculated from a matrix compatible with Model 2. This observation is consistent with mcGraw and Wong`s confidence limit formulas [6], unless the number of subjects n is too small. . . .