Table 1.
Symbols and abbreviations.
Table 2.
Experimental data matrix with n rows and k columns.
Fig 1.
Survey of models, mean square relations and ICC formulas.
Fig 2.
Relation between ICC models and ICC formulas.
Three statistical models used in the intraclass correlation theory are indicated: Model 1 (one-way model); Model 2 (two-way random model); and Model 3 (two-way mixed model). The figure shows the relation between these models and the three well-known sample ICC formulas, i.e. ICC(1), ICC(A,1) and ICC(C,1).
Fig 3.
Part of the computer printout from a SIMANOVA run using Model 1.
Corresponding ICC distributions are shown in Fig 4. See text for discussion.
Fig 4.
Probability distributions of ICC(1), ICC(A,1) and ICC(C,1) obtained with a simulation based on Model 1, i.e. in the absence of bias.
Input data and results from this simulation are shown in Fig 3. With Model 1, the simulated distributions are seen to be identical apart from small differences due to finite statistics (finite N).
Fig 5.
Probability distributions of ICC(1) values obtained with simulations using Model 1, showing the effect of increasing noise (error).
In all three cases n = 20, k = 3 and σr = 10. The standard deviation of the noise term is increased from σv = 5 (giving population ICC ρ1 = 0.8) to σv = 7.5 (ρ1 = 0.64) and σv = 10 (ρ1 = 0.5).
Fig 6.
Probability distributions of ICC(1) obtained with Model 1, showing the effect of increasing the number of subjects.
The number of subjects increases from n = 20 to n = 100 and n = 400. In all three cases k = 3 and the standard deviations of error and subject's score are 𝛔v = 𝛔r = 10, giving the population ICC = 0.5. As can be seen, an increasing n leads to a decrease in the width of the probability distribution.
Fig 7.
Probability distributions of ICC(1) obtained with Model 1, showing the effect of increasing the number of measurements k.
In all four distributions, n = 20 and 𝛔v = 𝛔r = 10 (giving population ICC = 0.5). Increasing k leads to a decreasing width.
Fig 8.
Graphic presentation of confidence limits.
The curves show the upper and lower 95% central range limits of the ICC(1) probability distributions as functions of the population ICC, using Model 1. The number of measurements is everywhere k = 3 while the number of subjects n range from 10 to 100. Read horizontally, the diagram provides the 95% confidence limits of the population ICC for a given sample ICC(1) value. For example, if n = 10 and the sample ICC(1) is found to be 0.70, then the confidence limits of the population ICC are graphically read to be approximately 0.38 and 0.91. The diagram is also valid for ICC(C,1), i.e. the consistency ICC obtained with Model 2 and Model 3.
Fig 9.
Probability distributions of ICC(A,1) and ICC(C,1) obtained with Model 2, showing the effect of increasing bias.
In all cases n = 20, k = 3, σr = 10 and σv = 5. The bias standard deviation σc is increased from σc = 0.1 to σc = 5 and then to σc = 10. As may be seen, the ICC(C,1) distributions are insensitive to bias. The ICC(A,1) distributions are, with increasing bias, shifted towards lower values and broadened.
Table 3.
A survey of features of Model 2.
Table 4.
Survey of model 2 for n = 20 and k = 3: the average ratio <ICC(C,1)/ICC(A,1)> and the probability for ICC(C,1) to be larger than ICC(A,1) in a single data matrix.
Fig 10.
Effect of fixed bias: ICC(A,1) and ICC(C,1) distributions simulated using Model 3.
Two simulated Model 3 cases are shown, (a) and (b). In both cases, n = 20, k = 3, σr = 10 and σv = 5. In case (a), the fixed bias values are c1 = 1, c2 = 6 and c3 = -1. In case (b), they are c1 = 10, c2 = 6 and c3 = -10. A larger spread among the fixed bias values gives a larger shift of the ICC(A,1) distribution towards lower values. The ICC(C,1) distributions are however insensitive to bias.
Table 5.
A clinical example from physiotherapy.