Intraclass Correlation Coefficient
An Intraclass Correlation Coefficient is a correlation coefficient that describes how strongly units in the same group resemble each other.
- See: Continuous Outcome Cluster Randomized Experiment Evaluation Algorithm, Within Group Variation, Covariance, Descriptive Statistic, Statistical Unit, ANOVA, Random Effects Model.
References
2013
- http://en.wikipedia.org/wiki/Intraclass_correlation
- In statistics, the 'intraclass correlation (or the intraclass correlation coefficient, abbreviated ICC)[1] is a descriptive statistic that can be used when quantitative measurements are made on units that are organized into groups. It describes how strongly units in the same group resemble each other. While it is viewed as a type of correlation, unlike most other correlation measures it operates on data structured as groups, rather than data structured as paired observations.
The intraclass correlation is commonly used to quantify the degree to which individuals with a fixed degree of relatedness (e.g. full siblings) resemble each other in terms of a quantitative trait (see heritability). Another prominent application is the assessment of consistency or reproducibility of quantitative measurements made by different observers measuring the same quantity.
- In statistics, the 'intraclass correlation (or the intraclass correlation coefficient, abbreviated ICC)[1] is a descriptive statistic that can be used when quantitative measurements are made on units that are organized into groups. It describes how strongly units in the same group resemble each other. While it is viewed as a type of correlation, unlike most other correlation measures it operates on data structured as groups, rather than data structured as paired observations.
- http://en.wikipedia.org/wiki/Intraclass_correlation#Early_ICC_definition:_unbiased_but_complex_formula
- QUOTE: For large K, this ICC is nearly equal to :[math]\displaystyle{ \frac{N^{-1}\sum_{n=1}^N(\bar{x}_n-\bar{x})^2}{s^2}, }[/math] which can be interpreted as the fraction of the total variance that is due to variation between groups. Ronald Fisher devotes an entire chapter to Intraclass correlation in his classic book Statistical Methods for Research Workers.
- http://en.wikipedia.org/wiki/Intraclass_correlation#Modern_ICC_definitions:_simpler_formula_but_positive_bias
- Beginning with Ronald Fisher, the intraclass correlation has been regarded within the framework of analysis of variance (ANOVA), and more recently in the framework of random effects models. A number of ICC estimators have been proposed. Most of the estimators can be defined in terms of the random effects model :[math]\displaystyle{ Y_{ij} = \mu + \alpha_j + \epsilon_{ij}, }[/math] where Yij is the ith observation in the jth group, μ is an unobserved overall mean, αj is an unobserved random effect shared by all values in group j, and εij is an unobserved noise term.[1] For the model to be identified, the αj and εij are assumed to have expected value zero and to be uncorrelated with each other. Also, the αj are assumed to be identically distributed, and the εij are assumed to be identically distributed. The variance of αj is denoted σα2 and the variance of εij is denoted σε2.
The population ICC in this framework is :<math> \frac{\sigma_\alpha^2}{\sigma_\alpha^2+\sigma_\epsilon^2}. </math An advantage of this ANOVA framework is that different groups can have different numbers of data values, which is difficult to handle using the earlier ICC statistics. Note also that this ICC is always non-negative, allowing it to be interpreted as the proportion of total variance that is "between groups." This ICC can be generalized to allow for covariate effects, in which case the ICC is interpreted as capturing the within-class similarity of the covariate-adjusted data values.[2]
- Beginning with Ronald Fisher, the intraclass correlation has been regarded within the framework of analysis of variance (ANOVA), and more recently in the framework of random effects models. A number of ICC estimators have been proposed. Most of the estimators can be defined in terms of the random effects model :[math]\displaystyle{ Y_{ij} = \mu + \alpha_j + \epsilon_{ij}, }[/math] where Yij is the ith observation in the jth group, μ is an unobserved overall mean, αj is an unobserved random effect shared by all values in group j, and εij is an unobserved noise term.[1] For the model to be identified, the αj and εij are assumed to have expected value zero and to be uncorrelated with each other. Also, the αj are assumed to be identically distributed, and the εij are assumed to be identically distributed. The variance of αj is denoted σα2 and the variance of εij is denoted σε2.
- ↑ Donner, Allan; Koval, John J (1980). "The Estimation of Intraclass Correlation in the Analysis of Family Data". Biometrics (International Biometric Society) 36 (1): 19–25. doi:10.2307/2530491. JSTOR 2530491. PMID 7370372.
- ↑ Stanish, William; Taylor, Noel (1983). "Estimation of the Intraclass Correlation Coefficient for the Analysis of Covariance Model". The American Statistician (American Statistical Association) 37 (3): 221–224. doi:10.2307/2683375. JSTOR 2683375.