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
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. 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
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.
This expression can never be negative (unlike Fisher's original formula) and therefore, in samples from a population which has an ICC of 0, the ICCs in the samples will be higher than the ICC of the population.
A number of different ICC statistics have been proposed, not all of which estimate the same population parameter. There has been considerable debate about which ICC statistics are appropriate for a given use, since they may produce markedly different results for the same data.
Read more about this topic: Intraclass Correlation
Famous quotes containing the words modern, simpler, formula, positive and/or bias:
“The critical method which denies literary modernity would appear—and even, in certain respects, would be—the most modern of critical movements.”
—Paul Deman (1919–1983)
“To translate, one must have a style of his own, for otherwise the translation will have no rhythm or nuance, which come from the process of artistically thinking through and molding the sentences; they cannot be reconstituted by piecemeal imitation. The problem of translation is to retreat to a simpler tenor of one’s own style and creatively adjust this to one’s author.”
—Paul Goodman (1911–1972)
“The formula for achieving a successful relationship is simple: you should treat all disasters as if they were trivialities but never treat a triviality as if it were a disaster.”
—Quentin Crisp (b. 1908)
“If all political power be derived only from Adam, and be to descend only to his successive heirs, by the ordinance of God and divine institution, this is a right antecedent and paramount to all government; and therefore the positive laws of men cannot determine that, which is itself the foundation of all law and government, and is to receive its rule only from the law of God and nature.”
—John Locke (1632–1704)
“The solar system has no anxiety about its reputation, and the credit of truth and honesty is as safe; nor have I any fear that a skeptical bias can be given by leaning hard on the sides of fate, of practical power, or of trade, which the doctrine of Faith cannot down-weigh.”
—Ralph Waldo Emerson (1803–1882)