COMPONENTS OF VARIANCE MODELS WITH TRANSFORMATIONS

Power transformations are a popular way to improve the agreement between the observations and the assumptions in a statistical model. In this paper it is assumed that the data, after appropriate power transformation Λ, satisfies a variance components model, with independent Gaussian components. The focus is on inference for quantities which have an interpretation regardless of the choice of Λ (Carroll & Ruppert, 1981) – in particular the intraclass correlation coefficient ρ, the predicted probability of a new observation being less than a specified value and the predicted quantile. It is shown that, in the case Λ= 0, the asymptotic variance of ρ is the same, whether or not one treats Δ as estimated or as known. This supports an empirical conjecture of Solomon (1985). For predicted probabilities and predicted quantiles the variance when A is estimated is shown to be only slightly greater than the variance assuming Δ is known, except in the tails of the distribution where there can be substantial difference between the two variances.