Regression Estimation Following the Square-Root Transformation of the Response

Authors: Gregoire, Timothy G.; Lin, Qi Feng; Boudreau, Johnathan; Nelson, Ross

Source: Forest Science, Volume 54, Number 6, December 2008 , pp. 597-606(10)

Publisher: Society of American Foresters

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Abstract:

In a variety of regression situations, there is interest in predicting the value of Y ², yet it is useful to model it using a square root transformation, such that Y rather than Y ² is regressed on one or more covariates. The back-transformation bias of the square root transformation of the response variable of interest is presented in detail. An unbiased estimator is presented: Ê[Y ²|x _∗] = μ_y|x∗ ² + σ − V(μ_y|x∗ ²). Its performance is compared against that of two biased estimators: Ê_b [Y ²|x _∗] = μ_y|x∗ ² + σ and Ê_p [Y ²|x _∗] = μ_y|x∗ ². The first two moments of these estimators are derived analytically and verified by means of a simulation study. Both biased estimators have lower mean square errors than the unbiased estimator. An example wherein aboveground biomass is the response variable is presented for illustration.

Keywords: back-transformation bias; nonlinearity

Document Type: Research article

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