Transformations to additivity in measurement error models

Abstract

In many problems, one wants to model the relationship between a response Y and a covariate X. Sometimes it is difficult, expensive, or even impossible to observe X directly, but one can instead observe a substitute variable W that is easier to obtain. By far, the most common model for the relationship between the actual covariate of interest X and the substitute W is W = X + U, where the variable U represents measurement error. This assumption of additive measurement error may be unreasonable for certain data sets. We propose a new model, namely h(W) = h(X) + U, where h(.) is a monotone transformation function selected from some family H of monotone functions. The idea of the new model is that, in the correct scale, measurement error is additive. We propose two possible transformation families H. One is based on selecting a transformation that makes the within-sample mean and standard deviation of replicated W's uncorrelated. The second is based on selecting the transformation so that the errors (U's) fit a prespecified distribution. Transformation families used are the parametric power transformations and a cubic spline family. Several data examples are presented to illustrate the methods.