Linear Transformation Model With Parametric Covariate Transformations

The traditional linear transformation model assumes a linear relationship between the transformed response and the covariates. However, in real data, this linear relationship may be violated. We propose a linear transformation model that allows parametric covariate transformations to recover the linearity. Although the proposed generalization may seem rather simple, the inferential issues are quite challenging due to loss of identifiability under the null of no effects of transformed covariates. This article develops tests for such hypotheses. We establish rigorous inferences for parameters and the unspecified transformation function when the transformed covariates have nonzero effects. The estimates and tests perform well in simulation studies using a realistic sample size. We also develop goodness-of-fit tests for the transformation and R-2 for model comparison. GAGurine data are used to illustrate the practical utility of the proposed methods.