Small area estimation based on linear mixed models can be inefficient when the underlying relationships are non-linear. In
this paper we introduce SAE techniques for variables that can be modelled linearly following a non-linear transformation. In
particular, we extend the model-based direct estimator of Chandra and Chambers (2005, 2009) to data that are consistent
with a linear mixed model in the logarithmic scale, using model calibration to define appropriate weights for use in this
estimator. Our results show that the resulting transformation-based estimator is both efficient and robust with respect to the
distribution of the random effects in the model. An application to business survey data demonstrates the satisfactory
performance of the method.