The Box-Cox power transformation model is considered for minimum sum of absolute errors (MSAE) regression. A long-tailed error distribution, specifically the Laplace distribution, is included and could accommodate observations that in the normal case are outlying or unduly influencing a choice of transformation. The log-likelihood procedure of Box and Cox (1964) for obtaining the optimal transformation parameter is adapted for Laplace errors and MSAE regression. Graphical methods for detecting the influence of individual observations on the choice of transformation are described. Application to examples illustrates that this approach can provide valuable additional information to the data analyst.