Threshold regression models are useful for studying threshold-dependent nonlinear relationships between outcomes and predictors of interest. The presence of the threshold parameter in the model complicates statistical inference since model identification can be problematic. Much work has been done on hypothesis testing for threshold regression models. However, most if not all this work has focused on nested problems, which limits the types of scientific questions that can be addressed. Motivated by an example from the prevention of mother-to-child transmission of HIV-1, we study a non-nested hypothesis testing problem that seeks to discriminate between a linear model and a hinge model, which is a special type of threshold model. To develop the test statistic, we follow the same approach as used in the nested testing problem; to obtain p-values, the approach is necessarily different. We study three parametric bootstrap procedures and find that fast double parametric bootstrap procedures have close to nominal type 1 error rates and are reasonably fast. Both linear regression and logistic regression are considered. We illustrate the proposed methods using the motivating example.