We propose a novel interpretation for a recently proposed Box-Cox transformation cure model, which leads to a natural extension of the cure model. Based on the extended model, we consider an important issue of model selection between the mixture cure model and the bounded cumulative hazard cure model via the likelihood ratio test, score test and Akaike's Information Criterion (AIC). Our empirical study shows that AIC is informative and both the score test and the likelihood ratio test have adequate power to differentiate between the mixture cure model and the bounded cumulative hazard cure model when the sample size is large. We apply the tests and AIC methods to leukemia and colon cancer data to examine the appropriateness of the cure models considered for them in the literature.