  In this paper new families of test statistics are introduced and studied for
the problem of comparing two treatments in terms of the likelihood ratio order.
The considered families are based on phi-divergence measures and arise as
natural extensions of the classical likelihood ratio test and Pearson test
statistics. It is proven that their asymptotic distribution is a common chi-bar
random variable. An illustrative example is presented and the performance of
these statistics is analysed through a simulation study. Through a simulation
study it is shown that, for most of the proposed scenarios adjusted to be small
or moderate, some members of this new family of test-statistic display clearly
better performance with respect to the power in comparison to the classical
likelihood ratio and the Pearson's chi-square test while the exact size remains
closed to the nominal size. In view of the exact powers and significance
levels, the study also shows that the Wilcoxon test-statistic is not as good as
the two classical test-statistics.
