A demonstration of the difference between the ODP approach and LR statistic. Suppose that hypothesis tests

*H*_{0} : μ=0 versus

*H*_{1} : μ≠0 are performed on μ

_{1}, μ

_{2},…, μ

_{m} based on respective datasets

**x**_{1},

**x**_{2},…,

**x**_{m}. Shown are the likelihood functions for test 5,

*L*(μ|

**x**_{5}) in red, and test 13,

*L*(μ|

**x**_{13}) in blue. Their maximum likelihood estimates are such that

, implying that they would produce equal LR statistics. The ODP utilizes information from all of the maximum likelihood estimates

, shown at the top of the plot. These tend to be more similar to

than

, lending greater evidence against the null hypothesis for test 13. The ODP quantifies this evidence by calculating the likelihood functions over all maximum likelihood estimates, shown as red dots for test 5 and in blue dots for test 13. It can be seen that

, implying that the ODP statistic for test 13 would be larger than that for test 5. This makes sense in that there are many more positive

than negative, so we should attribute stronger evidence against the null hypothesis to those tests with positive estimates. In more complex situations such as those encountered in gene expression studies, this aggregation of information becomes even more useful.

## PubMed Commons