Relative likelihood ratios for neutral comparisons of statistical tests in simulation studies

When comparing the performance of two or more competing tests, simulation studies commonly focus on statistical power. However, if the size of the tests being compared are either different from one another or from the nominal size, comparing tests based on power alone may be misleading. By analogy with diagnostic accuracy studies, we introduce relative positive and negative likelihood ratios to factor in both power and size in the comparison of multiple tests. We derive sample size formulas for a comparative simulation study. As an example, we compared the performance of six statistical tests for small-study effects in meta-analyses of randomized controlled trials: Begg's rank correlation, Egger's regression, Schwarzer's method for sparse data, the trim-and-fill method, the arcsine-Thompson test, and Lin and Chu's combined test. We illustrate that comparing power alone, or power adjusted or penalized for size, can be misleading, and how the proposed likelihood ratio approach enables accurate comparison of the trade-off between power and size between competing tests.