How Often Likelihood Ratios are Misleading in Sequential Trials

How often would investigators be misled if they took advantage of the likelihood principle and used likelihood ratios—which need not be adjusted for multiple looks at the data—to frequently examine accumulating data? The answer, perhaps surprisingly, is not often. As expected, the probability of observing misleading evidence does increase with each additional examination. However, the amount by which this probability increases converges to zero as the sample size grows. As a result, the probability of observing misleading evidence remains bounded—and therefore controllable—even with an infinite number of looks at the data. Here we use boundary crossing results to detail how often misleading likelihood ratios arise in sequential designs. We find that the probability of observing a misleading likelihood ratio is often much less than its universal bound. Additionally, we find that in the presence of fixed-dimensional nuisance parameters, profile likelihoods are to be preferred over estimated likelihoods which result from replacing the nuisance parameters by their global maximum likelihood estimates.