The likelihood difference heuristic and binary test selection given situation-specific utilities

Consider the task of selecting a medical test to determine whether a patient has a particular disease. Normatively, this requires taking into account (a) the prior probability of the disease, (b) the likelihood for each available test-of obtaining a positive result if the medical condition is present or absent, respectively, and (c) the utilities for both correct and incorrect treatment decisions based upon each possible test result. But these quantities may not be precisely known. Are there strategies that could help identify the test with the highest utility given incomplete information? Here, we consider the Likelihood Difference Heuristic (LDH), a simple heuristic that selects the test with the highest difference between the likelihood of obtaining a true positive and a false-positive test result, ignoring all other information. We prove that the LDH is optimal when the probability of the disease equals the therapeutic threshold, the probability for which treating the patient and not treating the patient have the same expected utility. By contrast, prominent models of the value of information from the literature, such as information gain, probability gain, and Bayesian diagnosticity, are not optimal under these circumstances. Further results show how, depending on the relationship of the therapeutic threshold and prior probability of the disease, it is possible to determine which likelihoods are more important for assessing tests' expected utilities. Finally, to illustrate the potential relevance for real-life contexts, we show how the LDH might be applied to choosing tests for screening of latent tuberculosis infection.