Parameterization of beta distributions for bias parameters of binary exposure misclassification in probabilistic bias analysis

Background: To account for bias due to misclassification of dichotomous variables, beta distributions may be assigned to bias parameters—such as the positive or negative predictive value (PPV or NPV)—based on data from an internal validation substudy. When zero cell frequencies occur in some strata of the validation substudy, due to small sample sizes and inefficient allocations, it may be helpful to assign prior distributions to the PPV and NPV.

Objectives: To mitigate sparse data problems that can arise when adjusting bias for a misclassified binary exposure in a validation substudy by assigning prior distributions to the PPV and NPV with and without continuity correction.

Methods: We simulated cohort studies of varying sizes, comprising a binary exposure with prevalence = 0.2 and outcome with baseline cumulative incidence = 0.1 and RR = 2.0, as well as internal validation substudies using a balanced design (i.e., same sample size for each stratum) for exposure classification. We conducted bias adjustment under five scenarios assigning prior distributions for NPV and PPV parameters: (1) conventional method (i.e., no prior distributions), (2) uniform prior beta (α = 1, β = 1), (3) Jeffreys prior beta (α = 0.5, β = 0.5), (4) Jeffreys prior as a continuity correction only when zero cells occurred, and (5) uniform prior as a continuity correction only when zero cells occurred. We also conducted sensitivity analyses for sparse validation data with different magnitudes of bias parameters (i.e., the sensitivity and specificity of exposure misclassification) and effect size. We evaluated performance by measuring coverage probability, bias, and MSE.

Results: For sparse validation data, methods (2)–(5) all had better coverage and lower MSE than the conventional method, among which using a uniform prior (2) yielded the best performance (coverage: 97.5% vs. 93.2%, MSE: 0.175 vs. 0.216 compared with no prior). Little difference between methods was observed when the validation substudy did not contain zero cells. We observed similar patterns when we increased the sensitivity of exposure classification from 0.60 to 0.75 or specificity from 0.98 to 1. However, when we decreased the true RR from 2 to 1, using a continuity correction factor of 1 rendered the lowest MSE (0.229 vs. 0.469 without a prior) with comparable coverage (99% vs. 88% without a prior) to using a uniform prior.

Conclusions: Using validation data as observed is not ideal for small validation substudies. Adding information by use of a Bayesian prior or continuity correction to the PPV and NPV will optimize the bias adjustment. If zero cells are expected in a validation substudy, using a uniform prior for the beta distribution of bias parameters is a principled and simple strategy to improve the validity of bias-adjusted measures.