System of Linear Equations to Derive Unreported Test Accuracy Counts for Meta-Analysis

Meta-analyses assessing test accuracy typically require extracting true positive (TP), false negative (FN), false positive (FP), and true negative (TN) counts from each study, commonly organized in a 2 × 2 table. However, many published test accuracy studies do not report all of these counts, which can limit the ability of a meta-analysis to fully capture the available evidence on the screening or diagnostic accuracy of a given test. Fortunately, test accuracy studies often report sufficient parameters, such as sensitivity and specificity, that enable the estimation of unreported counts. The relationships between these commonly reported parameters and the unreported cell counts may be expressed mathematically and organized into a system of four linear equations. The basic principles of solving such systems using matrix methods are introduced, accompanied by examples illustrating the development and solution of linear systems with unknown TP, FN, TN, and TN counts. Approaches for handling rounding errors of reported test accuracy parameters in publications are also demonstrated. Additionally, methods for obtaining a bound solution are explored in scenarios where the solution for missing test accuracy counts results in a system with three linear equations and four unknowns, leading to non-unique solutions. Simulation studies are conducted to assess the performance of these methods, and practical guidance for their implementation is provided. The Microsoft Excel spreadsheets and SAS and R code for the examples presented in this article are available in the Supporting Information.