Diagnostic tests can predict efficacy without randomization to intervention and control by solving simultaneous equations based on different test results: Required when assessing the effectiveness of test, trace and isolation

The efficacy of an intervention can be assessed by randomising patients to different diagnostic tests instead of to an intervention and control. This is based on a familiar view that treatment will be more effective if given to patients based on the results of appropriate diagnostic tests (i.e. the correct diagnosis) than if it given to those based on the results of inappropriate tests (i.e. the wrong diagnosis). The tests must have different characteristics e.g. sensitivities with respect to the outcome. This principle is applied by allocating an intervention to an individual if a test result is positive (or on one side of a threshold) and allocating to a control to an individual if the result is negative (or on the other side of the threshold). This can also be done with different dichotomizing thresholds for one test. The frequencies of the outcome in those with each of the four resulting observations are then used to calculate the risk reduction by solving a pair of simultaneous equations. This assumes that the risk reduction and the overall frequency of the outcome is the same in both groups created by randomization. The calculations are illustrated by using data from a randomized controlled trial that assessed the efficacy of the angiotensin receptor blocker in lowering the risk of diabetic nephropathy in patients conditional on different urinary albumin excretion rates. They are also illustrated with simulated data based on a suggested methodology for assessing the effectiveness of test, trace and isolation to reduce transmission of the SARS-Cov-2 virus using RT-PCR and LFD tests. This approach also allows the sensitivity and specificity of these tests with respect of the outcome of viral spreading to be determined irrespective of the efficacy of isolation.