Computation of efficiency by data envelopment analysis

Data envelopment analysis is a methodology to assess the relative efficiency of a set of decision-making units [38]. It does not depart from a defined production function as stochastic frontier analysis, but is a non-parametric method and can be used even on a small sample size. It has been widely used in health economics for efficiency assessment at levels ranging from single hospitals (microeconomic level) to health systems as a whole (macroeconomic level) [39–41]. A further advantage is the possible inclusion of multiple inputs and outputs valued either in natural/physical or monetary units.

In our case, the screening equipment and the antidotes represent the input. As the prices of both are known (antidotes) or can be reasonably assessed (equipment), the whole costs that have to be incurred can be expressed as a single input in a monetary value. In data envelopment analysis the best efficiency value for each decision making unit is computed by calculating weights for multiple inputs and outputs and optimal efficiency is often associated with a weight of zero for individual items for which a unit performs less well. This also means that efficiency corresponds to the concept of Farrel and not Pareto-Koopman [42]. This issue is avoided as in our case there is only a single input (total costs) and output (statistical lifetime saved), and efficiency is just computed as the ratio of productivity (statistical lifetime saved/total costs) relative to the highest productivity value for the scenario considered including all assumptions. For a given resource mix option i, this means put into a formula:

At the same time, the inverse of the highest productivity represents the costs of a statistical life year saved and therefore permits an assessment as in cost-effectiveness analysis.

In our data envelopment analysis, we used an input orientation as we consider that the output is set by the scenario and the highest medical benefit for the patients should be sought by implementing an “urgent treatment” strategy. Thus, only the investments in preparedness can be controlled. As accepting a reduction of the best achievable gain of lifetime is not an option for us, we considered only efficiency scores using a constant (CRS) and not a variable return on scale (VRS).

Our simulations are based on a binary distribution (no treatment or treatment), but we assume that all treated victims would absorb exactly the committed effective dose fixed as an indication threshold (and no higher dose) if remaining untreated. In a real setting, this is not realistic and there will be a more or less, but unpredictable variability in the committed effective doses absorbed by victims exceeding the dose threshold fixed. This means that the individual and average lifetime saved per person, as well as the total lifetime saved for all victims in reality will exceed our computed values. As long as the distribution of the committed effective doses remains constant in the victim population, this will however not affect efficiency values. But the costs to save a statistical life year will vary depending on the concrete distribution and the values of our computations are the highest costs for the dose threshold fixed (as the total lifetime saved is at a minimum). Our calculations for the costs of a saved statistical life year therefore lead to very conservative estimations.