LAUSR

We appreciate Dr. Gandjour’s interest in our recent article. Unfortunately, his claim that ‘‘the solution provided by Fairley et al. does not seem to adequately accommodate the presence of (semi-)fixed costs’’ appears to be based on a misreading of our work. In addition, he raises concerns about the work applying only in the realistic setting of a budgetary silo for research, uncoordinated with medical care budgets, which is less efficient than a theoretically possible centrally coordinated budget for research and medical care. The basis of this misunderstanding is related to a misinterpretation of our mixed-integer programming specification. In the optimization problem (OPT2), we do not treat fixed costs as ‘‘sunk’’ as Dr. Gandjour claims. Using a common approach to overcoming fixed costs in production problems, OPT2 includes a binary decision variable ys, where ys = 0 indicates no allocation of funding to study s and ys = 1 indicates at least a minimum level of funding such that the selected sample size has a positive expected net benefit of sampling (ENBS). Through the second and third constraints of OPT2, a decision of ys = 0 forces the selected sample size to also be zero, and a decision of ys = 1 forces the selected sample size to exceed a region of negative ENBS, which can occur due to fixed costs and/or from very small study benefits at small sample sizes. The possibility of a fixed cost and its impact on the budget are accounted for in the first constraint of OPT2, where ys is multiplied by the fixed cost (c f s ), resulting in the inclusion of the fixed cost when the decision has been made to select a positive ENBS sample size. In the original article, we provided 2 examples: 1 stylized example in which we did set fixed costs to zero and 1 realistic example in which each study has nonzero fixed costs. In his letter, Dr. Gandjour proposes changes in the curves presented in figure 3, the results of the stylized example, which he expects would occur if fixed costs were nonzero. Unfortunately, given his previously described misunderstanding, his proposed changes are only partially correct, as the y-axis intersection would not occur at the level of the 21 times the fixed cost. Following directly from the impact of the second and third constraints of OPT2, inclusion of a fixed cost would create an ENBS curve with the following features: 1) a point at (0,0), representing zero ENBS, no benefits, and no costs, for a study with no investment at all (i.e., g ns = 0); 2) a point at (1, Nsvs ns = 1 ð Þ cfs cns ns = 1 ð Þ), representing the ENBS of a study with a sample size of 1; and, 3) an ENBS curve parallel to the one shown, beginning at ns = 1, shifted downward by precisely the fixed cost (c f s ). The curve could be shifted down so far as to never have positive ENBS (and so the decision would be not to invest in collecting additional information at any sample size). The curve could also be shifted down such that the point on the ENBS curve where the marginal cost of one additional unit of information is equal to the shadow price of the budget constraint has negative ENBS; in this case, the optimal solution will also be not to invest in this study at any sample size (and this decision would be enforced by constraints 2 and 3 of OPT2). However, if there is a positive point on the ENBS curve at which the marginal cost of 1 additional unit of information is equal to the shadow price of the budget constraint, then there