Optimal selection of medical formularies

Abstract

The rising cost of drugs is drawing scrutiny from many policymakers to the design of medical formularies. Formulary design is difficult because of the need to account for heterogeneity in patient characteristics, and the potential variation of cost-effectiveness of drugs across multiple conditions that they might be used to treat. In this article, we formulate the problem of selecting a medical formulary to minimize the cost of providing the drugs in the formulary, as well as the negative consequences to both patients and providers of the restrictions of the formulary. We model multiple drug categories and random variations in patient characteristics. We assume that patients are advised by physicians who are informed about the relative effectiveness of drugs that are available, and always make a rational choice to obtain the best expected outcome for their treatment. We derive insights into the structure of the optimal formulary under both certain and uncertain information about drugs’ effectiveness. Our model shows that widely used heuristics for drug selection that employ average or incremental cost-effectiveness ratios can perform very poorly. The model also shows that it is much more cost-effective for the payer to keep physicians informed about the relative effectiveness of drugs so that they can help patients to make customized choices, rather than to restrict the formulary for quality-control purposes.

Appendix

Proof of Theorem 1.For , let

Let be an optimal solution of (Π) and let x* be the indicator-vector representation for . That is, if and otherwise.

Fix a category p. Let and . Assume that there is l such that c _lp ⩽c _up , , but increasing to 1 results in a suboptimal solution. Note that

We claim that . Assume otherwise. Let x° be obtained from x* by setting x _l * to 1. Note that x° is a feasible solution for (Π). For each x(t)=x*+t(x°−x*), t∈[0, 1], it is clear that . Therefore,

which is a contradiction because we assumed that increasing to 1 results in a suboptimal solution.

By inspecting the expression for , we see that if , then for all j such that c _pj >c _pl . Thus, . Let be obtained from x* by reducing x _u * to 0. Note that is a feasible solution for (Π). For each , t∈[0, 1], it is clear that . Therefore,

which is a contradiction because we assumed that x* is optimal. □

Proof of Corollary 2. If all drugs are specialized, then the problem (Π) separates into P independent subproblems. For each p, the p-th subproblem can be stated as

where is the set of drugs used by category p. By Corollary 1, the solution to (Π _p ) consists of some k _p of the lowest-cost drugs in . Therefore, k _p can be found using a linear search of at most steps.

The total number of search steps performed over all patient categories is  □

Proof of Theorem 2. Let be an optimal formulary. Let l be a bargain drug and assume that is suboptimal. Let x be the indicator vector for . As in the proof of Theorem 1, we have

As in the proof of Theorem 1, including l in would not increase the objective. This contradicts our assumption. □

Proof of Theorem 3. Let denote the set of all bargain drugs. By Theorem 2, we can assume without loss of generality that an optimal formulary has form . Hence, the problem (Π) can be reduced to

If all expensive drugs are specialized, then the problem (Π′) separates into P independent subproblems as in Corollary 1, each of which is solvable in linear time. That the solution to the problem (Π) consists of some k _p of the lowest-cost drugs for each category p can be proved similarly to Corollary 1. □

Proof of Theorem 4. Let F(x) be as in (8). Let be an optimal solution of (Π) and let x* be the indicator vector for . That is, x _j *=1 if and x _j *=0 otherwise.

Let u be any drug with in . Assume that x _l *=0, x _u *=1, l dominates u, but increasing x _l * to 1 results in a suboptimal solution. Note that

As in the proof of Theorem 1, . As l dominates u, this implies that . Therefore, as in the proof of Theorem 1, we have a contradiction. □

Proof of Theorem 5. Let us compare the value of a selection with that of the larger selection .

The selection is better if and only if

The inequality reduces to

It is easy to see that the right-hand side is non-negative. If for each p then the left-hand side is less than or equal to 0, so that the inequality above holds. □

Proof of Theorem 6. Let us compare the value of a selection with that of the larger selection . Similar to the proof of Theorem 5, the selection is better if and only if

It is easy to see that the right-hand side is positive. If b _p is sufficiently high, then the above inequality holds. □

Proof of Theorem 7. Let be an optimal formulary. Let F(x) be defined according to (8). Let x* be the indicator-vector representation for .

Fix a category p. Assume that there are more than one drug for p in . Let l be the highest-cost drug among these. Then in particular, c _pl >b _p , or l cannot be a bargain drug. Recall from Theorem 1 that

which is positive by the assumption of the theorem. Similar to the proof of Theorem 1, removing j from S results in a strictly better formulary, which is a contradiction. Hence, there can be at most one drug per category in the optimal formulary. The conclusion follows from Theorem 1. □

Proof of Proposition 1. Fix a category p. By Theorem 1, is equal to {[1]^p,[2]^p, …, [k]^p} for some k.

We will first construct μ′ such that From the definition of majorization, there is a l such that where l ⩽ k and Define μ′ by letting for all i<l, , and μ′ _pi =0 otherwise. Then .

We claim that . Indeed, we can write as

The inequality follows from the fact that for each j=1, …, k and imply that for each j=1, …, k.

By the claim, we have .

Next, let x* be the indicator-vector representation for . We will abuse notation and use F(μ′, x*) to mean . Similar to the proof of Theorem 1, note that is equal to

If then decreasing to 0 results in a better solution. Otherwise, increasing to results in a better solution. In the former case, as μ and μ′ are the same in components [1] to [l − 1], we have In the latter case, letting y* denote the modification of x*, we have by construction of y*,

Therefore,  □

Proof of Theorem 9. Note that the provisioning cost is unchanged with b for categories p ⩾ 2. For category 1, the expected cost is

We can write

which is concave in ν_1j. By Jensen’s inequality, this cost is less than or equal to

with the difference increasing as b increases. Therefore, the provisioning cost of decreases as b increases. □

Proof of Theorem 10. As in the proof of Theorem 9, the provisioning cost is unchanged with b for categories p ⩾ 2. For category 1, the expected cost is

We can write

which is convex in ν_1j. By Jensen’s inequality, this cost is less than or equal to

with the difference increasing as b increases. Therefore, the provisioning cost of increases as b increases. □

Proof of Theorem 11. Note that

The conditional expectation is less than or equal to by Jensen’s inequality, with the difference increasing as b increases.