# Overcoming Fixed Comparator Problems - Multiple Strategy Comparison of Costs and Effects with Flexible Axes on the C-DU Plane

Inference problems arising with fixed comparators on the C-E plane are solved by comparing GERD strategies under uncertainty on the cost-disutility (C-DU) plane using flexible axes of incremental costs relative to lowest costs strategy in each replicate (vertical axis) and incremental disutility relative to highest effect strategy (horizontal axis). As shown in Fig. 8.4, this enables immediate and full graphical effect and cost inference.

On the C-DU plane, the entire distribution of strategy B (maintenance PPI) lies on the horizontal axis with 0 incremental weeks with GERD (note framed from a disutility perspective) relative to the most effective strategy. Hence, strategy B has the lowest number of weeks over 12 months with GERD, or equivalently the highest weeks without GERD, for every one of the 1000 replicates. This stark contrast with the inability to distinguish between effects of strategies B, F, E and A in any given replicate on the C-E plane arises as flexible axes on the C-DU plane ensure that strategies are compared with the appropriate lowest cost and highest effect (lowest disutility) comparator on respective axes and for each replicate.

Similarly, either A or C are the least cost strategies as each have 0 incremental costs relative to the cheapest strategy and hence have replicates lying on the horizontal axis. In general the flexible axes on the C-DU plane, where DU across strategies is measured relative to the lowest DU or most effective strategy in each replicate

Fig. 8.4 Multiple strategy cost and effect inference with flexible axes on the C-DU plane (Source: Eckermann and Willan (2011)) and cost across strategies are measured relative to the least cost strategy in each replicate, allow appropriate graphical inference in relation to effects and cost from the proportion of distributions on vertical and horizontal axes. Nevertheless, as on the CE plane, covariance between cost and effects across replicates is still hidden on the C-DU plane, given neither map which replicates and their joint costs and effects link between strategies.

To inform societal decision making in relation to relative net benefit (cost effectiveness) across multiple strategies requires identifying the appropriate net benefit maximising comparator in any given replicate across potential threshold values (as with frontiers for deterministic analysis in Fig. 8.1). However, one needs to first consider what information is required to best inform such societal decision making under uncertainty.

The Arrow-Lind theorem (Arrow and Lind 1970) points to societal decision making (SDM) asymptotically approaching risk neutrality with risk spreading (diversification) across large numbers of government investment decisions and related patient populations, as argued by Claxton (1999). However, Graff, Zivin and Bridges (2002) suggested societal decision making in health care can be somewhat more risk averse than this might suggest, to the extent that patient outcomes may not be completely diversifiable and there are potential effects on private markets of public investment decisions in relation to health technology assessment. Nevertheless, in inform investment decisions under uncertainty, SDM can be characterized across these range of interpretations of the Arrow-Lind theorem for healthcare investment decisions as either asymptotically risk neutral or somewhat risk averse.

If SDM is risk neutral, then the investment objective without further research is to identify strategies maximising expected net benefit (ENB) across plausible threshold values. If SDM is somewhat risk averse, then at any given threshold value, the strategy with highest ENB will still be supported where that strategy also has as high or higher probability of maximising net benefit relative to other potentially optimal strategies. That will be the usual case except potentially over discrete regions of threshold values for effects where strategies vie for ENG maximisation, around which trades-offs can arise between the strategy maximising ENB at any given threshold value and another potentially optimal strategy (with somewhat risk- averse decision making) where they have higher probability of maximising NB at that threshold value.

The key implication is that for summary measures to best inform societal decision making primarily requires showing differences in ENB between strategies, regardless of whether SDM is risk neutral or somewhat risk averse. While expected values alone satisfy risk-neutral societal decision making, somewhat risk-averse decision making can additionally consider incremental probabilities of maximising net benefit across threshold regions but only over discrete threshold regions where trade-offs arise and only between potentially optimal strategies. In this respect, limitations of CEA curves, which only compare probabilities, were recognised by Fenwick et al. (2001) in pointing to the primary need for societal decision making to compare expected net benefit rather than the probability of maximising net benefit across multiple strate?gies. In Sect. 8.3, we identify expected net loss curves and frontiers (Eckermann et al. 2008; Eckermann and Willan 2011) as providing a first best solution to address the primary need of societal decision making to robustly compare expected net benefit across multiple strategies. Where decision making is somewhat risk averse, this primary need can be supplemented by presenting trade-offs over localised threshold regions where they arise, between incremental ENB and incremental probabilities of maximising NB. However, such probabilities should be informed by relevant bilateral CEA curves in comparison between potentially optimal strategies with the strategy maximising ENB to prevent confounding by other strategies inherent with multilateral CEA curves.

In later discussion, we consider the cost effectiveness acceptability frontier proposed by Fenwick et al. (2001) which presents probabilities of maximising NB for each strategy limited to restricted regions to indicate which strategies maximise ENB at any given threshold value. However, we show that such CEA frontiers are far from a first best solution in informing either primary or potential secondary needs of societal decision making under the Arrow-Lind theorem, facing many problems including: (i) the CEA frontier acting as a black box in lacking the ability to explain ENB optimisation or more generally compare relative ENB of strategies; (ii) decision maker conflation between probabilities presented in CEA curves or frontiers and expected values they want to primarily observe; and (iii) probability confounding in multiple strategy CEA curves or frontiers where societal decision making is somewhat risk averse and considers incremental probabilities alongside incremental ENB in trade-offs between potentially optimal strategies.

To begin identifying a first best solution, we first need to consider how differences in expected net benefit could be presented across multiple strategies, noting that INB curves introduced in Chap. 2 for two strategy comparisons, like the incremental cost effectiveness plane, are restricted to a single fixed comparator. Hence, the fixed strategy comparison that INB curves are based on does not accommodate the need to change comparator to the net benefit maximising strategy across replicates and for alternate threshold values. We need a more flexible comparator statistic than INB for multiple strategy comparisons.