## Two Way Sensitivity Analysis

Similarly, the physician may have only vague ideas concerning the actual costs of treating the patient with the PPI plus NSAID drug combination. In the middle panel of Figure 2-2, the drug costs of PPI plus NSAID were varied between $0 and $3,000. Again, a negative value on the y-axis indicates a cheaper COX-2 than the PPI plus NSAID strategy. As expected, any increase in PPI plus NSAID cost renders the COX-2 alternative increasingly more advantageous. Only with a PPI plus NSAID cost of less than $1,098 does the PPI plus NSAID treatment strategy represent the preferential treatment option.

As opposed to the one-way sensitivity analyses from above, in a subsequent two-way sensitivity analysis, one could also vary both types of drug costs simultaneously and check their joint influence on the cost difference between the two treatment options. Let us return to the decision tree of Figure 2-1. In the first step of a two-way sensitivity calculation, one would leave the COX-2 costs fixed at $1,200 and vary the NSAID plus PPI costs until the cost difference between the two treatment strategies (depicted in the first box on the left) becomes zero. This is achieved at NSAID plus PPI costs of $1,098. The pair of COX-2 and NSAID plus PPI costs ($1,200 and $1,098), which yield a zero net difference, are kept in a separate list. In a second step, one would choose a new value for the COX-2 costs, for example $1,600, and then restart the process of varying the NSAID plus PPI costs until the overall cost difference between the two treatment strategies again turns zero. The new pair of COX-2 and NSAID plus PPI costs ($1,600 and $1,498) is added to the list of pairs. These steps are repeated multiple times until a sufficient number of COX-2 and NSAID plus PPI cost pairs have been generated to plot the line shown in the right panel of Figure 2-2. In this particular example, a linear relationship for the COX-2 versus NSAID plus PPI costs is found, although nonlinear associations will frequently characterize other instances of a two-way sensitivity analysis.

COX-2 drug cost NSAID + PPI drug cost COX-2 drug cost

FIGURE 2-2. Sensitivity analysis of ulcer management. One-way sensitivity is shown in the left and middle panel; Two-way sensitivity analysis is shown in the right panel. COX-2 = cyclooxygenase-2-receptor antagonist; NSAID = nonsteroidal anti-inflammatory drug; PPI = proton pump inhibitor.

Each point on the line represents a combination of COX-2 and NSAID plus PPI costs, for which the decision tree yields an identical outcome for the two treatment strategies. The area beneath the black line includes all cost combinations of COX-2 and NSAID plus PPI, for which the PPI strategy is cheaper than the COX-2 strategy. This applies, for example, to a large variety of COX-2 and NSAID plus PPI cost pairs, such as $1,500 and $500, $2,000 and $500, or $2,500 and $1500. The white area above the black line represents all possible cost pairs for which the COX-2 strategy provides the cheaper treatment strategy. The line itself in this particular example depicts the equation NSAID + PPI = COX-2 - $102. In other words, the two treatment strategies yield identical outcomes, as long as the combined drug costs for NSAID plus PPI cost $102 less than the COX-2 costs alone.

Where does the $102 come from? Inspection of the original decision tree reveals that the $102 value stems from the 10% probability for the overall occurrence of complications multiplied by their respective expected value of $1,020. To reach equivalence between COX-2 and NSAID plus PPI, the NSAID plus PPI strategy needs to be cheaper by at least $102 to compensate for the additional costs of potential ulcer complications. The original decision tree was designed in such a way that any ulcer recurrence would automatically result in a switch from failed COX-2 to NSAID plus PPI. Obviously, if the COX-2 were to be continued even after instances of ulcer recurrences, and if it were to carry the same risk of future complications, the expected values of future complications in both treatment arms would cancel each other out. The two strategies would become equivalent if NSAID + PPI = COX-2. On the other hand, if COX-2 were associated with different probabilities for the occurrence of complications or different costs for the management of complications, the relationship between the two treatment options would change again. This last example serves as a reminder that the outcome of the decision tree depends on its overall structure and the type of question modeled by the tree.

Besides varying the costs or probabilities of the decision tree, as alluded to in the previous paragraph, one could also redraw parts of the tree or change its overall appearance. How far and how detailed should the medical history and the disease progression be followed into the future? The final outcomes of the present tree may seem somewhat arbitrary in that one could have easily proceeded further and spelled out many more details about the subsequent development of the patient's peptic ulcer. One could, for instance, subdivide pain into different types and severities or associate the hospital admission with far more detailed descriptions of the disease progression, such as ulcer bleeding, perforation, surgery, and their respective clinical outcomes. Because parameters on the far right of the decision tree become multiplied with an ever increasing number of probability values, they also tend to exert an ever decreas ing influence on the initial decision. For instance, in the overall process of averaging out the decision tree from right to left, the cost of death becomes multiplied once by 0.01 and twice by 0.1. In the final analysis, therefore, even doubling the annual cost of death from $30,000 to $60,000 only changes the COX-2 minus PPI cost difference from -$632 to -$659. Many of the parameters of a decision analysis exert little influence on its overall outcome, especially if they are located at the far end of the tree. As a general rule, therefore, it is not advisable to expand the tree too far into the future or include too many events that are associated with an a priori low probability value.

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