Threshold analysis in the presence of both the diagnostic and the therapeutic risk

Abstract

The well-established a priori probability of illness threshold in medical decision making, introduced by Pauker and Kassirer (N Engl J Med 293:229–234, 1975; N Engl J Med 302:1109–1117, 1980), involves the diagnostic risk only. We generalize the threshold analysis by adding the therapeutic risk, i.e., in accounting for the risk that a treatment might sometimes fail. We derive a priori probability of illness threshold as a function of the probability of successful treatment, as well as the inverted function, where the successful treatment probability threshold is a function of the a priori probability of illness. The thresholds in the general model are higher than those in the special cases where one of the two risks is absent. Applications show that the changes in the thresholds can be substantial. Our general model might explain empirical findings of much higher thresholds than the Pauker–Kassirer model suggests.

Appendix

Curvature of the threshold functions

The treatment threshold function [Eq. (4)], the test threshold function [Eq. (8)] and the test-treatment threshold function [Eq. (10)] are all negatively sloped and convex. The proofs are analogous. We present the proof for the treatment threshold function.

Equation (4) can be transformed to

$$p^{Rx} \left( \pi \right) = \frac{{\pi s_{\text{h}} + \left( {1 - \pi } \right)f_{\text{h}} }}{{\pi \left( {s_{\text{h}} + s_{\text{d}} } \right) + \left( {1 - \pi } \right)\left( {f_{\text{h}} - f_{\text{d}} } \right)}}.$$

(13)

Then, the first derivative is negative

$$\frac{{\partial p^{Rx} \left( \pi \right)}}{\partial \pi } = - \frac{{f_{\text{d}} s_{\text{h}} + f_{\text{h}} s_{\text{d}} }}{{\left( {\pi \left( {s_{\text{h}} + s_{\text{d}} } \right) + \left( {1 - \pi } \right)\left( {f_{\text{h}} - f_{\text{d}} } \right)} \right)^{2} }} < 0$$

(14)

as $s_{\text{d}} ,f_{\text{d}} ,s_{\text{h}} ,f_{\text{h}} > 0$. For the second derivative, we obtain

$$\frac{{\partial^{2} p^{Rx} \left( \pi \right)}}{{\partial \pi^{2} }} = 2\frac{{\left( {f_{\text{d}} s_{\text{h}} + f_{\text{h}} s_{\text{d}} } \right)\left( {s_{\text{h}} + s_{\text{d}} + f_{\text{d}} - f_{\text{h}} } \right)}}{{\left( {\pi \left( {s_{\text{h}} + s_{\text{d}} } \right) + \left( {1 - \pi } \right)\left( {f_{\text{h}} - f_{\text{d}} } \right)} \right)^{3} }} > 0,$$

(15)

which is positive provided that $s_{\text{h}} + f_{\text{d}} + s_{\text{d}} > f_{\text{h}}$. The latter inequality is satisfied for $0 \le p^{Rx} \left( \pi \right) \le 1$.