Treatment Decisions Without Diagnostic Tests

Abstract

We consider a medical case which requires a physician’s treatment decision where no diagnostic test is available. The physician knows the particular illness and its treatment, as well as a sick person’s utility gain and a healthy person’s utility loss from treatment. But he is uncertain whether the patient is actually sick or not. To aid his decision, we derive a lower boundary for the a priori probability of the illness at which treatment is indicated. We show that a risk-averse physician should treat at a lower a priori probability than a risk-neutral physician. We also analyze the therapeutic risk, i.e., the risk that treatment fails, and derive the success probability threshold, above which the physician will undertake treatment. In contrast to the diagnostic risk, the threshold for the successful treatment probability is higher for a risk-averse decision maker than for a risk-neutral one. Finally, we consider the diagnostic risk and the therapeutic risk simultaneously and study the thresholds in this extended model.

Exercises

• 1. This exercise is adopted from Eeckhoudt (2002), p. 21. After a physician has thoroughly examined a patient, he must decide for or against treatment. Assume that the patient’s utility can be measured in QALYs. If the patient is sick and does not undergo treatment, he obtains 10 QALYs. If the sick patient is treated, his utility increases by 15 QALYs. There is, however, a chance that the patient does not actually suffer from the particular illness. In this case, he obtains 50 QALYs if he is not treated. If the physician treats the patient despite his good health, a utility loss of 5 QALYs will ensue.

  • (a) How large are g and l in this situation?

  • (b) Calculate the treatment threshold.

  • (c) Imagine that this is not a single case, but that a physician’s practice features 100 patients from one population group with an a priori disease prevalence rate of 0.1, 200 patients from another group with a prevalence rate of 0.2, and 50 patients from a third group with an a priori rate of 0.6.

    • (1) How many patients will receive treatment if the physician follows the threshold rule?

    • (2) How large is the expected number of QALYs in the entire patient population if

      • No patient gets treatment?

      • All patients get treatment?

      • The physician follows the threshold rule?

  • (d) Assume that the effectiveness of treatment increases such that a sick patient undergoing treatment obtains 28 QALYs and a healthy patient obtains 47 QALYs after treatment.

    • (1) How does this affect the treatment threshold?

    • (2) How many patients are treated now?

    • (3) Calculate the new number of QALYs if

      • No patient gets treatment.

      • All patients get treatment.

      • The physician follows the threshold rule.

  • (e) Draw the equivalent of Fig. 4.2 for the situations before and after the improved treatment.

• 2. Show that even if the $E{U}_i^{+}(p)$ line has a negative slope, the treatment threshold is lower for a risk-averse decision maker than for a risk-neutral one.

• 3. The following values are given for these health states: ${h}_s^{+s}=36;$ ${h}_s^{-}=25;$ ${h}_s^{+f}=16$. A risk-neutral decision maker has the utility function $u(h)=0.1\cdot h+2.4$. The utility function of the risk-averse decision maker is $u(h)=\sqrt{h}.$

  • (a) Calculate the utility gains from successful treatment and the utility loss from a failure for both types of decision maker.

  • (b) At which success probability will the risk-neutral physician start to treat? What is the threshold for the risk-averse physician?

  • (c) Draw Fig. 4.9 for the above values.