Optimal Strategy for Multiple Diagnostic Tests

Abstract

This chapter deals with the optimal test strategy if multiple diagnostic tests are available and can be employed simultaneously or sequentially. This complicates the task for the decision maker. He has to decide on the positivity criterion for the composite test, which can be conjunctive or disjunctive, and on the order of the tests if he uses them sequentially. If testing is potentially harmful, it is easy to understand that sequential testing always dominates parallel testing.

Exercises

• 1. Assume there is a specific treatment for an illness from which a sick patient will benefit with 0.4 QALYs. But the patient might not actually suffer from the diagnosed illness and therefore experience a loss of 0.1 QALYs from unnecessary treatment. A diagnostic test available to the physician has Se = 0.6 and Sp = 1. Assume that the harm is equivalent to an additional QALY loss equal to l ^t = 0.01 for both the sick and the healthy. Equation (5.8), then, changes to

  $$ V I(p)=\left\{\begin{array}{l} p\cdot Se\cdot g-\left(1- p\right)\left(1- Sp\right) l-{l}^t\kern0.75em \mathrm{for}\kern1.5em 0\le p<{p}^{Rx},\\ {}- p\cdot g\left(1- Se\right)+\left(1- p\right) Sp\cdot l-{l}^t\kern0.75em \mathrm{for}\kern1.5em {p}^{Rx}\le p\le 1.\end{array}\right. $$

  • (a) Calculate the treatment threshold, the test and test-treatment thresholds, as well as the value of information for the situation in which a non-harmful test is available.

  • (b) Display the situation in a graph.

  • (c) How is the treatment decision affected if a test is harmful?

  A second test now becomes available. It has the same specificity but a higher sensitivity than the first test (Sp = 1 and Se = 0.8). The QALY loss for the second test is 0.02, which is higher than that of the first test.

  • (d) What is the minimum a priori probability of an illness at which the physician will prefer the second test?

  • (e) Calculate the overall sensitivity and specificity if the physician tests sequentially, starting with the first test and proceeding with the second test for patients who received a positive first result.

  • (f) How will the physician’s choice change if the QALY loss of the second test increases to 0.05?

• 2. Figure 7.3 reveals that one of the two individual tests is always inferior. Prove the following general proposition: If $V{I}_i(p)> V{I}_c(p)$ or $V{I}_i(p)>$ VI _d (p), then $V{I}_j(p)< V{I}_i(p),$ $i\ne j.$

• 3. Demonstrate the following two inequalities: ${p}_d^{tRx}>{p}_c^{tRx}>{p}^{Rx}$. You may follow the approach used in footnotes 2 and 4.

• 4. Assume two tests with the following properties: $S{e}_1=0.6,\ S{p}_1=0.9$ and $S{e}_2=0.7,\ S{p}_2=0.8.$ Furthermore, the utility gain for sick patients will be 4 QALYs and the utility loss to healthy patients 1 QALY.

  • (a) Calculate the overall sensitivity and specificity of the composite test (for both the conjunctive and the disjunctive positivity criterion).

  • (b) Calculate the positive and negative likelihood ratios of the two single tests, as well as the likelihood ratios of the composite test with conjunctive and disjunctive positivity criteria.

  • (c) Draw Fig. 7.3 for the given test properties.

  • (d) Which test strategy will be preferred at the a priori probabilities of 0.1, 0.5 and 0.9?

  • (e) Draw level curves of informational value for these a priori probabilities.

  • (f) In which prevalence range is which alternative dominant?

  • (g) Draw Fig. 7.6 for the given test properties.

• 5. Hull et al. (1981) analyzed diagnostic strategies for deep vein thrombosis (DVT). The table below shows the outcomes and costs generated by two alternative strategies: impedance plethysmography (IPG) alone versus IPG plus outpatient venography if IPG is negative.

  Program	Test result	Presence of DVT		Costs (US$)
  		yes	no
  IPG alone	$+$	142	5	321,488
  	$-$	59	272
  IPG plus venography	$+$	201	5	603,552
  if IPG is negative	$-$	0	272

  Perone et al. (2001) report the following (3-month) quality-adjusted expected survival probabilities for a 50-year-old patient with a life expectancy of 29 years: 99.518 % (with DVT and treatment), 97.5 % (with DVT and no treatment), 99.768 % (without DVT and treatment) and 100 % (without DVT and no treatment).

  • (a) Calculate the a priori probability of DVT, the sensitivity and specificity of the individual and the combined test.

  • (b) Calculate the QALY values for each scenario, as well as the utility gain and loss from treatment and the treatment threshold. For the given a priori probability, which is the best strategy without a test?

  • (c) Calculate the test and test-treatment thresholds for each individual test as well as the composite test. For the given a priori probability, which is the best strategy when tests are available?

  • (d) Assume that treatment of a patient costs 2755 US$ and that the shadow value of the budget constraint is 50,000 US$ per additional QALY. Which is the best strategy?