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Medical Decision Making
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.
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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 testtreatment thresholds, as well as the value of information for the situation in which a nonharmful 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.
(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?
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 (3month) qualityadjusted expected survival probabilities for a 50yearold 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 testtreatment 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?
$$ V I(p)=\left\{\begin{array}{l} p\cdot Se\cdot g\left(1 p\right)\left(1 Sp\right) l{l}^t\kern0.75em \mathrm{for}\kern1.5em 0\le p<{p}^{Rx},\\ {} p\cdot g\left(1 Se\right)+\left(1 p\right) Sp\cdot l{l}^t\kern0.75em \mathrm{for}\kern1.5em {p}^{Rx}\le p\le 1.\end{array}\right. $$
(a)
Calculate the treatment threshold, the test and testtreatment thresholds, as well as the value of information for the situation in which a nonharmful test is available.
(b)
Display the situation in a graph.
(c)
How is the treatment decision affected if a test is harmful?
(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?
(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.
(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?
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

(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 testtreatment 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?
1
2
3
4
Raiffa, H.—
Decision Analysis: Introductory lectures on choices under uncertainty—AddisonWesley Publishing Company, Reading, Massachusetts—1968, p. 271.
A test is (Pareto) efficient if there is no other test that is better than the other in terms of either sensitivity or specificity and that is at the same time not worse than the other.
\( L{R}_c^{+} L{R}_d^{+}>0 \) if
\( S{e}_1\cdot S{e}_2\left(1 S{p}_1\cdot S{p}_2\right)\left( S{e}_1+ S{e}_2 S{e}_1\cdot S{e}_2\right)\left(1 S{p}_1\right)\left(1 S{p}_2\right)>0. \) Transformation leads to
\( S{e}_1\cdot S{e}_2\left(1 S{p}_1+1 S{p}_2\right)\left( S{e}_1+ S{e}_2\right)\left(1 S{p}_1\right)\left(1 S{p}_2\right)>0. \) The next step results in
\( S{e}_1\left(1 S{p}_1\right)\left( S{e}_2+ S{p}_21\right)+ S{e}_2\left(1 S{p}_2\right)\left( S{e}_1+ S{p}_11\right)>0, \) which holds true for tests with discriminatory power.
Divide the numerator and denominator of
\( L{R}_d^{+} \) by the product
\( S{p}_1\cdot S{p}_2 \) to obtain
\( L{R}_d^{+}=\left(1/ S{p}_1\cdot S{p}_2 L{R}_1^{}\cdot L{R}_2^{}\right)/\left(1/ S{p}_1\cdot S{p}_21\right). \) As the negative likelihood ratios are smaller than one, the numerator exceeds the denominator, and hence
\( L{R}_d^{+}>1. \)
Ament, A., & Hasman, A. (1993). Optimal test strategy in the case of two tests and one disease.
International Journal of BioMedical Computing, 33(3–4), 179–197.
CrossRef
Drummond, M. F., O’Brien, B., Stoddart, G. L., & Torrance, G. W. (1987).
Methods for the economic evaluation of health care programmes. Oxford: Oxford University Press.
Hershey, J. C., Cebul, R. D., & Williams, S. V. (1986). Clinical guidelines for using two dichotomous tests.
Medical Decision Making, 6(2), 68–78.
CrossRef
Hull, R., Hirsh, J., Sackett, D. L., & Stoddart, G. L. (1981). Costeffectiveness of clinical diagnosis, venography and noninvasive testing in patients with symptomatic deepvein thrombosis.
New England Journal of Medicine, 304, 1561–1567.
CrossRef
Krahn, M. D., Mahoney, J. E., Eckman, M. H., Trachtenberg, J., Pauker, S. G., & Detsky, A. S. (1994). Screening for prostate cancer: A decision analytic view.
JAMA, 272(10), 773–780.
CrossRef
Perone, N., Bounameaux, H., & Perrier, A. (2001). Comparison of four strategies for diagnosing deep vein thrombosis: A costeffectiveness analysis.
The American Journal of Medicine, 110(1), 33–40.
CrossRef
Weinstein, M. C., Fineberg, H. V., Elstein, A. S., Frazier, H. S., Neuhauser, D., Neutra, R. R., et al. (1980).
Clinical decision analysis. Philadelphia: W.B. Saunders.
 Titel
 Optimal Strategy for Multiple Diagnostic Tests
 DOI
 https://doi.org/10.1007/9783662534328_7
 Autoren:

Stefan Felder
Thomas Mayrhofer
 Verlag
 Springer Berlin Heidelberg
 Sequenznummer
 7
 Kapitelnummer
 Chapter 7