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 tests.
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Notes
- 1.
Raiffa, H. – Decision Analysis: Introductory lectures on choices under uncertainty – Addison-Wesley Publishing Company, Reading, Massachusetts – 1968, p. 271.
- 2.
A test is (Pareto) efficient if there is no other test that is better in terms of either sensitivity or specificity and at the same time not worse on the other.
- 3.
\( LR_c^{+} - LR_d^{+} > 0 \) if \( S{e_1}S{e_2}\left( {1 - S{p_1}S{p_2}} \right) - \left( {S{e_1} + S{e_2} - S{e_1}S{e_2}} \right)\left( {1 - S{p_1}} \right)\left( {1 - S{p_2}} \right) > 0. \) Transformation leads to \( S{e_1}S{e_2}\left( {1 - S{p_1} + 1 - S{p_2}} \right) - \left( {S{e_1} + S{e_2}} \right)\left( {1 - S{p_1}} \right)\left( {1 - S{p_2}} \right) > 0. \) The next step results in \( S{e_1}\left( {1 - S{p_1}} \right)\left( {S{e_2} + S{p_2} - 1} \right) + S{e_2}\left( {1 - S{p_2}} \right)\left( {S{e_1} + S{p_1} - 1} \right) > 0, \) which holds true for tests with discriminatory power.
- 4.
Divide the numerator and denominator of \( LR_d^{+} \) by the product \( S{p_1}S{p_2} \) to yield \( LR_d^{+} = {{{\left( {{{1} \left/ {{S{p_1}S{p_2} - LR_1^{-} LR_2^{-} }} \right.}} \right)}} \left/ {{\left( {{{1} \left/ {{S{p_1}S{p_2} - 1}} \right.}} \right).}} \right.} \) As the negative likelihood ratios are smaller than one, the numerator exceeds the denominator, and hence \( LR_d^{+} > 1. \)
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Felder, S., Mayrhofer, T. (2011). Optimal Strategy for Multiple Diagnostic Tests. In: Medical Decision Making. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18330-0_7
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DOI: https://doi.org/10.1007/978-3-642-18330-0_7
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