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The Optimal Cutoff Value of a Diagnostic Test

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Abstract

We introduce a world where the number of available tests is infinite. This is the case if the test outcome has to be determined by the decision maker by setting a cutoff value. A good example is the prostate-specific antigen (PSA) test for the detection of prostate cancer in men. The analysis of the blood sample results in a PSA value which the physician judges to be positive or negative, depending on his chosen cutoff value. We demonstrate how the optimal cutoff value depends on the a priori probability of the illness as well as on the utility of and potential harm from testing and treatment. This chapter also introduces a novel use of the receiver operating characteristic (ROC) curve which is well known in clinical epidemiology.

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Notes

  1. 1.

    ‘Thus the will is revealed only in the act.’—Schopenhauer, A.—Die Welt als Wille und Vorstellung—Brockhaus Verlag, Leipzig—1859, p. 281.

  2. 2.

    The cost of testing does, however, play a role for the optimal cutoff value in sequential tests. The choice of the cutoff value for the first test must consider the cost of the second test, since it influences the probability that the patient will be subject to the second test.

  3. 3.

    Strictly speaking, the difference between the expected utility of testing and the expected utility of ‘no treatment’ is negative and the value of information is zero, as the decision maker would not employ the test in this prevalence range. For simplicity, we abstract from this complication again.

  4. 4.

    Felder et al. (2003), © 2003 by SAGE, reprinted by permission of SAGE Publications.

  5. 5.

    Mansfield et al. (1999) report termination rates after terminal diagnosis for five diagnoses in a systematic literature review. They were highest following a prenatal diagnosis for Down’s syndrome (92 %) and for Klinefelter syndrome (58 %). The inclusion of alternative ‘treatment’ options would extend our analysis, as the harm from amniocentesis is then relevant both in the case of positive and negative test outcomes.

  6. 6.

    Intangible costs include pain and anguish, or more generally the loss of quality of life due to an illness.

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Author information

Authors and Affiliations

  1. Department of Business and Economics, University of Basel, Basel, Switzerland

    Stefan Felder

  2. School of Business Studies, Stralsund University of Applied Sciences, Stralsund, Germany

    Thomas Mayrhofer

Authors
  1. Stefan Felder
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  2. Thomas Mayrhofer
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Exercises

Exercises

  1. 1.

    Assume a patient with an a priori probability of an illness of 20 %. If the patient is sick and gets treatment, he will gain 8 QALYs. If the patient is healthy but is treated he will lose 1 QALY. To detect the illness, the physician can use a diag nostic test with endogenous test characteristics that are summarized in the following ROC curve: \( S e=\sqrt[10]{1- Sp}. \)

    1. (a)

      Calculate the sensitivity and specificity using the Manhattan norm and the weighted Manhattan norm (see Box 8.1).

    2. (b)

      Show that the indifference curves of the Manhattan norm have a slope equal to one.

    3. (c)

      Calculate the sensitivity and specificity using the concept of the optimal cutoff value.

    4. (d)

      Draw the ROC curve for the diagnostic test and include the different optima.

    5. (e)

      Compare the results of (b) and (c) and comment.

  2. 2.

    Diagnostic misclassification can be due not only to overlapping distributions of the marker values for the sick and the healthy, but also to measurement errors. Assume that the true measurement variable has only two values, one for the sick and one for the healthy, so that the observed distribution of the variables is due only to measurement errors. Let there be two tests that differ with respect to the variance of the distributions of the measurement error. Test 1 has a lower vari ance than test 2 for both f s (x) and f h (x).

    1. (a)

      Draw ROC curves for the two tests and explain the difference.

    2. (b)

      What can you say about the optimal cutoffs for a given prevalence rate p and utility gains and losses from treatment g and l?

  3. 3.

    Harris et al. (2001) discuss the implication of pregnant women’s preferences for prenatal testing for chromosomal disorders. They argue that given \( g=1.27, \) the maternal age for administering amniocentesis immediately should be lowered to about 30. Remember that for \( g=1, \) Pauker and Pauker (1979) calculate an optimal age of around 35, which actually informed the official recommendations given in many countries.

    1. (a)

      Restate the argument of Harris et al. (2001).

    2. (b)

      Why were Pauker and Pauker possibly right in 1979? And why were Harris et al. maybe wrong in 2001? Discuss the role of biochemical testing in this context.

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Felder, S., Mayrhofer, T. (2017). The Optimal Cutoff Value of a Diagnostic Test. In: Medical Decision Making. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53432-8_8

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  • DOI: https://doi.org/10.1007/978-3-662-53432-8_8

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