Cost-benefit considerations of the biased diagnostician

Amnon Sonnenberg

doi:10.1515/dx-2024-0031

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Cost-benefit considerations of the biased diagnostician

Amnon Sonnenberg

Veröffentlicht/Copyright: 22. Juli 2024

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Aus der Zeitschrift Diagnosis Band 11 Heft 4

Abstract

Objectives

In the cognitive process of establishing a diagnosis, the performance of a diagnostician can be characterized in terms of sensitivity and specificity. The aims of the present study are to analyze in quantitative terms how cognitive bias affects the performance of a diagnostician, and how a diagnostician’s biased decision making is further influenced by personal cost-benefit considerations.

Methods

The test matrices of two sequential diagnostic tests are manipulated according to the rules of linear algebra, using multiplication of the second with the first test matrix to calculate their joint test characteristics. The decision tree and receiver operating characteristic (ROC) of a biased and unbiased diagnostician are used to calculate which combination of test characteristics maximizes the expected utility value.

Results

Biased diagnosticians cannot establish a diagnosis beyond their own limited or distorted level of understanding. An unbiased and a biased diagnostician alike adjust their choice of test characteristics according to their different cost-benefit estimation of the various test outcomes. From the perspective of an unbiased diagnostician, the choices made by a biased diagnostician appear to invert reality. However, the same appearance of inverted reality is perceived by the biased diagnostician, judging the choices made by the unbiased diagnostician.

Conclusions

As a general principle, human testers cannot test beyond their own level of understanding. They only see what they know. As they base their judgment on preconceived notions about the utilities associated with different test outcomes, human testers also tend to only know what they want to know.

Keywords: cognitive bias; medical decision making; receiver operating characteristic; sensitivity; specificity; test characteristics

Corresponding author: Amnon Sonnenberg, MD, MSc, The Portland VA Medical Center and the Division of Gastroenterology and Hepatology, Oregon Health & Science University, P3-GI, 3710 SW US Veterans Hospital Road, Portland, OR 97239, USA, E-mail: sonnenbe@ohsu.edu

Research ethics: Not applicable. No patients or patient records were used in the present analysis.
Informed consent: Not applicable to the present theoretical analysis.
Author contributions: The author, Amnon Sonnenberg, has accepted responsibility for the entire content of this manuscript and approved its submission. Amnon Sonnenberg is the sole author responsible for study design, data analysis, and writing of the manuscript.
Competing interests: The author states no conflict of interest.
Research funding: None declared.
Data availability: Not applicable.

Calculating the cut-off value of an unbiased diagnostician

According to the decision tree of Figure 2, the overall expected utility (U_exp) of the entire decision tree corresponds to the sum of the utilities associated with the four terminal branches of the tree, weighted by their individual probability values of occurrence, as depicted by Eq. (6) from above and repeated below as Eq. (A1):

(A1) U exp = p ∙ t p ∙ U t p + p ∙ f n ∙ U f n + ( 1 − p ) ∙ f p ∙ U f p + ( 1 − p ) ∙ t n ∙ U t n

Using the two relationships fn = 1 − tp and tn = 1 − fp, equation (A1) is converted into Eq. (A2), which represents U_exp as a function exclusively dependent on tp and fp, that is, U _exp= f(tp, fp).

(A2) U exp = p ∙ t p ∙ U t p + p ∙ ( 1 − t p ) ∙ U f n + ( 1 − p ) ∙ f p ∙ U f p + ( 1 − p ) ∙ ( 1 − f p ) ∙ U t n

Some simple algebraic manipulations applied to Eq. (A2) result in equation (A3) shown below.

(A3) U exp = p ∙ t p ∙ U t p + p ∙ U f n − p ∙ t p ∙ U f n + f p ∙ U f p − p ∙ f p ∙ U f p + ( 1 − p ) ∙ U t n − ( 1 − p ) ∙ f p ∙ U t n

The second and the sixth term do not contain tp or fp, would not vary with changing tp or fp values, and are therefore considered representing constants. These two terms are summed up as value C in the subsequent analysis. After some additional algebraic manipulations, equation (A3) changes into Eq. (A4), as shown below.

(A4) U exp = p ∙ t p ∙ ( U t p − U f n ) + ( 1 − p ) ∙ f p ∙ ( U f p − U t n ) + C

Following the rules of calculus, to find the combination of fp and tn values, which maximize U_exp, one calculates the first derivative of Eq. (A4), and sets it equal to 0.

(A5) d U exp d f p = p ∙ d t p d f p ( U t p − U f n ) + ( 1 − p ) ∙ ( U f p − U t n ) = 0

The term dtp/dfp corresponds to the slope of the ROC curve at the cut-off point, where the corresponding tp and fp values maximize U_exp.

(A6) d t p d f p = ( 1 − p ) p ∙ ( U t n − U f p ) ( U t p − U f n )

Calculating the cut-off value of a biased diagnostician

A biased and unbiased diagnostician may look at the same decision tree and start with the same equation (A1) for the overall expected utility value (U_exp) of the tree. However, rather than being concerned about diagnosing D⁺, a completely biased diagnostician may focus on its absence or the presence of an alternative diagnosis D⁻. Accordingly, the biased diagnostician may consider an alternative ROC curve, with the fraction of tn being plotted against fn. The biased diagnostician would be interested in finding the optimized cut-off value for the alternative ROC curve. Using the two relationships tp = 1 − fn and fp = 1 − tn, equation (A1) is now being converted into Eq. (A7), which represents U_exp as function exclusively dependent on tn and fn, that is, U _exp= f(tn, fn).

(A7) U exp = p ∙ ( 1 − f n ) ∙ U t p + p ∙ f n ∙ U f n + ( 1 − p ) ∙ ( 1 − t n ) ∙ U f p + ( 1 − p ) ∙ t n ∙ U t n

Few simple algebraic manipulations applied to Eq. (A6) result in Eq. (A8) shown below.

(A8) U exp = p ∙ U t p − p ∙ f n ∙ U t p + p ∙ f n ∙ U f n + ( 1 − p ) ∙ U f p − ( 1 − p ) ∙ t n ∙ U f p + ( 1 − p ) ∙ t n ∙ U t n

The first and fourth term do not contain tn or fn, would not vary with changing tn or fn values, and are therefore considered representing two constants. These two terms are summed up as value C in the subsequent analysis. After some additional algebraic manipulations, Eq. (A8) changes into Eq. (A9), as shown below.

(A9) U exp = p ∙ f n ∙ ( U f n − U t p ) + ( 1 − p ) ∙ t n ∙ ( U t n − U f p ) + C

Following the rules of calculus, to find the combination of fn and tn values, which maximize U_exp, one calculates the first derivative of Eq. (A9), and sets it equal to 0.

(A10) d U exp d f n = p ∙ ( U f n − U t p ) + ( 1 − p ) ∙ d t n d f n ( U t n − U f p ) = 0

The term dtn/dfn corresponds to the slope of the alternative ROC curve. Its value as shown below indicates the slope of the ROC curve at the cut-off point, where the corresponding tn and fn values maximize U_exp according to the biased diagnostician.

(A11) d t n d f n = p ( 1 − p ) ∙ ( U t p − U f n ) ( U t n − U f p ) ≈ B e n e f i t exp C o s t exp

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Received: 2024-02-12

Accepted: 2024-06-18

Published Online: 2024-07-22

Published in Print: 2024-11-26

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https://doi.org/10.1515/dx-2024-0031

Schlagwörter für diesen Artikel

cognitive bias; medical decision making; receiver operating characteristic; sensitivity; specificity; test characteristics