How accurate should a precision medical test be to meet cost-effectiveness

Abstract

Precision medical tests play a critical role in tailoring treatment plans to individual patients and optimizing medical resource utilization. This paper establishes a quantitative model to analyze the relationship between the accuracy of a precision medical test and its cost-effectiveness. Two theoretical lower bounds on test accuracy are proposed to be required for a testing strategy to be considered cost-effective. The first bound ensures that the intervention matched to a test result remains locally optimal for the reported patient class, thereby satisfying a local efficiency criterion. The second bound ensures that the testing strategy yields a higher expected net monetary benefit for the entire population compared to no testing, satisfying a global efficiency criterion. A special case analyzing sensitivity and specificity for binary test results illustrates these thresholds intuitively. Numerical simulations support these theoretical findings. The results illustrate how accuracy requirements depend on class prevalence, treatment benefits, and test costs, and may serve as a preliminary screening criterion for further evaluation.

Introduction

Cost-effectiveness analysis (CEA) is widely used in health economics to evaluate healthcare interventions and inform health technology assessment (HTA). By comparing costs and health outcomes across alternative strategies, CEA provides an important framework for guiding healthcare resource allocation. Although CEA was originally developed primarily for evaluating therapeutic interventions, it has increasingly been applied to diagnostic and medical testing technologies in recent years. In particular, CEA has been deployed in numerous HTAs evaluating diagnostic and testing strategies, including studies such as Thomas et al.  [1], Jacobsen et al.  [2], and Choi et al.  [3]. At the same time, several reviews have noted that the use of CEA in the evaluation of diagnostic and testing technologies has grown substantially, although important methodological issues remain insufficiently addressed (Fang et al.  [4]; Yang et al.  [5]; Snowsill  [6]).

From a theoretical perspective, the economic evaluation of diagnostic tests is closely related to decision theory and treatment-selection models under uncertainty. Diagnostic tests can be interpreted as information-generating mechanisms that update beliefs about a patient’s underlying health state and guide treatment choices. In Bayesian decision frameworks, test results modify the posterior probability of disease and therefore influence the optimal intervention decision for a patient. These treatment-selection models provide a conceptual foundation for understanding how diagnostic information affects clinical decision making and healthcare resource allocation, and they have been widely discussed in the decision-analytic health economics literature (e.g., Nixon et al.  [7]; Felder and Mayrhofer  [8]). Building on these theoretical foundations, decision-analytic modelling has become a standard methodological approach for evaluating diagnostic and testing strategies. Decision trees and state-transition models are commonly used to connect test performance, treatment pathways, and long-term health outcomes. These modelling approaches allow researchers to synthesize evidence from diagnostic accuracy studies and treatment effectiveness studies to estimate the cost-effectiveness of alternative testing strategies. Previous methodological work has emphasized the importance of explicitly incorporating diagnostic accuracy, clinical pathways, and downstream health outcomes when conducting economic evaluations of diagnostic technologies (Yang et al.  [9]; Siebert et al.  [10]; Koffijberg et al.  [11]; di Ruffano et al.  [12]). Another important methodological perspective is provided by the value-of-information (VOI) framework. VOI analysis evaluates the economic value of reducing uncertainty in healthcare decision making through additional information, such as diagnostic testing or further research. Measures including the expected value of perfect information (EVPI) and the expected value of sample information (EVSI) quantify the potential benefits of improved information for healthcare decisions (Siebert et al. [13]; Mohiuddin et al. [14]). Within this framework, the value of diagnostic tests depends on the information they generate and the extent to which this information improves treatment decisions and healthcare resource allocation. These methodological considerations have become increasingly important with the emergence of precision medicine. Precision medical tests, including genomic sequencing, proteomic analysis, and biomarker-based diagnostics, aim to identify individual biological characteristics and guide targeted therapeutic interventions. By enabling more personalized treatment strategies, these tests have the potential to improve clinical outcomes and optimize the allocation of healthcare resources. However, considering their high costs and uncertainty, only a fraction of emerging precision medical technologies are likely to be cost-effective and adopted by healthcare systems (Ling et al.  [15]; Kasztura et al.  [16]). In precision medical testing, accurate patient classification is essential for matching individuals to appropriate targeted interventions. Therefore, diagnostic test accuracy plays a critical role in determining the value of testing strategies (Gavan et al. [17]; Mattos-Arruda and Siravegna  [18]; Chen et al.  [19]). The importance of incorporating test accuracy into cost-effectiveness analysis has long been recognized in the literature. More than two decades ago, Hernandez  [20] emphasized the central role of diagnostic accuracy in the economic evaluation of laboratory tests. Subsequent systematic reviews and methodological guidance have further highlighted the need to incorporate diagnostic accuracy when reporting CEA results (Van den Bruel et al.  [21]; Seo and Strong  [22]; Snowsill  [6]). In addition, several disease-specific economic evaluations have shown that the cost-effectiveness of testing strategies can be highly sensitive to diagnostic accuracy parameters (Novielli et al.  [23]; Novielli et al.  [24]; Garg et al.  [25]; Biltaji et al.  [26]).

Despite these advances, most existing studies focus on specific disease contexts or empirical modelling applications. While the importance of diagnostic accuracy is widely acknowledged, there remains a notable lack of general theoretical frameworks that quantitatively link test accuracy to cost effectiveness under a unified set of assumptions. Existing work tends to analyze the value of diagnostic information within narrowly defined clinical scenarios, yet it rarely addresses how the effects of test accuracy interact with heterogeneous patient subgroups in a general and structurally explicit manner. Moreover, the literature predominantly adopts a forward approach that estimates cost effectiveness for a given test accuracy profile, with little attention to the inverse question of determining the minimum accuracy required for a test to be cost effective under specified conditions. Collectively, these gaps motivate the need for a comprehensive analytical framework that not only establishes formal bounds on accuracy from both local and global efficiency perspectives but also systematically incorporates subgroup heterogeneity to inform the development and evaluation of precision medical tests.

To bridge this gap, our study establishes a generalized model to quantify the accuracy requirements of precision medical tests to achieve cost-effectiveness. Within the assumptions of our model, we derive two theoretical lower bounds that test accuracy must exceed: 1) The Lower Bound for Matching Intervention, which represents the minimum accuracy required to ensure that the intervention matched to a test result is the locally optimal choice for that reported class; and 2) The Lower Bound for Overall Effectiveness, which represents the minimum accuracy required for the testing strategy to maximize the expected net monetary benefit of the entire population relative to no testing. These derivations offer theoretical insights into the boundary conditions for developing and evaluating diagnostic tests. We also analyze a special case involving sensitivity and specificity for binary test results, a common scenario in clinical practice such as genetic screening or infectious disease testing. This specific analysis yields more intuitive accuracy thresholds relevant to standard medical screening evaluations.

Our results highlight how insufficient test accuracy can lead to suboptimal resource allocation and provide a theoretical perspective on the economic challenges of implementing precision medicine in small patient populations, such as those with rare diseases. Through numerical simulations, we verify the theoretical results and explore the impact of test accuracy on expected net monetary benefits and the proportion of patients receiving matched interventions. Our findings propose a set of theoretical criteria to assist in assessing whether a precision medical test possesses the fundamental accuracy potential to be cost-effective within a stylized model under some assumptions.

Model settings

To illustrate the relationship between test accuracy and CEA results, we utilize a heterogeneous model framework. We assume that patients can be divided into a finite number of distinct classes. For each patient class and each possible intervention, the health outcome might differs, which allows for heterogeneous treatment effects across class-intervention pairs. Furthermore, this paper uses net monetary benefit (NMB) as a measure to assess which health intervention is more cost-effective, from the point of view of classical CEA.

General settings

Consider a patient population where each patient belongs to one of distinct, mutually exclusive classes (e.g., defined by a specific genetic mutation, disease subtype, or risk category). For this patient population, there is an intervention pool with optional interventions available for them.

For each class , there exists a matched medical intervention (e.g., a targeted therapy) defined as follows.

Definition 1

The -th intervention is defined to be (best) matched to the patients in class if and only if

1

where denotes the health outcome, measured in Quality-Adjusted Life Years (QALYs), for the patients in class using the -th intervention, and denotes the cost of the -th intervention. Here, is a willingness-to-pay threshold used to combine QALYs and costs.

Remark 1

It could be easily seen that Eq. (1) means that for class , the -th intervention is more cost-effectiveness than any other optional intervention, by showing that

2

which coincides with the form of ICER in classical CEA analyses.

Remark 2

”In general, and may differ. Generally, if , it means that there exist at least one intervention in the pool which is not optimal for any patient class. And also, if , it represents the case that more than one patient class shares the same optimal intervention.”

Through this definition and further we assume that there is no redundant intervention in the optional pool, a Coherent Local Optimal Matching Function could be defined as follows.

Definition 2

Coherent Local Optimal Matching A function is defined as a coherent local optimal matching function (CLOM) if it satisfies

3

Remark 3

The economic intuition of CLOM is simply helps to find out the unique most cost-effectiveness intervention, for patient class .

According to the Net Monetary Benefit (NMB) framework, we denote the NMB of giving intervention to a patient from class as

4

The distribution structure of the intrinsic characteristics of patients in the entire population can be observed through retrospective statistics over time. The population proportion of class is , forming a probability distribution. The societal objective is to maximize the expected NMB across the population along with some necessary conditions.

Medical test, accuracy and strategy

In abstract terms, a Medical Test is a patient classifier that identifies any patient into one of N patient classes based on multiple pieces of information. Because of the partial information issues and other uncertainties, any medical test could make mistakes randomly. So the outcome for a medical test for any patient could be a random variable. Mathematically, there follows

Definition 3

A Medical Test is defined as a random function where for all , is a random variable taking values in , and is the set of all random variables on . Besides, we use to represent the cost of any one patient taking this medical test.

This means that a patient in any class taking the medical test will be assigned to a nominated class . Of course, if the test gives out the right answer, it will show that , otherwise . As there is uncertainty, the probabilistic properties of matters.

Accordingly, the distribution of could always be expressed as

5

where represents the probability of test giving the right answer on class , and is the probability of wrongly assign the class patient into class . Clearly, there is . Here, s are called Accuracy Vector of test .

Remark 4

The parameters and directly determine the likelihood of correct and incorrect classification. Positive and negative predictive values could be seen as derived quantities that depend on both these accuracy parameters and the others, which are not separately modeled as inputs in this framework. Moreover, real-world tests faces the alignment between predicted probabilities and observed frequencies, which is also not incorporated, as we assume the test produces a hard classification for simplification.

Generally, a patient’s real class cannot be directly observed. The only observable result is the patient’s medical test outcome . Any intervention decision can only be made based on the outcome following the test. An intervention strategy is kind of making decisions according to the outcome of the medical test, which is shown as the following definition.

Definition 4

An intervention strategy is defined as to give the patient with test outcome the -th intervention.

Remark 5

It is worthy to note that the choosing of intervention is made on the test outcome instead of the real patient class, because the real class is unobservable.

Expected NMB with and without the medical test

With all the settings above, the expected NMB per patient using the test with strategy is denoted as . One can easily show that it could be expressed as

6

where denotes the conditional expectation with respect to class . Although complex, we can decompose the expression using in Lemma 1.

Lemma 1

has a decomposition with , which is

7

where is part of NMB that the class of patients with result contributes and could be expressed as

8

Proof of Lemma 1

Initially, we can reform the expression for by implementing the total probability formula, subsequently sorted according to the result of . This approach enables us to have

9

where

10

Moreover,

11

Remark 6

This is very easy to understand in the NMB framework. In the Eq. (8), the first part of the summation could be interpreted as the NMB that are correctly tested into class , which is , and with probability . Then the second part is a summation, integrating the NMB that are incorrectly tested into class , which is and with probability . It is worth noting that if we assume , then would be relatively large but not the other ones. This contribute a lot in the future development of theorems in this paper.

For the case that decisions are made without the test, a single intervention must be chosen for all patients. The expected NMB in this case for a strategy assigning intervention to everyone is denoted as . It is easy to show that

12

Decision making

In an economic perspective, medical decision makers must select the most cost-effective intervention from available treatments to make a decision on whether or not medical tests should be present. This actually requires a comparison between the optimal NMB performance with and with out the medical tests. Hence, analyzing the NMB output under the most cost-effective intervention for each situation is crucial before comparing them.

Decision making without the test

When medical testing is absent, medical decision makers cannot differentiate between various types of patients. Although in clinical practice, patients still could receive various treatments because of availability, coverage of insurance, or clinician’s preference, but these situations are independent of the category of patients. This means that the various treatments could be regarded as random as refers to the category, which could be proved to be not more cost effective than the unique intervention strategy in expectation. This means that it is reasonable to assume that the physician must select a single intervention as the most cost-effective option for all patients. Thus, comparing the expected NMB results of the interventions for the entire population would suffice to determine the best intervention alternative in this case. Without considering patient characteristics to dictate treatment decisions, the optimal strategy is the specific intervention that offers the highest expected NMB for the general patient population among the options. The expression for this strategy is

13

The forthcoming Theorem 1 presents the most cost-effective intervention selection for the situation without medical testing.

Theorem 1

Without a medical test, the optimal intervention should be chosen as

14

where which is the average QALY outcome on the whole patient population using intervention .

Proof of Theorem 1

This could be easily shown by noting that

15

From Theorem 1, it is clear that is the most cost effective intervention for the whole patient population, which is very intuitive in the case without any medical test. With this, the expected optimal NMB in the absence of medical testing could be measured as

16

which serves as the decision benchmark to determine whether to use a medical test. This means that for a medical test to be cost-effective, the NMB of the entire population of patients must exceed this benchmark.

Decision making with the test

Now we consider the optimal expected NMB outcome with the medical test. This means that we need to find the optimal strategy given that the medical test is taken. This is defined as the following.

Definition 5

The optimal strategy given that the medical test is taken is defined as

17

Thus, the optimal expected NMB outcome with the medical test could be expressed as , which is a functional of the test function .

The necessary conditions of a precision medical test to be cost-effective contains two major parts. The first one is localized, aims to maintain local optimum standards. The core of this standard is to provide fair, consistent, and optimalThe necessary conditions of a precision medical test to be cost-effective contains two major parts. The first one is localized, aims to maintain local optimum standards. The core of this standard is to provide fair, consistent, and optimal treatment opportunities for all patients based on the results of precision medical tests. For example, if treatment is determined as the most effective for patients in class , then it is necessary for all patients classified as by the test to receive treatment , even if some may actually belong to other categories due to errors in the test. This means that for any nominated class of patients, the optimal strategy should suggest the most cost-effective intervention for class . The second part reflects the overall economic standard. It requires that the optimal expected NMB of adopting precision medical tests must exceed the NMB when no test is performed. This ensures that the introduction of medical testing brings additional overall economic value to the healthcare system. treatment opportunities for all patients based on the results of precision medical tests. For example, if treatment is determined as the most effective for patients in class , then it is necessary for all patients classified as by the test to receive treatment , even if some may actually belong to other categories due to errors in the test. This means that for any nominated class of patients, the optimal strategy should suggest the most cost-effective intervention for class . The second part reflects the overall economic standard. It requires that the optimal expected NMB of adopting precision medical tests must exceed the NMB when no test is performed. This ensures that the introduction of medical testing brings additional overall economic value to the healthcare system.

Based on these conditions, the cost-effectiveness of a precision medical test could be defined as follows.

Definition 6

A precision medical test is cost-effective (w.r.t. no test) if and only if the following conditions are satisfied.

• Local Cost-effective Axiom

  For any nominated patient class , which means , the expected NMB of adopting the optimal strategy suggested according to the precision medical tests to patient class , , must exceed the expected NMB on using any other strategy, which is

  18

• Global Cost-effective Axiom

  The optimal expected NMB of adopting precision medical tests must exceed the NMB when no test is performed, which is

  19

Remark 7

From the standpoint of coherent optimality, the Local Cost-effective Axiom protects some patients from the fully Bayesian optimal decision rule. In fact, under the fully posterior-optimal Bayesian decision rule, a relatively noisy test may yield suboptimal interventions for some patients who are correctly classified. This is because the Bayesian optimal decision rule demands the absolutely optimal intervention for each reported class, even when a reported class may be composed of multiple distinct true classes, thereby leading to an intervention that is optimal for the class as a whole but not necessarily for each correctly classified member. In such a case, we can neither ask the correctly classified patients to bear the cost of an inaccurate test, nor can we impose the ethical cost on the entire reported group. Therefore, we must require that the test result satisfy the Local Cost-effective Axiom to ensure that, even under a fully Bayesian optimal decision rule, the “matching” intervention for every reported group is the locally optimal one.

In this section, we first discuss the necessary condition of “Local Cost-effective Axiom”, which will lead to a “Lower Bound for Matching Intervention”. The following Theorem 2 provides the minimum accuracy level required for the necessary condition of “Local Cost-effective Axiom” to hold.

Theorem 2

Lower Bound for Matching Intervention The necessary condition for the “Local Cost-effective Axiom” to be hold is that

20

Proof of Theorem 2

For any , according to Eq. (3), is the unique solution satisfies Eq. (18). This means that

21

Then based on the definition of , there should be

22

Moreover, given Eq. (8), it means that

23

Then we get that for all , there is

24

which is

25

As this holds for all , we finally get

26

Although the mathematical expression appears complex, Theorem 2 conveys a clear economic interpretation. It implies that to satisfy the Local Cost-effective Axiom, the accuracy of the medical test for any patient class must exceed a lower bound. The economic rationale behind this bound is not complicated. A closer examination of the numerator and denominator on the right-hand side reveals that the term in the denominator

27

represents the difference in expected net monetary benefit for patients in class when they receive their optimal strategy compared to any other strategy. Since is the optimal strategy, this difference is always positive. In contrast, the term in the numerator

28

represents the increase in expected net monetary benefit for patients in class (with ) when they receive an alternative strategy instead of the strategy .

From an economic perspective, this expression can be understood as follows. When patients from class are correctly identified as belonging to class by the test, administering yields a gain in net monetary benefit. However, when patients from class are incorrectly classified as class , administering may lead to a reduction in net monetary benefit. The lower bound essentially ensures that the proportion of correctly identified cases is sufficiently high relative to the overall population, such that the gain from correct classifications outweighs the loss from incorrect ones, thereby satisfying the Local Cost-effective Axiom.

The following discussion uses corollaries under symmetric or degenerate scenarios to further illustrate the intuition behind Theorem 2.

Corollary 1

Given the following symmetric or degenerate conditions,

• Precision rate for any class is equal and errors are equally distributed, which is

  29
• Degenerate the NMB outcome into 2 state, correct and incorrect intervention, which is

  30

  where .

The lower bound in Theorem 2 is expressed as

31

Proof of Corollary 1

Under these conditions, Eq. (20) could be expressed as

32

This could be further transformed to

33

which is

34

Finally, we get

35

Corollary 1 gives a simplified situation that the accuracy should satisfy in Theorem 2, it provides a unified hurdle rate for the degenerated .

It is worth noting that

36

is a decreasing function of , and the summation of s should be 1. This gives us two trivial cases. The first is that all s are equal. In this case, there is

37

which means that the accuracy should exceed 50%, which is a very common situation for any medical test.

The second gives more insight. As the hurdle rate is a decreasing function of , the smallest determines the value. Given that

38

it means that it is nearly impossible for a test to satisfy the require of this lower bound.

This finding aligns with the observed scarcity of effective precision medical tests for rare diseases in real-world settings. The preceding discussion shows that in the context of rare diseases, the distribution of test results becomes highly skewed, with a very small prior probability for certain class. This drives the lower bound discussed in this section up close to 1, imposing exceptionally stringent requirements on the corresponding precision medical test. Such high accuracy is not only technically challenging to achieve but also imposes substantial cost burdens. As will be shown in subsequent sections, higher testing costs in turn create even greater demands on test accuracy. The theorems and corollaries presented in this section thus provide a quantitative theoretical foundation for this phenomenon.

Finally, once this lower bound is satisfied, we can calculate the expected NMB for the case that faces the Local Cost-effective Axiom with the medical test, which is

39

Cost-effectiveness analysis for the medical test

Based on the theorems above, we have derived the optimal expected NMB with and without medical testing. Based on this, we can analyze the cost effectiveness of precision medical testing by comparing the difference in the expected optimal NMB, to make it fit into the requirement of Global Cost-effective Axiom. Theorem 3 below gives the necessary and sufficient conditions for a medical test to satisfy Global Cost-effective Axiom.

Theorem 3

Lower Bound for Overall Effectiveness The sufficient and necessary condition for the test, , to satisfy Global Cost-effective Axiom, which is, could be expressed as

40

where is the optimal intervention without the test, see Theorem 1, and is the expectation operator with distribution .

Proof of Theorem 3

Using Eq. (39) and Theorem 1, there is

41

This means that is equivalent to

42

which is

43

and finished the proof.

Theorem 3 provides the conditions necessary and sufficient for a medical test to be cost-effective in the form of a lower bound requirement for a weighted summation of the accuracy. This means that the optimal expected NMB with testing will exceed that without testing if and only if the medical test’s accuracy meets the criteria in Theorem 3.

Remark 8

Similar to the previous theorem, the algebraic complexity of Theorem 3 does not obscure its clear economical essence. In essence, when no medical test is performed, the globally optimal strategy selects an intervention that yields acceptable outcomes for the majority of patient classes. That is, while may not be more cost-effective than the optimal intervention for each specific class, it remains superior to most other alternatives. Consequently, in the inequality that characterises the condition in Theorem 3, the left-hand side represents the net monetary benefit gain from switching from to when the test result is correct. The right-hand side captures the loss incurred from switching from to other interventions when the test result is incorrect, along with the test cost . The inequality essentially requires that the gain from correct classification must outweigh the sum of the losses from misclassification and the cost of the test. This aligns closely with standard economic intuition.

The following discussion uses corollaries under symmetric or degenerate scenarios to further illustrate the intuition behind Theorem 3.

Corollary 2

Given the following symmetric or degenerate conditions,

• Precision rate for any class is equal and errors are equally distributed, which is

  44
• Degenerate the NMB outcome into 2 state, correct and incorrect intervention, which is

  45

  where .

The lower bound in Theorem 3 is expressed as

46

where .

Proof of Corollary 2

Under these conditions, Eq. (41) could be expressed as

47

which finished the proof by .

Corollary 2 helps to understand Theorem 3 much easier. It shows that three components decide this kind of lower bound of accuracy. First, the cost of the test. The more expensive the test is, the higher the bound is. Second, the NMB difference between “correct” and “incorrect” intervention. The larger the difference is, the lower the bound is. Third, the benefited group by choosing , which is in this case. The better performs, the higher the bound is. All these are trivial in this simple case, but tells and verifies the rules that Theorem 3 discovers.

In summary, the following theorem gives the overall condition for a precise medical test to be cost effective, which is the combination of Theorem 2 and Theorem 3.

Theorem 4

The sufficient and necessary condition for the test, , to satisfy both Local Cost-effective Axiom and Global Cost-effective Axiom is given as

• Lower Bound for Matching Intervention:

  48
• Lower Bound for Overall Effectiveness:

  49

  where is the optimal intervention without the test, see Theorem 1, and is the expectation operator with distribution .

Proof of Theorem 4

Just combine Theorem 2 and Theorem 3.

The case of sensitivity and specificity

In this section, we discuss the special case that is more common in general medical test studies. In this case, the class number is set to , which means that only two kinds of patient are in the target population: one is called the “True Positive” class, marked as , and the other is called the “True Negative”, marked as . This is the simplest case in clinical decision making, which often occurs when deciding whether a patient has a specific disease or is at high risk.

We consider a medical test with just two types of results: one gives a positive result, called the “Reported Positive (j = 1)” and the other gives a negative result, called the “Reported Negative (j = 0)”. In this case, the accuracy vector is measured by two variables, one is named sensitivity, , which means that . The other is named as specificity, , which is defined as means that .

It is very clear that sensitivity is the proportion of true positive tests out of all patients with a condition, and specificity is the percentage of true negatives out of all subjects who do not have a disease or condition. An accurate medical test requires both the sensitivity and specificity to be high, which is often conflicted.

Moreover, we assume that the treatment for class , the true positive class, there exist one conditionally cost-effective intervention, which will bring the true positive class additional NMB. However, if the intervention is given to the true negative class, it will bring a additional NMB. We can always assume , because brings’do nothing’ cases and could be fixed by trivial transform.

Note that in this binary case, the number of interventions may be larger than 2, but we focus on the two interventions (treatment and no treatment) that are relevant for the analysis. The general definitions of and apply.

The following corollary gives the lower bound for matching interventions for the case .

Corollary 3

The lower bound for matching interventions for the case is

50

Proof of Corollary 3

Bring in all the settings into Theorem 2.

Corollary 3 gives the lower bound that tries to ensures that the true positive class should receive treatment and the true negative ones should not receive treatment. It is more clear if we assume , then Corollary 3 becomes , a very trivial special case. Moreover with , there is . This means that the accuracy of the medical test should be higher than the randomly guessed.

For the test overall to be cost-effective, we have the following corollary from Theorem 3.

Corollary 4

The lower bound for the test to be cost effective for the case is

51

Proof of Corollary 4

Noting that

52

Then we use Theorem 3 to complete this proof.

The Corollary 4 gives the benchmark that accuracy must satisfies. Combined with Corollary 3 and Corollary 4, we can obtain the general lower bound of the accuracy of precision medical tests when , as is shown in Theorem 5.

Theorem 5

The lower bound of the accuracy of medical test to be both cost-effectiveness and intervention matching when is given as

53

Proof of Theorem 5

Noting that the condition in Corollary 4 is more compact than the ones in Corollary 3, the proof can be shown.

Theorem 5 shows that the overall lower bound is determined by the relationship among three components, and . It is worth noting that represents the total net monetary benefit gain from moving the true positive class from no intervention to receiving the intervention, whereas represents the total net monetary benefit gain from moving the true negative class from receiving the intervention to no intervention. Theorem 5 implies that for a medical test to be cost-effective, a weighted sum of its accuracy must exceed the larger of these two increments plus an additional testing cost. It is also clear that the larger , the more expensive the test is and the more difficult it is to be cost-effective. We use Figure 1 to give a more intuitive sense.

Fig. 1.

Lower bound with different and

In Fig. 1, we use and for simple. Figure 1 provides the schematic showing the relationship between the overall lower bound with and .

In Fig. 1, each subfigure presents a plane spanned by and , where distinct coloured regions indicate the parameter combinations under which the test satisfies the cost-effectiveness criterion for different values of , with cooler tones corresponding to higher . Across the subfigures, the value of increases progressively from left to right.

As can be observed from Fig. 1, the region in which the test meets the cost-effectiveness requirement is located in the upper right corner of the plane spanned by and , with its lower bound characterised by Theorem 5. Moreover, as increases, the area of this region shrinks and approaches the upper right corner, corroborating the assertion that a higher test price necessitates greater accuracy. A cross-sectional comparison across the subfigures reveals that the boundary becomes steeper as increases, implying that a larger imposes a more stringent requirement on . In other words, when the positive effect of the corresponding treatment is more pronounced relative to its adverse effects, the importance of for the test becomes greater.

Figure 1 also provides another key insight. For a typical test, and form a trade-off curve that encloses a convex region in the lower left corner. This curve can be tangent to the boundary of the region identified in Fig. 1, corresponding to a specific value of that represents the maximum price at which a test with a given receiver operating characteristic curve can be priced. This observation warrants further investigation in future research.

A numerical simulation example

In this section, we employ numerical simulation to illustrate and confirm the previously mentioned findings. First, we will use one unique fixed case to show the relationship between the lower bounds and the cost-effectiveness of the precise medical test. Second, more sampled simulations are made to verify the correctness of the theorems.

Sample case

Generally, we define a set of patient classes with corresponding medical interventions, assign costs and QALY outcomes, and calculate the expected NMBs for both scenario with and without the medical test with different accuracy parameters. This enables us to derive a precise correlation between the accuracy of a medical test and its CEA outcomes.

To balance the variety of patient classes alongside comprehensible simulation results, we use a simple symmetric setting in the unique sample case. We take the number of patient classes , and an optional intervention pool with medical interventions. Each patient with the most suitable intervention achieves an expected NMB output of and for the rest. Furthermore, we assume Table 1 to demonstrate the composition of the patients in the classes and the respective cost level of the most fitting intervention for each class.

Table 1.

The probability of patients in corresponding classes

Patient Class	1	2	3	4
	0.10	0.20	0.30	0.40

For the medical test, in this sample case we use the simplified assumption, let all and all . All values of from to would be tested, and simulations will show the NMB performance of all classes and give evidence to understand the bounds and accuracies. The results are shown in Fig. 2.

Fig. 2.

Curves of accuracy vs. Expected NMB under different with corresponding cost-effectiveness decisions

Figure 2 presents the results of the simulation designed to illustrate the relationship between test accuracy and cost-effectiveness. The figure consists of five subfigures. The four left panels correspond to each test result class , plotting the conditional expected net monetary benefit for each possible intervention as a function of test accuracy , where the solid curve represents the intervention matched to the result class and the dashed curves represent the non-matched interventions. In each of these panels, a vertical dashed line indicates the local lower bound derived from Theorem 2, marking the minimum accuracy above which the matched intervention becomes the optimal choice for that particular result class. The rightmost panel displays the total expected net monetary benefit for the entire population when the test is used, plotted against , alongside horizontal dashed lines representing the optimal net monetary benefit without testing plus the test cost for three different values of . The intersection of the total expected net monetary benefit curve with each horizontal reference line determines the global lower bound of accuracy required for the test to be cost-effective, as characterised by Theorem 3, and these bounds are marked with vertical dashed lines in corresponding colours. The simulation demonstrates that as increases, the total expected net monetary benefit rises monotonically, and the test becomes cost-effective only when exceeds a threshold that increases with , consistent with the theoretical predictions. Furthermore, the local bounds for each result class vary with class probability, reflecting the differential impact of misclassification costs across subgroups.

Figure 2 provides a comprehensive visual summary of the simulation results and substantiates three key findings. First, the local lower bounds depicted in the left four panels and the global lower bounds in the rightmost panel align precisely with the theoretical thresholds derived in Theorem 2 and Theorem 3, respectively, confirming that the analytical expressions correctly characterise the accuracy requirements for satisfying each axiom. Second, for accuracy levels exceeding these bounds, the simulation confirms that both the local and global cost-effectiveness conditions are satisfied: in each diagnostic result class, the matched intervention yields the highest conditional expected net monetary benefit once the local bound is surpassed, and the total expected net monetary benefit with testing exceeds the optimal no–test benchmark once the global bound is met. Third, the interplay between the two axioms reveals that satisfying the global cost-effectiveness condition does not automatically guarantee that the local condition holds; even for a very inexpensive test with , the test must still respect the local lower bounds for each result class to ensure that the intervention assignment remains optimal at the subgroup level. This highlights the importance of jointly considering both criteria when evaluating the cost-effectiveness of precision medical tests.

Multiple case simulation

To support more general conclusions, it is necessary to relax these symmetry and simplifying assumptions. However, this also renders the corresponding presentation and visualisation of the results considerably more complex. For the subsequent multiple case simulation, we maintain while allowing , , , , , and to be generated randomly, with the sole objective of verifying whether the two lower bound theorems hold under general conditions.

Figure 3 presents the results of a comprehensive simulation designed to verify the theoretical lower bounds under general conditions beyond the symmetric setting. In this simulation, all model parameters are randomly generated for each of 200 independent cases. The number of patient classes and interventions is set to . Intervention costs are drawn from a uniform distribution on . Health outcomes measured in quality-adjusted life years are drawn uniformly from . Class proportions are generated from a uniform distribution on and then normalized to sum to one. The test cost is sampled uniformly from . For each class , the test accuracy is drawn uniformly from and then clipped to to avoid numerical extremes. The misclassification probabilities for are generated by first drawing positive values uniformly and then scaling them so that their sum equals , ensuring a valid confusion matrix.

Fig. 3.

Multiple case simulation performance

For each randomly generated scenario, the local lower bound for each diagnostic result class is computed according to Theorem 2, and the global lower bound condition is evaluated using the inequality derived in Theorem 3. The simulation then computes two key performance metrics for each class: the difference between the expected net monetary benefit when assigning the matched intervention and the best non–matched alternative, plotted against the difference between the actual accuracy and its theoretical local lower bound. For the global analysis, the difference between the total expected net monetary benefit with testing and the optimal no–test benchmark is plotted against the left–hand side minus right–hand side of the inequality in Theorem 3, where a positive value indicates that the test satisfies the global cost–effectiveness condition.

The four left panels correspond to the four diagnostic result classes, each showing individual simulation runs as points. Points with above the local lower bound (i.e., ) are coloured blue, while those below the bound are coloured red. A vertical dashed line at zero separates the two regimes. In every panel, all blue points lie strictly above the horizontal axis, meaning that whenever the local accuracy condition is satisfied, the matched intervention indeed yields a higher conditional expected net monetary benefit than any other intervention. Conversely, red points are predominantly located below the horizontal axis, indicating that when the local bound is violated, the matched intervention is no longer optimal. The rightmost panel displays the global performance in a similar manner. Here the horizontal axis represents the left–hand side minus right–hand side of the global inequality from Theorem 3; a positive value on this axis implies that the theoretical condition for global cost–effectiveness is met. The vertical axis shows the actual difference between the total expected net monetary benefit with testing and the optimal no–test benchmark. Points with a positive horizontal coordinate (blue) are all located above the horizontal axis, confirming that whenever the global condition holds, the test is indeed cost–effective. Red points with a negative horizontal coordinate are below the axis, indicating that when the condition fails, the test does not achieve global cost–effectiveness.

These simulation results provide strong empirical support for the theoretical bounds. The local performance panels demonstrate that Theorem 2 correctly identifies the threshold above which the matched intervention becomes the optimal choice for each result class, regardless of the randomness in parameters. The global performance panel similarly validates Theorem 3, showing that the inequality derived therein accurately separates cost–effective from non–cost–effective tests across a wide range of randomly generated scenarios. The simulation thus confirms that the two lower bounds are not only mathematically exact under the model assumptions but also robust to arbitrary heterogeneity in patient populations, test characteristics, and cost structures.

In summary, the numerical simulations presented in this section serve to validate the theoretical results derived in the preceding sections. The sample case, under symmetric assumptions, illustrates the direct relationship between test accuracy and cost–effectiveness, demonstrating how the local lower bounds for each diagnostic result class and the global lower bound for the overall population emerge naturally from the model and align with the predictions of Theorem 2 and Theorem 3. The multiple case simulation extends this validation to a more general setting by randomly generating all model parameters, including class probabilities , intervention costs , health outcomes , test accuracy , misclassification probabilities , and test cost . Across 200 independent scenarios, the simulation confirms that the matched intervention becomes optimal precisely when exceeds its theoretical local bound, and that the test achieves global cost–effectiveness exactly when the inequality in Theorem 3 holds. These findings collectively establish that the two lower bounds are not only mathematically exact under the model assumptions but also robust to arbitrary heterogeneity in the underlying parameters, providing a solid quantitative foundation for evaluating the cost–effectiveness of precision medical tests.

Conclusion and discussion

This study establishes a quantitative framework to propose the theoretical relationship between the accuracy of precision medical tests and their cost-effectiveness within a decision-analytic model. By deriving analytical lower bounds on test accuracy, we identify two critical thresholds that must be exceeded for a testing strategy to be economically viable. The first threshold, defined in Theorem 2, represents the minimum accuracy required to satisfy the Local Cost-effective Axiom, ensuring that the intervention matched to a specific test result remains the optimal choice for that subgroup despite misclassification risks. The second threshold, derived in Theorem 3, corresponds to the Global Cost-effective Axiom, requiring that the net monetary benefit of the entire population under the testing strategy exceeds the optimal outcome achievable without testing. The numerical simulations confirm that these bounds accurately demarcate the transition points where testing shifts from being economically detrimental to beneficial under the specified model conditions.

The theoretical results reveal that the accuracy requirements depend on a range of factors beyond the test’s own performance. The prevalence of each patient class, the cost and effectiveness of the subclass interventions, and the cost of the test all play critical roles. In particular, the Lower Bound for Matching Intervention shows that even if a test is highly accurate overall, it may fail to be locally optimal for a specific diagnostic result class if the class proportion is small or if the harm from misclassifying patients into that class is large. In fact, the introduction of the Local Cost-effective Axiom, leading the Lower Bound for Matching Intervention, constitutes an important complement to conventional global cost-effectiveness analysis of medical testing. As discussed in Remark 7, the fully posterior-optimal Bayesian decision rule may yield a globally optimal intervention, but when test accuracy is low, this global optimum can be locally suboptimal and, in extreme cases, may even lead to contradictory conclusions. The Local Cost-effective Axiom does not challenge the fully posterior-optimal Bayesian decision rule. Rather, it adds an extra safeguard on top of this Bayesian rule by ensuring a certain consistency between the global optimum and all theoretically locally optimal interventions. This guarantees that the intervention recommended by the test will not give rise to ethical or clinical implementation challenges.

This provides a potential theoretical explanation for why precision medicine can be particularly challenging to implement for rare diseases or small patient subgroups. When the prior probability of a class is very low, the required accuracy approaches , implying that only near perfect tests can justify assigning the matched intervention to that class. This insight aligns with the observed scarcity of cost-effective diagnostic tools for rare conditions and highlights a fundamental economic barrier to the development of such tests.

The Lower Bound for Overall Effectiveness further reinforces the importance of test cost and the performance of the no test strategy. When the optimal intervention without testing already yields high net monetary benefit for a large portion of the population, the threshold accuracy for the test to be cost-effective becomes more stringent. Conversely, when the no test strategy performs poorly because the population is heterogeneous and no single intervention is suitable for the majority, even a moderately accurate test can provide substantial value. This trade off is also captured in the corollary for the symmetric binary case, where the required accuracy is expressed as , showing that the bound increases with test cost and with the proportion of patients already benefiting from the no test strategy, while decreasing with the incremental net monetary benefit of a correct match.

The findings suggest several considerations for health technology assessment and clinical practice on medical tests. For decision makers, the framework offers a structured way to set accuracy thresholds for test adoption as a starting point for discussion, based on the clinical context rather than relying on arbitrary rules of thumb. For test developers, the analysis provides a tool to evaluate the feasibility of a test given its expected accuracy and cost, and to understand how changes in the test’s performance characteristics affect its potential value. For rare diseases, the results underscore the difficulty of achieving cost-effectiveness under current pricing models, suggesting that alternative mechanisms such as differential pricing or bundled payment arrangements may be necessary to create sustainable incentives for test development. Although constrained by the limitation of the stylized framework, these findings could be interpreted as theoretical insights, rather than direct policy recommendations.

It is fundamental to recognize that the expected net monetary benefit framework employed in this analysis is theoretically equivalent to the standard incremental cost-effectiveness ratio approach used in health technology assessment. Both methodologies rely on the same underlying axioms of rational choice under uncertainty and will yield identical adoption decisions when applied consistently to the same data. Within this unified theoretical landscape, our modeling strategy for a single diagnostic event can be conceptualized as a flattened representation of the standard decision-tree structure. By algebraically expanding every possible downstream outcome branch of a conventional decision tree into a single generalized expectation formula, we preserve the full logical integrity of the diagnostic pathway while enabling the derivation of explicit analytical bounds. This structural equivalence ensures that our results are not an alternative to standard decision modeling but rather a rigorous algebraic compression of the same logic tailored for theoretical insight.

While standard practices often augment the net monetary benefit framework with probabilistic sensitivity analysis and cost-effectiveness acceptability curves to account for parameter uncertainty, such approaches are mathematically equivalent to introducing risk-adjusted penalty terms based on specific risk measures into the objective function. Incorporating these risk measures would fundamentally alter the optimization landscape by shifting the focus from expected value maximization to risk-aware decision making. Although this extension is highly relevant for practical application, it introduces a layer of complexity regarding the choice of risk metrics and confidence levels that falls outside the primary scope of establishing baseline accuracy thresholds. Therefore, the present analysis focuses strictly on the expected value case to define the fundamental limits of cost-effectiveness, leaving the integration of risk-adjusted criteria and the construction of acceptability curves for future research dedicated to robustness under deep uncertainty.

Furthermore, the scope of this paper is deliberately restricted to the evaluation of a single diagnostic test leading to a one-time treatment allocation. Clinical pathways involving sequential testing, iterative monitoring, or dynamic treatment selection where subsequent decisions depend on prior results constitute a distinct class of problems requiring dynamic optimization theory. Extending the static bounds derived here to such multi-stage settings would necessitate the formulation of Bellman equations or Markov decision processes to capture the value of information over time. While the single-test thresholds presented in this work provide the necessary foundational conditions for any individual step within a larger sequence, the optimal policy for a dynamic pathway cannot be determined by simply applying these static bounds in isolation. Addressing the complexities of sequential decision making and the interaction between multiple tests represents a significant avenue for future investigation that warrants a separate and dedicated theoretical treatment.

Several other limitations should be acknowledged. The model assumes that patients can be partitioned into a finite number of discrete classes and that health outcomes and costs are deterministic within each class. This is because that the scope of our analysis focuses primarily on class-specific accuracy and does not formally incorporate these broader test-performance concepts in a substantial way into the model framework. In practice, continuous measures of disease severity or treatment response are common, and extending the analysis to consider positive and negative prediction values would be a valuable direction for future work. The analysis also assumes that the test results are conditionally independent of the outcomes given the true class, which is a standard assumption in diagnostic accuracy studies but may not capture all complexities of real world testing. Additionally, the framework does not consider the possibility of sequential testing or dynamic decision making where multiple tests are used over time. Incorporating these features would make the model more applicable to complex clinical pathways. Finally, the analysis focuses on the net monetary benefit criterion, which represents a societal perspective under a fixed willingness to pay threshold. Alternative perspectives, such as those of payers or patients, may require different formulations.

In conclusion, this paper provides a theoretical foundation for understanding the accuracy requirements of precision medical tests through the derivation of local and global cost-effectiveness bounds. By explicitly defining the conditions under which testing adds economic value, we offer a structured lens for evaluating the viability of new diagnostic technologies. The results underscore that test accuracy cannot be evaluated in isolation but must be considered in the context of disease prevalence, the magnitude of treatment benefits, and the cost of misclassification. While the stylized nature of the model necessitates caution in translating these bounds directly into policy thresholds, the underlying economic logic remains robust and consistent with established decision-analytic principles. Health technology assessment agencies and test developers may consider using these insights as preliminary screening and supportive tools. Future work should focus on generalizing these results to asymmetric error structures, integrating risk-adjusted performance measures, and extending the framework to sequential decision environments to enhance its utility in real-world health economic planning.

Author contributions

Ermo Chen: Conceptualization, Formal analysis, Methodology, Writing – original draft, Writing – review & editing. Xinyue Zeng: Conceptualization, Formal analysis, Methodology, Writing – original draft, Writing – review & editing. Huajie Jin: Validation, Writing – review & editing, Supervision. All authors have read and approved the final manuscript.

Funding

No funding has been received for the conduct of this study.

Data availability

No datasets were generated or analysed during the current study.

Code availability

The code in this study is available on request from the corresponding author.

Declarations

Ethical approval

Not applicable.

Consent for publication

Not applicable.

Consent for participate

Not applicable.

Competing interests

The authors declare no competing interests.