How Accurate Should a Medical Test be to Meet Cost-Effectiveness

1. Introduction

Cost-effectiveness analysis (CEA) is a commonly used tool to evaluate interventions and medications in health economics, providing valuable information through health technology assessments (HTA). Gradually, researchers have discovered its potential application in diagnostic and medical tests. In recent years, CEA has been deployed in numerous HTAs for diagnosis and medical tests, including studies such as Thomas et al. [1], Jacobsen et al. [2], and Choi et al. [3]. However, some studies (Fang et al. [4], Yang et al. [5], and Snowsill [6]) indicate that the application of CEA to the diagnosis and testing of HTAs is becoming more popular, but various crucial factors such as optional interventions, additional information, and test accuracy have not been adequately considered.

In this context, precision medical tests have emerged as advanced diagnostic tools that use techniques such as genomic sequencing, proteomics, and biomarker analysis to accurately identify individual biological characteristics, guiding personalized treatment decisions. These tests are crucial for implementing targeted therapies, improving the accuracy of clinical decision making, and optimizing the allocation of healthcare resources. However, Ling et al. [7] and Kasztura et al. [8] note that, considering the high cost and uncertainty, only a fraction of these new technologies are likely to be cost-effective and adopted by healthcare systems, raising concerns about the efficient use of research funding and healthcare resources.

Designs for CEA analysis of diagnosis and testing require specific considerations given their distinct characteristics. Mushlin et al. [9] emphasizes the importance of focusing on factors such as the clinical pathway, the baseline probability of disease, and intermediate outcome measures. Koffijberg et al. [10] further points out that when evaluating the cost-effectiveness of new diagnostic test technologies, it is also necessary to consider the characteristics, costs, and effects of the latest diagnostic tests. In addition to the above, Smith et al. [11] emphasizes the importance of incorporating valuable new evidence and iterating the analysis methodology in the cost-effectiveness analysis process of diagnostic tests.

In precision medical tests, patient identification and targeted interventions are key, which requires careful consideration of the accuracy of the diagnostic tests used for patient classification (Gavan et al. [12], Mattos-Arruda and Siravegna [13], Chen et al. [14]). This underscores the importance of the accuracy of the test in CEA, a consensus that has existed for a long time in the academic community. More than two decades ago, Hernandez [15] emphasized the importance of test accuracy in CEA. Subsequent systematic reviews by Van den Bruel et al. [16] further demonstrated the need to analyze test and diagnostic accuracy when reporting CEA results. Seo and Strong [17] presents a practical model guide for CEA of diagnostic and testing techniques, emphasizing the importance of accuracy in the analysis process. Similarly, Snowsill [6] emphasizes the need to fully consider the impact of test accuracy on results during model construction. The sensitivity of the results to the accuracy parameters is also reflected in some disease-specific CEA studies, including Novielli et al. [18], Novielli et al. [19], Garg et al. [20], Biltaji et al. [21], and other studies. However, most existing research focuses on specific applications or qualitative discussions, lacking a quantitative framework to systematically analyze the relationship between test accuracy and cost-effectiveness under general assumptions.

To bridge this gap, our study establishes a generalized model to quantify the accuracy requirements of precision medical tests to achieve cost-effectiveness.

Fundamentally, we derive a theoretical lower bound that combined by two bounds that the accuracy of precision medical tests must exceed: 1) Lower Bound for Matching Intervention, which is the minimum accuracy required to ensure that appropriate treatment matching is provided to each patient category based on the test results; 2) Lower Bound for Overall Effectiveness, which is the minimum accuracy required to make the test cost-effective for the entire patient population. The derivation of these two lower bounds offers theoretical guidance for developing, pricing, and procuring diagnostic tests. In addition, we develop the special case of sensitivity and specificity, that is, the case in which only binary test results are considered. This situation is common in clinical practice, such as genetic screening and the testing of infectious diseases. By analyzing this particular case, we obtain more specific accuracy thresholds, which are significant for evaluating the cost-effectiveness of standard medical screening tests.

Our results not only reveal unfair medical decision making that can be caused by insufficient test accuracy, but also provide a new economic perspective to explain the underdiagnosis of rare diseases and the challenges of implementing precision medicine in small groups of patients. Through numerical simulations, we further verify the theoretical results and explore the impact of test accuracy on expected net monetary benefits and the proportion of patients receiving appropriate interventions. In summary, our finding helps to provide a theoretical criteria tool for determining if a precision medical test is cost effective.

2. Model Settings

To illustrate the relationship between test accuracy and CEA results, we utilize a simplified model. We assume that patients can be divided into finite classes, each of which is suited to a particular medical intervention, and there are only two possible health outcomes: receiving the appropriate intervention or not. 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.

2.1. General Settings

Consider a patient population in which each patient has unique intrinsic and unobservable characteristics that can be categorized into N classes. Each class of patients has a unique medical intervention that is the most appropriate. It is assumed that the most suitable medical intervention can lead to the gain of $Q^{+}$ quality adjusted life years (QALY) for a patient belonging to the respective class, while an unsuitable one can only allow the acquisition of $Q^{-}$ QALY. Specifically, the QALY returns obtained from intervention j for patients in class i, $q(i,j)$, can be expressed as

$$\forall i,j\in \{1,2,...,N\},q(i,j)=\left\{\begin{array}{cc}{Q}^{+},\hfill & \hfill j=i,\\ Q^{-},\hfill & \hfill j\ne i.\end{array}\right.$$

(1)

That is, patients belonging to the class i gain $Q^{+}$ QALY when they undergo their designated intervention i, while the use of interventions not designed for their class can only result in $Q^{-}$ QALY, with $Q^{+}>Q^{-}$. To clarify, in clinical practice, $Q^{+}$ and $Q^{-}$ could be positive or negative, according to the outcome of different interventions.

The cost of an intervention is denoted as $c\left(j\right)$, where j refers to the intervention. The NMB approach combines the QALY and the cost of an intervention with a monetary constant $\lambda$. That is, the NMB of using intervention j for patients in class i, $S^i\left(j\right)$, can be expressed as

$$S^i\left(j\right)=\lambda q(i,j)-c\left(j\right),\forall i,j\in \{1,2,...,N\}.$$

(2)

The distribution structure of the intrinsic characteristics of patients in the entire population can be observed through retrospective statistics over time. Let $p\left(i\right)$ be a discrete distribution that represents the probability of patients being assigned to a specific class i, where i is one of the classes. The primary objective of medical intervention decisions is to maximize the expected NMB of patients for the entire population.

2.2. The Medical Test and the Accuracy

Suppose a medical test is available to identify intrinsic characteristics of patients that incur a cost of $c_0$, as previously mentioned. However, the medical test is not always accurate, with a level of accuracy denoted as $\alpha$. In other words, the probability that the medical test correctly identifies the patients in class i is $\alpha$. For simplicity, we assume that errors are equally likely to be assigned to each of the other classes. Denote the random outcome of the medical test as $T\left(·\right)$. Then, there exists

$$\forall i,j\in \{1,2,...,N\},\mathbb{P}(T\left(i\right)=j)=\left\{\begin{array}{cc}\alpha ,\hfill & \hfill j=i,\\ {\displaystyle \frac{1-\alpha }{N-1}},\hfill & \hfill j\ne i.\end{array}\right.$$

(3)

In particular, a patient’s own intrinsic type i cannot be directly observed. The only observable result is the patient’s medical test, which produces a result of $T\left(i\right)$. Following the test, any intervention decision can only be made based on the outcome $T\left(i\right)$. However, the result $T\left(i\right)$ is a random variable. This is the primary factor that contributes to the effect of medical test accuracy on the evaluation results of CEA.

Let $\mathit{J}=\{j_1,j_2,...,j_N\}$ be an intervention strategy based on the medical test, where $j_i$ denotes the intervention assigned to patients with a test result of i. To be clear, we notice that $\mathit{J}$ is actually a mapping rule, mapping the patient category i to intervention $j_i$. It is important to note that the class of patients with a test result of i may not necessarily correspond to patients of intrinsic type i. If the test is performed and the intervention strategy $\mathit{J}$ is chosen, the expected NMB per unit for all patients is indicated as $S_T\left(\mathit{J}\right)$. If no medical test is performed, a unique intervention will be assigned to all patients. The expected NMB per unit, in this case, is denoted as $S_{NT}\left(\mathit{J}\right)$.

2.3. Expected NMB with the Medical Test

This subsection focuses on the expressions of $S_{NT}\left(\mathit{J}\right)$ and $S_T\left(\mathit{J}\right)$. Based on the definition, one can easily show that the expected NMB per unit, $S_T\left(\mathit{J}\right)$, could be expressed as

$$S_T\left(\mathit{J}\right)=\mathbb{E}\left[\lambda \mathbb{E}\left[q(i,j_{T\left(i\right)})\right|i]-\mathbb{E}\left[c\left(j_{T\left(i\right)}\right)\right|i]\right]-c_0,$$

(4)

where $E\left[·\right|i]$ denotes the conditional expectation with respect to class i. Although complex, we can decompose the expression using $T\left(·\right)$ in Lamma 1.

Lamma 1.

$S_T\left(\mathit{J}\right)$ has a decomposition with the result of $T\left(·\right)$, which is

$$S_T\left(\mathit{J}\right)={\displaystyle \sum _{i=1}^N}{S}_T^i\left(j_i\right),$$

(5)

where $S_T^i\left(j_i\right)$ is part of NMB that the class of patients with result $T\left(·\right)=i$ contributes and could be expressed as

$$\left\{\begin{array}{cc}\lambda \left(\alpha p\left(i\right)Q^{+}+(1-p\left(i\right)){\displaystyle \frac{1-\alpha }{N-1}}{Q}^{-}\right)-(c\left(i\right)+c_0)\left(\alpha p\left(i\right)+(1-p\left(i\right)){\displaystyle \frac{1-\alpha }{N-1}}\right),\hfill & \hfill j_i=i,\\ \lambda \left(p\left(i_0\right){\displaystyle \frac{1-\alpha }{N-1}}{Q}^{+}+\alpha p\left(i\right)Q^{-}+(1-p\left(i_0\right)-p\left(i\right)){\displaystyle \frac{1-\alpha }{N-1}}{Q}^{-}\right)-(c\left(i_0\right)+c_0)\left(\alpha p\left(i\right)+(1-p\left(i\right)){\displaystyle \frac{1-\alpha }{N-1}}\right),\hfill & \hfill j_i=i_0\ne i.\end{array}\right.$$

(6)

Proof. 

potL1Proof of Lamma 1

Initially, we can reform the expression for $S_T\left(\mathit{J}\right)$ by implementing the total probability formula, subsequently sorted according to the result of $T\left(·\right)$. This approach enables us to derive:

$$\begin{array}{cc}\hfill S_T\left(\mathit{J}\right)=& {\displaystyle \sum _{i=1}^N}p\left(i\right)\left(\lambda \mathbb{E}\left[q(i,j_{T\left(i\right)})\right|i]-\mathbb{E}\left[c\left(j_{T\left(i\right)}\right)\right|i]\right)-c_0\hfill \\ \hfill =& {\displaystyle \sum _{i=1}^N}p\left(i\right)\left({\displaystyle \sum _{k=1}^N}\mathbb{P}(T\left(i\right)=k)\left(\lambda q(i,j_k)-c\left(j_k\right)\right)-c_0\right)\hfill \\ \hfill =& {\displaystyle \sum _{k=1}^N}{\displaystyle \sum _{i=1}^N}p\left(i\right)\mathbb{P}(T\left(i\right)=k)\left(\lambda q(i,j_k)-c\left(j_k\right)-c_0\right)={\displaystyle \sum _{k=1}^N}{S}_T^k\left(j_k\right),\hfill \end{array}$$

(7)

where

$$S_T^i\left(j_i\right)={\displaystyle \sum _{k=1}^N}p\left(k\right)\mathbb{P}(T\left(k\right)=i)\left(\lambda q(k,j_i)-c\left(j_i\right)-c_0\right).$$

(8)

Moreover,

$$\begin{array}{cc}\hfill S_T^i\left(j_i\right)=& {\displaystyle \sum _{k=1}^N}p\left(k\right)\mathbb{P}(T\left(k\right)=i)\left(\lambda q(k,j_i)-c\left(j_i\right)-c_0\right)\hfill \\ \hfill =& p\left(i\right)\mathbb{P}(T\left(i\right)=i)\left(\lambda q(i,j_i)-c\left(j_i\right)-c_0\right)+{\displaystyle \sum _{k\ne i}}p\left(k\right)\mathbb{P}(T\left(k\right)=i)\left(\lambda q(k,j_i)-c\left(j_i\right)-c_0\right)\hfill \\ \hfill =& \alpha p\left(i\right)\left(\lambda q(i,j_i)-c\left(j_i\right)-c_0\right)+{\displaystyle \sum _{k\ne i}}p\left(k\right){\displaystyle \frac{1-\alpha }{N-1}}\left(\lambda q(k,j_i)-c\left(j_i\right)-c_0\right).\hfill \end{array}$$

(9)

Next, if $j_i=i$,

$$\begin{array}{cc}\hfill S_T^i\left(j_i\right)=& \alpha p\left(i\right)\left(\lambda Q^{+}-c\left(i\right)-c_0\right)+{\displaystyle \sum _{k\ne i}}p\left(k\right){\displaystyle \frac{1-\alpha }{N-1}}\left(\lambda Q^{-}-c\left(i\right)-c_0\right)\hfill \\ \hfill =& \lambda \left(\alpha p\left(i\right)Q^{+}+(1-p\left(i\right)){\displaystyle \frac{1-\alpha }{N-1}}{Q}^{-}\right)-(c\left(i\right)+c_0)\left(\alpha p\left(i\right)+(1-p\left(i\right)){\displaystyle \frac{1-\alpha }{N-1}}\right),\hfill \end{array}$$

(10)

and else if $j_i=i_0\ne i$,

$$\begin{array}{cc}\hfill S_T^i\left(j_i\right)=& \alpha p\left(i\right)\left(\lambda Q^{-}-c\left(i\right)-c_0\right)+{\displaystyle \sum _{k\ne i,i_0}}p\left(k\right){\displaystyle \frac{1-\alpha }{N-1}}\left(\lambda Q^{-}-c\left(i\right)-c_0\right)+p\left(i_0\right){\displaystyle \frac{1-\alpha }{N-1}}\left(\lambda Q^{+}-c\left(i\right)-c_0\right)\hfill \\ \hfill =& \lambda \left(p\left(i_0\right){\displaystyle \frac{1-\alpha }{N-1}}{Q}^{+}+\alpha p\left(i\right)Q^{-}+(1-p\left(i_0\right)-p\left(i\right)){\displaystyle \frac{1-\alpha }{N-1}}{Q}^{-}\right)-(c\left(i_0\right)+c_0)\left(\alpha p\left(i\right)+(1-p\left(i\right)){\displaystyle \frac{1-\alpha }{N-1}}\right).\hfill \end{array}$$

(11)

Lamma 1 presents the result that a patient’s expected NMB when using the medical test, depending on the decision strategy $\mathit{J}$. On contrary, in cases where the test is not used, we can express $S_{NT}\left(\mathit{J}\right)$ as

$$S_{NT}\left(\mathit{J}\right)=S_T\left(\mathit{J}\right)+c_0,\forall \mathit{J}\in {\mathcal{J}}_{NT}.$$

(12)

In this instance, ${\mathcal{J}}_{NT}$ refers to the set of unique strategies where $j_i=k$ for all i.