贝叶斯更新与序贯检验：克服筛选检验的推理局限性
Bayesian Updating and Sequential Testing: Overcoming Inferential Limitations of Screening Tests
摘要：
贝叶斯定理赋予作为疾病流行率函数的筛查测试的准确性固有的局限性。我们在以前的工作中已经表明，检测系统可以容忍患病率的显著下降，直到某个被称为$prevalence$ $threshold$的明确定义的点，低于该点，阳性筛查试验的可靠性急剧下降。在此，我们建立了一个数学模型，以确定序贯测试是否克服了上述贝叶斯限制，从而提高了筛选测试的可靠性。我们表明，对于接近$k$的$ρ$预期阳性预测值，$n_i$所需的阳性测试迭代次数为：$ n_i =\lim_{ρ\to k}\left\lceil\frac{ln\left[\frac{ρ(ϕ-1)}{ϕ(ρ-1)}\right]}{ln\left[\frac{a}{1-b}\right]}\right\rceil$其中$n_i$ =达到预期阳性预测值$ρ$所需的测试迭代次数，a =灵敏度，b =特异性，$ϕ$ =疾病患病率，$k$ =常数。基于上述推导，我们提供了获得$ρ(ϕ)$为50、75、95和99 $\%$所需的测试迭代次数的参考表，作为各种灵敏度、特异性和疾病患病率水平的函数。
英文摘要：
Bayes' Theorem confers inherent limitations on the accuracy of screening tests as a function of disease prevalence. We have shown in previous work that a testing system can tolerate significant drops in prevalence, up until a certain well-defined point known as the $prevalence$ $threshold$, below which the reliability of a positive screening test drops precipitously. Herein, we establish a mathematical model to determine whether sequential testing overcomes the aforementioned Bayesian limitations and thus improves the reliability of screening tests. We show that for a desired positive predictive value of $ρ$ that approaches $k$, the number of positive test iterations $n_i$ needed is: $ n_i =\lim_{ρ\to k}\left\lceil\frac{ln\left[\frac{ρ(ϕ-1)}{ϕ(ρ-1)}\right]}{ln\left[\frac{a}{1-b}\right]}\right\rceil$ where $n_i$ = number of testing iterations necessary to achieve $ρ$, the desired positive predictive value, a = sensitivity, b = specificity, $ϕ$ = disease prevalence and $k$ = constant. Based on the aforementioned derivation, we provide reference tables for the number of test iterations needed to obtain a $ρ(ϕ)$ of 50, 75, 95 and 99$\%$ as a function of various levels of sensitivity, specificity and disease prevalence.
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