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In this manuscript, we present various proposed methods estimate the prevalence of disease, a critical prerequisite for the adequate interpretation of screening tests. To address the limitations of these approaches, which revolve primarily around their a posteriori nature, we introduce a novel method to estimate the pretest probability of disease, a priori, utilizing the Logit function from the logistic regression model. This approach is a modification of McGee's heuristic, originally designed for estimating the posttest probability of disease. In a patient presenting with $n_\theta$ signs or symptoms, the minimal bound of the pretest probability, $\phi$, can be approximated by: $\phi \approx \frac{1}{5}{ln\left[\displaystyle\prod_{\theta=1}^{i}\kappa_\theta\right]}$ where $ln$ is the natural logarithm, and $\kappa_\theta$ is the likelihood ratio associated with the sign or symptom in question.

We define the information threshold as the point of maximum curvature in the prior vs. posterior Bayesian curve, both of which are described as a function of the true positive and negative rates of the classification system in question. The nature of the threshold is such that for sufficiently adequate binary classification systems, retrieving excess information beyond the threshold does not significantly alter the reliability of our classification assessment. We hereby introduce the "marital status thought experiment" to illustrate this idea and report a previously undefined mathematical relationship between the Bayesian prior and posterior, which may have significant philosophical and epistemological implications in decision theory. Where the prior probability is a scalar between 0 and 1 given by $\phi$ and the posterior is a scalar between 0 and 1 given by $\rho$, then at the information threshold, $\phi_e$: $\phi_e + \rho_e = 1$ Otherwise stated, given some degree of prior belief, we may assert its persuasiveness when sufficient quality evidence yields a posterior so that their combined sum equals 1. Retrieving further evidence beyond this point does not significantly improve the posterior probability, and may serve as a benchmark for confidence in decision-making.

The accuracy of binary classification systems is defined as the proportion of correct predictions - both positive and negative - made by a classification model or computational algorithm. A value between 0 (no accuracy) and 1 (perfect accuracy), the accuracy of a classification model is dependent on several factors, notably: the classification rule or algorithm used, the intrinsic characteristics of the tool used to do the classification, and the relative frequency of the elements being classified. Several accuracy metrics exist, each with its own advantages in different classification scenarios. In this manuscript, we show that relative to a perfect accuracy of 1, the positive prevalence threshold ($\phi_e$), a critical point of maximum curvature in the precision-prevalence curve, bounds the $F{_{\beta}}$ score between 1 and 1.8/1.5/1.2 for $\beta$ values of 0.5/1.0/2.0, respectively; the $F_1$ score between 1 and 1.5, and the Fowlkes-Mallows Index (FM) between 1 and $\sqrt{2} \approx 1.414$. We likewise describe a novel $negative$ prevalence threshold ($\phi_n$), the level of sharpest curvature for the negative predictive value-prevalence curve, such that $\phi_n$ $>$ $\phi_e$. The area between both these thresholds bounds the Matthews Correlation Coefficient (MCC) between $\sqrt{2}/2$ and $\sqrt{2}$. Conversely, the ratio of the maximum possible accuracy to that at any point below the prevalence threshold, $\phi_e$, goes to infinity with decreasing prevalence. Though applications are numerous, the ideas herein discussed may be used in computational complexity theory, artificial intelligence, and medical screening, amongst others. Where computational time is a limiting resource, attaining the prevalence threshold in binary classification systems may be sufficient to yield levels of accuracy comparable to that under maximum prevalence.

In previous work by this author, the screening paradox - the loss of predictive power of screening tests over time $t$ - was mathematically formalized using Bayesian theory. Where $J$ is Youden's statistic, $b$ is the specificity of the screening test and $\phi$ is the prevalence of disease, the ratio of positive predictive values at subsequent time $k$, $\rho(\phi_{k})$, over the original $\rho(\phi_{0})$ at $t_0$ is given by: $\zeta(\phi_{0},k) = \frac{\rho(\phi_{k})}{\rho(\phi_{0})} =\frac{\phi_k(1-b)+J\phi_0\phi_k}{\phi_0(1-b)+J\phi_0\phi_k}$ Herein, we modify the traditional Kermack-McKendrick SIR Model to include the fluctuation of the positive predictive value $\rho(\phi)$ (PPV) of a screening test over time as a function of the prevalence threshold $\phi_e$. We term this modified model the SIR-P model. Where a = sensitivity, b = specificity, $S$ = number susceptible, $I$ = number infected, $R$ = number recovered/dead, $\beta$ = infectious rate, $\gamma$ = recovery rate, and $N$ is the total number in the population, the predictive value $\rho(\phi,t)$ over time $t$ is given by: $\rho(\phi,t) = \frac{a[\frac{\beta IS}{N}-\gamma I]}{ a[\frac{\beta IS}{N}-\gamma I]+(1-b)(1-[\frac{\beta IS}{N}-\gamma I])}$ Otherwise stated: $\rho(\phi,t) = \frac{a\frac{dI}{dt}}{ a\frac{dI}{dt}+(1-b)(1-\frac{dI}{dt})}$ where $\frac{dI}{dt}$ is the fluctuation of infected individuals over time $t$., Comment: arXiv admin note: text overlap with arXiv:2011.06032

From the fundamental theorem of screening (FTS) we obtain the following mathematical relationship relaying the pre-test probability of disease $\phi$ to the positive predictive value $\rho(\phi)$ of a screening test: $\displaystyle\lim_{\varepsilon \to 2}{\displaystyle \int_{0}^{1}}{\rho(\phi)d\phi} = 1$ where $\varepsilon$ is the screening coefficient - the sum of the sensitivity ($a$) and specificity ($b$) parameters of the test in question. However, given the invariant points on the screening plane, identical values of $\varepsilon$ may yield different shapes of the screening curve since $\varepsilon$ does not respect traditional commutative properties. In order to compare the performance between two screening curves with identical $\varepsilon$ values, we derive two geometric definitions of the positive likelihood ratio (LR+), defined as the likelihood of a positive test result in patients with the disease divided by the likelihood of a positive test result in patients without the disease, which helps distinguish the performance of both screening tests. The first definition uses the angle $\beta$ created on the vertical axis by the line between the origin invariant and the prevalence threshold $\phi_e$ such that $LR+ = \frac{a}{1-b} = cot^2{(\beta)}$. The second definition projects two lines $(y_1,y_2)$ from any point on the curve to the invariant points on the plane and defines the LR+ as the ratio of its derivatives $\frac{dy_1}{dx}$ and $\frac{dy_2}{dx}$. Using the concepts of the prevalence threshold and the invariant points on the screening plane, the work herein presented provides a new geometric definition of the positive likelihood ratio (LR+) throughout the prevalence spectrum and describes a formal measure to compare the performance of two screening tests whose screening coefficients $\varepsilon$ are equal.

Bayes' Theorem imposes inevitable limitations on the accuracy of screening tests by tying the test's predictive value to the disease prevalence. The aforementioned limitation is independent of the adequacy and make-up of the test and thus implies inherent Bayesian limitations to the screening process itself. As per the WHO's $Wilson-Jungner$ criteria, one of the prerequisite steps before undertaking screening is to ensure that a treatment for the condition screened exists. However, in so doing, a paradox, henceforth termed the screening paradox, ensues. If a disease process is screened for and subsequently treated, its prevalence would drop in the population, which as per Bayes' theorem, would make the tests' predictive value drop in return. Put another way, a very powerful screening test would, by performing and succeeding at the very task it was developed to do, paradoxically reduce its ability to correctly identify individuals with the disease it screens for in the future. Where $J$ is Youden's statistic (sensitivity [$a$] + specificity [$b$] - 1), and $\phi$ is the prevalence, the ratio of positive predictive values at subsequent time $k$, $\rho(\phi_{k})$, over the original $\rho(\phi_{0})$ at $t_0$ is given by: $\zeta(\phi_{0},k) = \frac{\rho(\phi_{k})}{\rho(\phi_{0})} =\frac{\phi_k(1-b)+J\phi_0\phi_k}{\phi_0(1-b)+J\phi_0\phi_k}$ In this manuscript, we explore the mathematical model which formalizes said screening paradox and explore its implications for population level screening programs. In particular, we define the number of positive test iterations (PTI) needed to reverse the effects of the paradox as follows: $n_{i\phi_e}=\left\lceil\frac{ln\left[\frac{\omega\phi_e\phi_k-\omega\phi_e}{\omega\phi_e\phi_k-\phi_k}\right]}{2ln\omega}\right\rceil$ where $\omega$ is the square root of the positive likelihood ratio (LR+).

Using Bayes' Theorem, we derive generalized equations to determine the positive and negative predictive value of screening tests undertaken sequentially. Where a is the sensitivity, b is the specificity, $\phi$ is the pre-test probability, the combined positive predictive value, $\rho(\phi)$, of $n$ serial positive tests, is described by: $\rho(\phi) = \frac{\phi\displaystyle\prod_{i=1}^{n}a_n}{\phi\displaystyle\prod_{i=1}^{n}a_n+(1-\phi)\displaystyle\prod_{i=1}^{n}(1-b_n)}$ If the positive serial iteration is interrupted at term position $n_i-k$ by a conflicting negative result, then the resulting negative predictive value is given by: $\psi(\phi) = \frac{[(1-\phi)b_{n-}]\displaystyle\prod_{i=b_{1+}}^{b_{(n-1)+}}(1-b_{n+})}{[\phi(1-a_{n-})]\displaystyle\prod_{i=a_{1+}}^{a_{(n-1)+}}a_{n+}+[(1-\phi)b_{n-}]\displaystyle\prod_{i=b_{1+}}^{b_{(n-1)+}}(1-b_{n+})}$ Finally, if the negative serial iteration is interrupted at term position $n_i-k$ by a conflicting positive result, then the resulting positive predictive value is given by: $\lambda(\phi)= \frac{\phi a_{n+}\displaystyle\prod_{i=a_{1-}}^{a_{(n-1)-}}(1-a_{n-})}{\phi a_{n+}\displaystyle\prod_{i=a_{1-}}^{a_{(n-1)-}}(1-a_{n-})+[(1-\phi)(1-b_{n+})]\displaystyle\prod_{i=b_{1-}}^{b_{(n-1)-}}b_{n-}}$ The aforementioned equations provide a measure of the predictive value in different possible scenarios in which serial testing is undertaken. Their clinical utility is best observed in conditions with low pre-test probability where single tests are insufficient to achieve clinically significant predictive values and likewise, in clinical scenarios with a high pre-test probability where confirmation of disease status is critical.

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 $\rho$ that approaches $k$, the number of positive test iterations $n_i$ needed is: $ n_i =\lim_{\rho \to k}\left\lceil\frac{ln\left[\frac{\rho(\phi-1)}{\phi(\rho-1)}\right]}{ln\left[\frac{a}{1-b}\right]}\right\rceil$ where $n_i$ = number of testing iterations necessary to achieve $\rho$, the desired positive predictive value, a = sensitivity, b = specificity, $\phi$ = disease prevalence and $k$ = constant. Based on the aforementioned derivation, we provide reference tables for the number of test iterations needed to obtain a $\rho(\phi)$ of 50, 75, 95 and 99$\%$ as a function of various levels of sensitivity, specificity and disease prevalence.

The relationship between a screening tests' positive predictive value, $\rho$, and its target prevalence, $\phi$, is proportional - though not linear in all but a special case. In consequence, there is a point of local extrema of curvature defined only as a function of the sensitivity $a$ and specificity $b$ beyond which the rate of change of a test's $\rho$ drops precipitously relative to $\phi$. Herein, we show the mathematical model exploring this phenomenon and define the $prevalence$ $threshold$ ($\phi_e$) point where this change occurs as: $\phi_e=\frac{\sqrt{a\left(-b+1\right)}+b-1}{(\varepsilon-1)}$ where $\varepsilon$ = $a$+$b$. Using its radical conjugate, we obtain a simplified version of the equation: $\frac{\sqrt{1-b}}{\sqrt{a}+\sqrt{1-b}}$. From the prevalence threshold we deduce a more generalized relationship between prevalence and positive predictive value as a function of $\varepsilon$, which represents a fundamental theorem of screening, herein defined as: $\displaystyle\lim_{\varepsilon \to 2}{\displaystyle \int_{0}^{1}}{\rho(\phi)d\phi} = 1$ Understanding the concepts described in this work can help contextualize the validity of screening tests in real time, and help guide the interpretation of different clinical scenarios in which screening is undertaken.

Adequate fetal and neonatal development depend upon the presence of a normal acid-base environment during pregnancy and the smooth transition from intra-uterine to extra-uterine life. Current methods to assess fetal pH and acid-base status are invasive and carry significant maternal and fetal risks. Given these limitations, obstetrical care providers developed the electronic fetal monitoring (EFM) system, a non-invasive tool, which evaluates beat-to-beat fetal heart rate (FHR) patterns in order to predict fetal oxygenation status in real-time. Every year, about 85 percent of the approximately 4 million live births in the United States are evaluated using EFM. Unfortunately, though there is ample physiological evidence that FHR patterns are inextricably linked to fetal acid-base status, the use of EFM has not been shown to reliably predict neonatal pH, nor has it reduced the incidence of adverse perinatal outcomes, including long-term neurological morbidity and cerebral palsy (CP). The poor specificity associated with the current interpretation of the EFM therefore leads to a paradox we have henceforth defined as the Obstetrical Paradox. In this study, we develop and seek to determine whether a novel, non-invasive method known as the FETAL technique (Fourier Evaluation of Tracings and Acidosis in Labor), which applies the Fourier Transform to EFM tracings and determines the spectral frequency distributions of the FHR, improves the assessment of the fetal pH in real time. We hypothesize that the improvement in the sensitivity and specificity of the EFM with the use of the FETAL technique will lead to a significant reduction in the rate of neonatal hypoxic injury and in the rate of caesarean deliveries for suspected fetal distress. The implications of a successful application of the FETAL technique would have paradigm-shifting consequences in the provision of modern obstetrical care.

Citation metrics are analytic measures used to evaluate the usage, impact and dissemination of scientific research. Traditionally, citation metrics have been independently measured at each level of the publication pyramid, namely at the article-level, at the author-level, and at the journal-level. The most commonly used metrics have been focused on journal-level measurements, such as the Impact Factor and the Eigenfactor, as well as on researcher-level metrics like the Hirsch index (h-index) and i10 index. On the other hand, reliable article-level metrics are less widespread, and are often reserved to non-standardized and non-scientific characteristics of individual articles, such as views, citations, downloads, and mentions in social and news media. These characteristics are known as 'altmetrics'. However, when the number of views and citations are similar between two articles, no discriminating measure currently exists with which to assess and compare each articles' individual impact. Given the modern, exponentially growing scientific literature, scientists and readers of Science need optimized, reliable, objective methods for managing, measuring and comparing research outputs and individual publications. To this end, I hereby describe and propose a new standardized article-level metric henceforth known as the 'Individual Impact Index Statistic', or $i^3$ for short. The $i^3$ is a weighted algorithm that takes advantage of the peer-review process, and considers a number of characteristics of individual scientific publications in order to yield a standardized and readily comparable measure of impact and dissemination. The strengths, limitations, and potential uses of this novel metric are also discussed., Comment: The i3 is provisionally patent protected by the U.S Patent and Trademark Office (USPTO: #62/506,119). The manuscript is comprised of 14 pages including references, and includes 4 figures