Notes

A Note on Medical Diagnosis

Diagnosis in medicine involves identifying the unique characteristics of a disease through abduction, deduction, and induction. The term 'diagnosis,' derived from the Greek 'διάγνωσις' meaning 'discernment', highlights the critical process of distinguishing between healthy and diseased states in individuals. It can be defined as the stochastic mapping of symptoms, signs, and findings from laboratory and medical imaging onto a specific disease condition using medical knowledge.

A. Threshold-Based Diagnosis

In threshold-based diagnosis, diagnostic tests are used to classify individuals into either diseased or non-diseased populations. This method involves setting a diagnostic threshold or cut-off point to dichotomize results, even though the probability distributions of the test measurements in these populations overlap. Despite the inherent uncertainty introduced by these overlapping distributions, this approach significantly impacts medical decision-making by linking a continuum of evidence to binary clinical decisions, such as treatment choices.

Diagnostic Accuracy Measures

For the patients’ safety, it is imperative that their diagnostic accuracy, that is the correctness of this classification, is rigorously evaluated. From the many diagnostic accuracy measures appearing in the literature a few are used for evaluating diagnostic accuracy in clinical research and practice^{1, 2}. These include:
1. Sensitivity (Se), specificity (Sp) diagnostic odds ratio, and likelihood ratio for positive or negative result, which are defined conditionally on the true disease status and are prevalence invariant.
2. Overall diagnostic accuracy, which is defined conditionally on the true disease status and is prevalence-dependent.
3. Positive predictive and negative predictive value, which are defined conditionally on the test outcome and are prevalence-dependent.
Receiver operating characteristic (ROC) curves are also used to evaluate the diagnostic performance of a screening or diagnostic test. ROC curves are plots of Se against 1-Sp.

B. Bayesian Diagnosis

Bayesian diagnosis leverages Bayes' theorem to update the probability of a disease after obtaining test results. This theorem relates the posterior probability P(H|E) of a hypothesis H given evidence E to the likelihood P(E|H) of the evidence given the hypothesis.

Formal Bayesian Inference

Formal Bayesian inference begins with a prior distribution that reflects initial beliefs about the parameters of interest before observing the evidence. This prior is updated with the likelihood function (representing the new evidence) using Bayes' theorem to produce the posterior distribution, which combines prior knowledge with new data.

Prior Distribution

Priors represent researchers' beliefs about parameters before considering the evidence and can be informative, weakly informative, or diffuse, depending on the level of certainty they convey.

Likelihood Function

The likelihood function indicates how probable the observed evidence is given different parameter values, playing a key role in updating the prior to form the posterior distribution.

Posterior Distribution

The posterior distribution results from combining the prior and likelihood functions, reflecting updated knowledge about the parameters after considering the observed evidence.

Workflow

A typical Bayesian workflow includes specifying the prior distribution, determining the likelihood function, and combining both using Bayes' theorem to obtain the posterior distribution. This process involves model checking, refinement, and sensitivity analysis to ensure robust Bayesian inferences.

Empirical Bayesian Inference

Empirical Bayesian inference simplifies the Bayesian framework by using data to estimate the prior distribution³, making it practical when prior information is scarce. This approach treats the prior as an unknown to be estimated from the data, facilitating real-time data integration in medical diagnostics.

Workflow

The empirical Bayesian workflow typically involves:
1. Collecting a large dataset and conducting preliminary statistical analyses to understand the data's distributions and characteristics,
2. Estimating prior distributions using empirical data through methods such as maximum likelihood, and
3. Applying Bayes' theorem with the estimated prior distributions to compute posterior probabilities, thereby integrating the observed data, such as diagnostic test results.

C. Uncertainty in Medical Diagnosis

Measurement uncertainty in diagnostic tests affects threshold-based diagnostic accuracy measures^{1, 4} and Bayesian posterior probability for disease⁵.

Theodora Chatzimichail, M.R.C.S.,
tc@hcsl.com

Related Publications

1. Chatzimichail T, Hatjimihail AT. A Software Tool for Exploring the Relation between Diagnostic Accuracy and Measurement Uncertainty. Diagnostics. 2020;10(9):610.
Abstract
2. Kroll MH, Bipasa B, Budd JR,Durham P, Gorman RT, Gwise TE, Abdel-Baset H, Hatjimihail AT, Hilden J, Kyunghee S. Assessment of the Diagnostic Accuracy of Laboratory Tests Using Receiver Operating Characteristic Curves; Approved Guideline. Second Edition. CLSI document EP24-A2. Clinical and Laboratory Standards Institute; 2011:1-60.
Abstract
3. Chatzimichail T., Hatjimihail AT. A Bayesian Inference Based Computational Tool for Parametric and Nonparametric Medical Diagnosis. Diagnostics. 2020;13(19):3135.
Abstract
4. Chatzimichail T, Hatjimihail AT. A Software Tool for Calculating the Uncertainty of Diagnostic Accuracy Measures. Diagnostics. 2021;11(3):406.
Abstract
5. Chatzimichail T, Hatjimihail AT. A Software Tool for Estimating Uncertainty of Bayesian Posterior Probability for Disease. Diagnostics. 2024;14(4):402.
Abstract