Audit test denominators without diagnosing a person

Define the host evidence

A detection or classification test can have different error probabilities depending on the underlying state. Sensitivity is the fraction of state-positive records classified positive; specificity is the fraction of state-negative records classified negative. The fraction of positive results that actually have the state uses a different denominator and also depends on the state’s frequency in the tested population.

Connect mechanism and inference

The original fictional population has 1000 records and sensitivity 0.8. With state prevalence p and specificity s, expected true positives are 800p, false negatives 200p, false positives 1000(1−p)(1−s) and true negatives 1000(1−p)s. These are expected contingency counts, not an actual diagnostic dataset or a named disease/test.

Positive predictive fraction divides true positives by all positive results. Changing prevalence changes that fraction even if sensitivity and specificity remain fixed. The calculation assumes these error properties apply to the same sampled population and state definition; selection, measurement dependence and uncertain reference standards can invalidate such transport.

This lesson teaches conditional probabilities at the fictional population level. It cannot interpret a person’s test result, diagnose infection, establish infectiousness or recommend treatment. Actual medical interpretation needs the relevant validated test, clinical context and qualified care rather than these arbitrary parameters.

Reconstruct the original record

At p=0.1, s=0.9, expected counts are 80 true positives,20 false negatives,90 false positives and 810 true negatives. Positive predictive fraction is 80/(80+90)=8/17≈0.470588, not sensitivity 0.8. Both probabilities are correct but condition on different groups.

Predict and investigate

Predict first: Why can a fixed sensitivity coexist with a changing fraction of true states among positive results?

Construct the full two-by-two table before dividing. Change only prevalence, then only specificity, and preserve the denominator and population assumptions for every comparison.

Defend a changed case

Apply the mechanism to a changed boundary

Use p=0.3, s=0.9. Find all four counts and the positive predictive fraction.

Retain the first response, units, uncertainty and any prior feedback exposure in the notebook before comparing this public explanation.

Compare a worked response

True positives 240, false negatives 60, false positives 70 and true negatives 630 total 1000. The positive predictive fraction is 240/310=24/31≈0.774194. This remains a fictional population statement, not a patient result.

Retain bounded evidence

Preserve original assumptions, first responses, source identity and feedback exposure in the Microbiology evidence workspace. These are fictional nonclinical records, not personal medical advice, actual patient data, diagnosis, treatment or an invitation to collect biological samples. Actual practical and independent learner competence require separate evidence.

Primary mechanism context, checked 31 October 2026. Original teaching and numerical records; no source clinical cases or procedures copied.