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The false positive rate is $\mathbb{P}(\text{positive result}\mid \text{truly negative class})$. In screening, it measures erroneous positives among genuinely disease-free individuals.

Let $ϵ$ be the false positive rate for a genuinely cancer-free person, encompassing the harmless-tumour misclassification, and assume screening errors are conditionally independent given the latent states. A pre-clinical case tests positive with probability one; a clinical case is separately recognized. Thus a negative screen identifies latent state $1$, whereas a positive pre-clinical screen can arise from state $1$ or $2$. This is a Hidden Markov model with a screening misclassification model.

Conditional on the negative baseline screen, the state at year $0$ is $1$. The negative screen at year $2$ requires staying in state $1$ for two years and then testing negative. From that state, the positive screen at year $4$ can arise either with probability $ϵ$ while still disease-free or through pre-clinical disease. Consequently the conditional likelihood isIf the study instead specifies an initially disease-free patient and includes the baseline screening result in the likelihood function, its probability contributes a further factor $1-ϵ$, givingThe formulas use the positive emission for a pre-clinical screen, with no clinical diagnosis in this record. A persistent harmless-tumour subtype producing correlated screening errors would need an additional latent-state or emission specification; its likelihood is not determined by the single parameter $ϵ$.

Define the evaluation universe first, including which sufficiently covered loci were eligible for both methods. Relative to the reference assay's labels,The true positive rate is the fraction of reference-positive loci retained by xPore; the false positive rate is the fraction of reference-negative loci incorrectly retained. The latter is not the fraction of xPore calls that fail validation.

An imperfect reference assay can miss real modifications. “False positive” here therefore means disagreement with that reference, not independently demonstrated absence of a modification. Restricting the universe differently for the two denominators would invalidate the comparison.

Suppose exactly one of $N$ equally likely candidates is the source of an observation. A test is always positive for that source and is positive with probability $p$ for any nonsource candidate. For one randomly selected candidate, Bayes' theorem gives $P(G\mid +)=(1/N)/[(1/N)+p(1-1/N)]$, proving the displayed result. Even a small false positive rate can leave a substantial chance of a different source when there are many alternatives. The model concerns one selected candidate with the stated prior; searching many candidates and selecting a match requires accounting for that selection procedure.

The fraction of predicted positives that are genuinely positive. It depends on prevalence as well as the true positive rate and false positive rate.

A plot of true positive rate against false positive rate while a decision threshold varies over a fixed evaluation population. A random ranking has an expected diagonal curve; a perfect ranking reaches the upper-left corner.

In a screening misclassification model, true disease states are latent and screen results are emissions. A cancer-free state may emit a positive result with a false positive rate, while a perfectly sensitive pre-clinical state always emits a positive result. Conditional independence of emissions given latent states is an additional model assumption, not a consequence of a marginal false-positive rate.