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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 $ϵ$.

One analysis can treat a reported symptom-onset date as the exact $H\to I$ transition time. That adds an exactly observed infection-time density to the likelihood, but assumes symptoms begin immediately at infection, every relevant episode is symptomatic, and dates are recalled and reported without error.

A more realistic analysis treats true infection as a latent transition and symptom onset as a noisy observation. A reporting-delay distribution, and possibly probabilities of asymptomatic infection and non-reporting, can be added to a Hidden Markov model. Weekly tests then interval-censor the state transition while the symptom date refines its distribution. This approach uses more information but requires an identifiable and correctly specified symptom-delay and reporting model.