Systematic errors in estimates of $R_t$ from symptomatic cases in the presence of observation bias

We consider the problem of estimating the reproduction number $R_t$ of an epidemic for populations where the probability of detection of cases depends on a known covariate. We argue that in such cases the normal empirical estimator can fail when the prevalence of cases among groups changes with time. We propose a Bayesian strategy to resolve the problem, as well as a simple solution in the case of large number of cases. We illustrate the issue and its solution on a simple yet realistic simulation study, and discuss the general relevance of the issue to the current covid19 pandemic.