  For many diseases, logistic and other constraints often render large
incidence studies difficult, if not impossible, to carry out. This becomes a
drawback, particularly when a new incidence study is needed each time the
disease incidence rate is investigated in a different population. However, by
carrying out a prevalent cohort study with follow-up it is possible to estimate
the incidence rate if it is constant. In this paper we derive the maximum
likelihood estimator (MLE) of the overall incidence rate, $\lambda$, as well as
age-specific incidence rates, by exploiting the well known epidemiologic
relationship, prevalence = incidence $\times$ mean duration ($P = \lambda
\times \mu$). We establish the asymptotic distributions of the MLEs, provide
approximate confidence intervals for the parameters, and point out that the MLE
of $\lambda$ is asymptotically most efficient. Moreover, the MLE of $\lambda$
is the natural estimator obtained by substituting the marginal maximum
likelihood estimators for P and $\mu$, respectively, in the expression $P =
\lambda \times \mu$. Our work is related to that of Keiding (1991, 2006), who,
using a Markov process model, proposed estimators for the incidence rate from a
prevalent cohort study \emph{without} follow-up, under three different
scenarios. However, each scenario requires assumptions that are both disease
specific and depend on the availability of epidemiologic data at the population
level. With follow-up, we are able to remove these restrictions, and our
results apply in a wide range of circumstances. We apply our methods to data
collected as part of the Canadian Study of Health and Ageing to estimate the
incidence rate of dementia amongst elderly Canadians.
