The unknown denominator problem in population studies of disease frequency

Problems related to unknown or imprecisely measured populations at risk are common in epidemiologic studies of disease frequency. The size of the population at risk is typically conceptualized as a denominator to be used in combination with a count of disease cases (a numerator) to calculate incidence or prevalence. However, the size of the population at risk can take other epidemiologic properties in relation to an exposure of interest and the count outcome, including confounding, modification, and mediation. Using spatial ecological studies of injury incidence as an example, we identify and evaluate five approaches that researchers have used to address "unknown denominator problems": ignoring, controlling for a proxy, approximating, controlling by study design, and measuring the population at risk. We present a case example and recommendations for selecting a solution given the data and the hypothesized relationship between an exposure of interest, a count outcome, and the population at risk.