# Maximum likelihood estimation of age-specific incidence rate from prevalence

## TL;DR

This paper introduces a new method to estimate age-specific incidence rates of chronic diseases using maximum likelihood, based on prevalence and mortality data.

## Contribution

The paper presents a novel maximum likelihood approach for incidence estimation with confidence intervals, replacing resampling techniques.

## Key findings

- Maximum likelihood estimation using a binomial likelihood function can replace resampling techniques for incidence rate estimation.
- The method is applicable in scenarios with non-differential and differential mortality.
- Results are demonstrated using historical data on breathlessness and diabetes.

## Abstract

Usually, age-specific incidence rates of chronic diseases are estimated from longitudinal studies that follow participants over time and record incident cases. However, these studies can be cost- and time-expensive and are prone to loss to follow up. An alternative method allows incidence estimation based on aggregated data from (cross-sectional) prevalence and mortality studies using relations between incidence, prevalence and mortality described by the illness-death model and a related partial differential equation. Currently, adequate options for the assessment of the accuracy of the achieved incidence estimates are missing and bootstrap resampling methods are used instead. Therefore, we developed novel ways to estimate incidence rates based on the maximum likelihood principle with corresponding confidence intervals. Historical data about breathlessness in British coal miners and diabetes in Germany are used to illustrate the applicability of this method in scenarios with non-differential and differential mortality. We have two scenarios of available data in the case of differential mortality: mortality of diseased and all-cause mortality, or all-cause mortality and mortality rate ratio. Our results show that estimation of incidence rates and corresponding confidence intervals of chronic conditions based on aggregated data with the maximum likelihood method using a binomial likelihood function is possible and can replace resampling techniques.