# Can I use logistic (or Poisson) regression to model epidemiological data with incomplete (but generalizable) knowledge about the population?

My question has three levels of complexity, I will try to explain one by one. Please accept I am no professional in statistics and still trying to broaden my understanding.

1) I would like to investigate the effect of continuous variable (X1) on incidence of a disease. I have complete data on positive cases (time and age of diagnosis, and value of X). However, for the negative cases (without the disease) I have only epidemiological data - exact numbers of people from statistical office and the distribution of X (mean and SD) taken from populational study performed a few years ago. However, I lack the knowledge on specific value of X for each of the people without the disease. Can I create an "artificial population" with particular N, mean and SD and then use it in regression analysis?

2) If the disease I am modelling is rare (0-heavy) should I use Poisson regression or negative binominal model?

3) If I have more data I would like to include in the analysis (sex and city/rural residence) can I do this in the same way as in 1) i.e. distribute males and females and city and rural residence randomly (but summing to the numbers I have from statistical office) among this "artificial population"?

4) Can I really use the statistical data on X from a study performed a few years ago? It assessed the value of X in a representative sample of population and created percentile charts which are still used to assess X in patients. However, I am afraid that with years the population may have changed and I may have some false positive findings when comparing the patients (in whom, the X change over time, if one exists, is reflected in gathered data) to fixed-in-time population distribution.

A Bayesian approach to include your historical data in the analysis is to use power priors. A power prior can be specified setting a parameter $a_{0}$, which will weight your historical data. The power prior distribution of your parameter of interest so called $\theta$ is given by:
$\pi(\theta|D_{0}, a_{0}) \propto L(\theta| D_{0})^{a_{o}} \pi_{0}(\theta|c_{0})$
Where $L(\theta| D_{0})$ is the likelihood of the current study, $c_{0}$ is the hyperprior parameter that controls the impact of the $\pi_{0}(\theta|c_{0})$ and $a_{o}$ is a scalar that controls the influence of the historical data.