#### Abstract

**Aims**: At CPDD 2015, we applied parametric Hill functions to estimate the probability of drug dependence in relation to the duration of drug-taking experience. A problem we and others have encountered in the estimation of risk of becoming a drug dependence case is an observed point estimate of zero – the so-called “zero-numerator problem.” This problem can be easily observed in certain low risk subgroups even when the sample is large (e.g., the incidence of heroin dependence among 12 year old newly incident heroin users) or with small subgroup sample sizes. In these instances, tan observed zero point estimate does not necessarily imply zero risk of developing dependence for the subgroup. Here, our aim is to describe our approach to a potential solution to the zero-numerator problem based on a Bayesian model in conjunction with parametric Hill functions.

**Methods**: The traditional frequentist statistical approach can provide an estimate for the 95% upper bound of an incident rate even with the observed zero in the numerator. A Bayesian approach is required if estimation of the incident rate itself is of interest. The Bayesian approach demands specification of a prior distribution for the risk parameter. In this work, we are exploring the sensitivity of the Hill function parameter estimates to the choice of a particular informative prior distribution across a range of estimated chances of developing drug dependence very soon after onset of drug use.

**Conclusions**: Whereas we frame our work in relation to risk of developing drug dependence syndromes, the zero-numerator problem often is faced in other contexts (e.g., pharmacokinetics, toxicology). Our approach, combining Bayesian statistics in conjunction with Hill functions, is expected to provided a useful solution to these zero numerator problems.

#### Document Type

Presentation

#### Publication Date

6-14-2016

#### Funding Information

The research was supported via a National Institute on Drug Abuse Senior Scientist Award JCA: K05DA015799.

#### Repository Citation

Vsevolozhskaya, Olga A.; Wagner, Fernando A.; and Anthony, James C., "Dealing with Zero-Numerators in Estimating Drug-Dependence Chances: A Bayesian Approach" (2016). *Biostatistics Presentations*. 5.

http://uknowledge.uky.edu/biostatistics_present/5

## Notes/Citation Information

A poster presentation at the College on Problems of Drug Dependence 78th Annual Scientific Meeting in Palm Springs, CA.