Statistics in Biosciences

, Volume 1, Issue 1, pp 65–79

On Analysis of the Difference of Two Exposure-Adjusted Poisson Rates with Stratification: From Asymptotic to Exact Approaches


    • Clinical BiostatisticsMerck Research Laboratories
  • William W. B. Wang
    • Clinical BiostatisticsMerck Research Laboratories

DOI: 10.1007/s12561-009-9007-0

Cite this article as:
Chan, I.S.F. & Wang, W.W.B. Stat Biosci (2009) 1: 65. doi:10.1007/s12561-009-9007-0


Long-term studies are frequently conducted to assess the treatment effect on rare diseases or the safety of a new treatment. To account for differential follow-up often encountered in long-term studies, exposure-adjusted incidence rates are used in evaluating the treatment effect. The difference of rates is sometimes used to quantify the treatment’s public health impact because the reciprocal of this difference can be interpreted as “the number needed to treat (or number needed to vaccinate) in order to cure (or prevent) 1 case of disease.” In this paper we focus on the stratified analysis of the difference of two exposure-adjusted rates in the setting of superiority, noninferiority and super-superiority hypothesis testing. After a brief review of asymptotic methods, we derive an exact method that guarantees control of the type I error. But it is conservative for noninferiority and super-superiority testing because of the search of the maximum tail probability over a multidimensional nuisance parameter space. Then, we present an approximate exact method where the p-value is estimated at the maximum likelihood estimates of the nuisance parameter. This method is identical to exact method for superiority testing and reduces the conservatism for noninferiority and super-superiority testing. In addition, a quasi-exact method is discussed. A real-life vaccine clinical trial example is used to illustrate these methods. Finally, we compare the performance of these methods via empirical studies and make some general practical recommendations.


Approximate exact methodConstrained maximum likelihood estimateExact testPoisson rateRisk differenceStratified analysis

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© International Chinese Statistical Association 2009