Small sample saddlepoint confidence intervals in epidemiology

In section 1, we develop a novel method of confidence interval construction for directly standardized rates. These intervals involve saddlepoint approximations to the intractable distribution of a weighted sum of Poisson random variables and the determination of hypothetical Poisson mean values for each of the age groups. Simulation studies show that, in terms of coverage probability and length, the saddlepoint confidence interval (SP) outperforms four competing confidence intervals obtained from the moment matching (M8), gamma-based (G1,G4) and ABC bootstrap (ABC) methods. In section 2, we first consider Brillinger\u27s classical model for a vital rate estimate with a random denominator. We derive statistical properties for this rate estimate and investigate difficulties encountered while trying to perform statistical inference about its expected value. Since inference about this expected value is not possible, we consider instead confidence intervals for covariance of the bivariate Poisson distribution which underlies Brillinger\u27s model, on the way to proposing a new model which is a modification of Brillinger\u27s model and which has numerous theoretical and computational advantages over the latter. A simulation study for our new model shows that in terms of coverage probability, our novel two-dimensional mid-P Clopper-Pearson type confidence interval (CP2) outperforms the Clopper-Pearson type interval with no mid-P correction (CP0) and the Clopper-Pearson type interval with a classical one-dimensional mid-P correction (CP1). In addition, CP2 was found to be more or less equivalent, in terms of coverage probabilities, to CDF0, CDF1 and CDF2 which are the CDF pivot methods with no mid-P correction, a one-dimensional mid-P correction and a two-dimensional mid-P correction, respectively. Furthermore, method CP2 performed essentially as well as the saddlepoint approximation (SP0) to the CDF0 method. Finally, all of the above-mentioned methods (CDF0, CDF1, CDF2, CP0, CP1, CP2 and SP0) substantially outperform the large sample (LS) method of confidence interval construction, in terms of coverage probability --Abstract, page iii