A geometric confidence ellipse approach to the estimation of the ratio of two variables

One is often interested in the ratio of two variables, for example in genetics, assessing drug effectiveness, and in health economics. In this paper, we derive an explicit geometric solution to the general problem of identifying the two tangents from an arbitrary external point to an ellipse. This solution permits numerical integration of a bivariate normal distribution over a wedge-shaped region bounded by the tangents, which yields an evaluation of the tangent slopes as confidence limits on the ratio of the component variables. After suitable adjustment of the confidence coverage of the ellipse, these confidence limits are shown to be equivalent to those from Fieller's method. However, the geometric approach allows additional interpretation of the data through identification of the points of tangency, the ellipse itself, and expressions for the coverage probability of the confidence interval. Numerical evaluations using the theoretical expressions for the geometric confidence intervals (but ignoring sample variation in the underlying parameters) suggested that they perform well overall and are slightly conservative. Simulations that do take account of sample variation in the underlying parameters again suggested that the intervals perform well overall, although here they are slightly anti-conservative. Coverage probabilities for the confidence intervals were only weakly dependent on the distance and correlation of the ellipse, but there were asymmetries in the failure rates of the upper and lower confidence limits in some configurations. The probability of no real solution existing was also evaluated. These ideas are illustrated by a practical example.     

Analysis of recurrent event data under the case-crossover design with applications to elderly falls

The case-crossover design is useful for studying the effects of transient exposures on short-term risk of diseases or injuries when only data on cases are available. The crossover nature of this design allows each subject to serve as his/her own control. While the original design was proposed for univariate event data, in many applications recurrent events are encountered (e.g. elderly falls, gout attacks, and sexually transmitted infections). In such situations, the within-subject dependence among recurrent events needs to be taken into account in the analysis. We review three existing conditional logistic regression (CLR)-based approaches for recurrent event data under the case-crossover design. A simple approach is to use only the first event for each subject; however, we would expect loss of efficiency in estimation. The other two reviewed approaches rely on independence assumptions for the recurrent events, conditionally on a set of covariates. Furthermore, we propose new methods that adjust the CLR using either a within-subject pairwise resampling technique or a weighted estimating equation. No specific dependency structure among recurrent events is needed therein, and hence, they have more flexibility than the existing methods in the situations with unknown correlation structures. We also propose a weighted Mantel-Haenszel estimator, which is easy to implement for data with a binary exposure. In simulation studies, we show that all discussed methods yield virtually unbiased estimates when the conditional independence assumption holds. These methods are illustrated using data from a study of the effect of medication changes on falls among the elderly.