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.