MOVER-R confidence intervals for ratios and products of two independently estimated quantities

Abstract

The purpose of this note is to show how the MOVER algorithm for a ratio of two independently estimated quantities, recently published by Donner and Zou,1 requires modification to cope with the general case in which the lower and upper confidence limits for the numerator can take either sign. The MOVER2 (Method Of Variance Estimates Recovery) algorithm as originally formulated relates to a difference (or sum) of two independently estimated parameters. Suppose that for i = 1 and 2, θi has maximum likelihood estimate and 1 − α confidence limits Li and Ui. Then the MOVER interval for the difference θ1 − θ2, called MOVER-D is delimited by The MOVER algorithm does not explicitly take into account how the limits (Li, Ui) were calculated, but obviously for validity we require that for each parameter, . It has been used extensively to derive intervals for various quantities, notably a difference of independent proportions.3 It is particularly useful for quantities derived from proportions as it preserves boundary-respecting properties. Donner and Zou4,5 provided a theoretical justification. Li and Ui are used to derive local variance estimates for θi, separately for θi ranging from Li to and from to Ui. For the lower tail of the interval for θi, the variance estimate recovered from Li is simply ( − Li)2/z2. Similarly, the variance estimate appropriate for the upper tail is (Ui − )2/z2. These recovered local variance estimates are then used to construct lower and upper limits for θ1 − θ2 in an obvious manner. Obviously these algorithms require great care with regard to signs and which limit is which.