Ratios : A short guide to confidence limits and proper use

Researchers often calculate ratios of measured quantities. Specifying confidence limits for ratios is difficult and the appropriate methods are often unknown. Appropriate methods are described (Fieller, Taylor, special boot-strap methods). For the Fieller method a simple geometrical interpretation is given. Monte Carlo simulations show when these methods are appropriate and that the most frequently used methods (index method and zero–variance method) can lead to large liberal deviations from the desired confidence level. It is discussed when we can use standard regression or measurement error models and when we have to resort to specific models for heteroscedastic data. Finally, an old warning is repeated that we should be aware of the problems of spurious correlations if we use ratios. 3 In a number of situations, researchers are interested in the ratio of two measured quantities. For example, in bioassay researchers are interested in the potency of a drug relative to a standard drug (Finney, 1978). Similarly, whenever we calculate " percentage change " or " relative change " we calculate a ratio (Miller, 1986). Another example is inverse prediction in regression analysis. Assume researchers fit the linear model: y i = α + βx i + i (with i = 1,. .. , n) and then want to predict at which x 0 to expect a certain y 0 value. This calculates as x 0 = (y 0 − ˆ α)/ ˆ β which is again a ratio of the random parameter estimatesˆα andˆβ. Inverse prediction is often used in calibration Specifying confidence limits for ratios is a well–know problem in statistics with a number of unusual properties. The classic solution to this problem is called " Fieller's theorem " Quite surprisingly, however, this issue seems to be largely unknown in psychology and the cognitive neurosciences. For example, none of the above cited studies used Fieller's method. Most studies unquestioningly used a method which I will call the " index " method and which turns out to require very specific assumptions about the distribution of numerator and denominator of the ratio. If these assumptions are not met, the method can lead to confidence limits with much too small coverage. Other studies used another ad–hoc method (the " zero–variance " method), which is even more problematic. The index method is closely related to the use of indices which are determined on a per observation basis and then processed further as if they …