Power of the Cochran-Armitage trend test when exposure scores are based on empirical quantiles of exposure

Epidemiologists often categorize exposures based on quantiles of exposure and use the Cochran-Armitage trend test based on such categories to detect associations between disease and exposure. Power calculations typically assume that the population quantiles are known, but in practice quantiles are often estimated from the sample data. We evaluated the power of the Cochran-Armitage trend test for cohort designs and for case-control designs in which sample quantiles of exposure in the cohort or in controls from a case-control study, respectively, are used to define the cut-points that separate exposure score categories. We give the asymptotic formulas for size and power for the Cochran-Armitage test based on empirical quantiles separately for cohort and case-control designs, together with efficient simulation methods to estimate size and power. Numerical results indicate that estimation of sample quantiles has only a slight effect on power for cohort studies with at least four categories or with more than 280 subjects. However, estimating quantiles can reduce power appreciably in smaller studies with fewer than four exposure categories. For case-control studies of rare diseases, the power loss is limited with more than 120 cases plus controls if the odds ratio comparing the highest exposure category to the lowest category is greater than 0.5. However, if that odd ratio is smaller than 0.5, only samples with more than 360 cases plus controls can guarantee a small loss of power, and increasing the number of exposure categories does not eliminate the loss of power.