Ratio estimates, the delta method, and quantal response tests for increased carcinogenicity

Bieler, G., & Williams, R. (1993). Ratio estimates, the delta method, and quantal response tests for increased carcinogenicity. Biometrics, 49(3), 793-801.

This paper demonstrates the use of the delta method for estimating the variance of ratio statistics derived from animal carcinogenicity experiments. The Cochran-Armitage test (Cochran, 1954, Biometrika 10, 417-451; and Armitage, 1955, Biometrics 11, 375-386) is routinely applied to carcinogenicity data as a test for linear trend in lifetime tumor incidence rates. The computing formula for this test derives from the assumption that the denominators of the quantal response rates are fixed. However, when time-at-risk weights are introduced to correct for treatment-related differences in survival, the denominators of the quantal response rates are subject to random variation. The delta method and weighted least squares techniques are applied here to approximate the variance of such ratio statistics and test for a linear dose-response relationship among treatments. This technique is compared to that of Bailer and Portier (1988, Biometrics 44, 417-431), who introduced a survival-adjusted quantal response test for trend in lifetime tumor incidence rates. Their test modifies the usual Cochran-Armitage computing formula by weighting the denominators of the response rates to reflect less-than-whole-animal contributions to risk. Within the framework of a weighted least squares linear regression model that underlies the Cochran-Armitage test, the time-at-risk weights of Bailer and Portier are incorporated using the delta method. Although the delta method approach is slightly more computationally intensive, small-sample simulations indicate that it has superior operating characteristics over the Poly-3 trend test of Bailer and Portier when background tumor incidence rates are low (under 3%) and survival patterns differ markedly across treatments.(ABSTRACT TRUNCATED AT 250 WORDS)