Variance functions and the minimum detectable concentration in assays

Assay data are often fit by a nonlinear regression model incorporating heterogeneity of variance, as ion radioimmunoassay, for example. Typically, the standard deviation of the response is taken to be proportional to a power theta of the mean. There is considerable empirical evidence suggesting that for assays of a reseasonable size, how one estimates the parameter theta does not greatly affect how well one estimates the the mean regression function. An additional component of assay analysis is the estimation of auxillary constructs such as the minimum detectable concentration, for which many definitions exist; we focus on one such definition. The minimum detectable concentration depends both on theta and the mean regression function. We compare three standard method of estimating the parameter theta due to Rodbard (1978), Raab (1981a) and Carroll and Ruppert (1982b). When duplicate counts are taken at each concentration, the first method is only 20% efficient asymptotically in comparison to the third, and the resulting estimate of the minimum detectable concentration is asymptotically 3.3 times more variable for first than the third. Less dramatic results obtain for the second estimator compared to the third; this estimator is still not efficient, however. Simulation results and an example are supportive of the asymptotic theory. Keywords: Least squares method.