Measuring Parallelism, Linearity, and Relative Potency in Bioassay and Immunoassay Data

ABSTRACT

There is often a need to determine parallelism or linearity between pairs of dose–response data sets for various biological applications. This article describes a technique based on a modification of the well-known extra-sum-of-squares principle of statistical regression. The standard extra-sum-of-squares method uses an F-distributed ratio as a statistic and an F-test based on this statistic as the parallelism test. It is shown here that this metric does not directly measure the parallelism between the two curves and can often vary in opposition to actual parallelism. To overcome this problem, a metric based on a chi-square test applied directly on the chi-square–distributed extra-sum-of-squares statistic is developed, which is shown to correspond directly to parallelism. This parallelism metric does not suffer from the shortcomings of the conventional F-test–based metric, and is a more reliable and appropriate measure of parallelism. The article also shows that the choice of curve model has a large effect on the sensitivity of either metric, and that using an asymmetric model, such as the asymmetric five-parameter logistic function, a generalization of the commonly used symmetric four-parameter logistic function, is necessary when working with asymmetric dose–response data. The effect of noise, as well as the importance of correct weighting on the parallelism metrics and the relative potency, is also studied.

Key Words:

• Bioassay
• Five parameter logistic
• Four parameter logistic
• Immunoassay
• Linearity
• Nonlinear weighted regression
• Parallelism
• Relative potency

Notes

The curve fit p values are the p values of a chi-square test on their RSSEs (Gottschalk and Dunn 18 in press a).

Eleven concentrations between 0.1 and 5 (dilution factor  = 1.479) were used to generate data set 1, and 11 concentrations between 0.0533 and 2.667 (dilution factor  = 1.479) were used to generate data set 2. The simulation parameters used to generate data sets 1 and 2 were (1.7, −10, 1, 0.2, 0.15), (1.7, −10, 0.67, 0.2, 0.15), (0.0001, 1.5) and (1.7, −10, 1, 0.2, 0.15), (1.7, −16, 0.67, 0.2, 0.15), (0.0001, 1.5), respectively. The parameters of the expected variance model used to calculate the weights for the regressions were optimal, as discussed in Appendix C. In the Parallelism columns, the X² Prob is listed above the F Prob. Based on the ratio of the c parameters, the relative potency is about 1.5.