A Ballooned Beta-Logistic Model

ABSTRACT

The beta distribution is a simple and flexible model in which responses are naturally confined to the finite interval (0,1). Its parameters can be related to covariates such as dose and gender through a regression model. The ballooned beta-logistic (BBL) model expands the response boundaries from (0,1) to (L,U), where L and U are unknown parameters. Under the BBL model, expected responses follow a logistic function that can be made equal to that of the four-parameter logistic (4PL) model. But the distribution of responses differs from the classical 4PL model that has additive normal errors. In contrast, the BBL model naturally has bounded responses and inhomogenous variance. The asymptotic normality of maximum likelihood estimators (MLEs) is obtained even though the support of this nonregular regression model depends on unknown parameters. We find MLEs converge faster to L and U than extreme values do at the minimum and maximum concentrations. Given enzyme-linked immunosorbent assay data from multiple plates, we study a motivating validation objective, which is to set suitability criteria for estimates of L and U; after this, plates with boundary estimates outside these limits would be considered “reference failures.” We show the BBL model has advantages over the 4PL model. Supplementary materials for this article are available online.

KEYWORDS:

• Dose–response
• Laboratory assay
• Logistic regression
• Nonregular maximum likelihood
• Validation study

Acknowledgment

Authors thank Valerii Fedorov for initially suggesting this research.