Proast models

In Proast, a family of (nested) dose-response models are available that can be used for describing the change in any continuous endpoint as a function of dose. The likelihood ratio test is used to select one of the available models (model selection to prevent overparameterization).

Model 1: $y = a$ with $a > 0$
Model 2: $y = a \cdot e x p (x / b)$ with $a > 0$
Model 3: $y = a \cdot e x p (\pm (x / b)^{d})$ with $a > 0, b > 0, d \geq 1$
Model 4: $y = a [c - (c - 1) e x p (- x / b)]$ with $a > 0, b > 0, c > 0$
Model 5: $y - (c - 1) e x p (- (x / b)^{d})]$ with $a > 0, b > 0, c > 0, d \geq 1$

where $y$ is any continuous endpoint and $x$ denotes the dose. In all models parameter $a$ represents the level of the endpoint at dose 0, and $b$ is considered as the parameter reflecting the efficacy of the substance or the sensitivity of the subject. At high doses model 4 and 5 level of to the value $a \cdot c$ , so the parameter $c$ can be interpreted as the maximum relative change. Model 3 and 5 have the flexibility to mimic threshold-like responses. All these model are nested to each other, except models 3 and 4, which both have three parameters.

In all models the parameter $a$ is constrained to being positive for obvious reasons (it denotes the value of the endpoint at dose 0). The parameter $d$ is constrained to values larger than (or equal to) 1, to prevent the slope of the function at dose 0 being infinite, which seems biologically implausible. The parameter $b$ is constrained to be positive in all models. Parameter $c$ in models 4 and 5 determines whether the function increases or decreases, by being larger or smaller than unity, respectively. To make model 3 a decreasing function a minus sign has to be inserted in the exponent (Slob (2002), Slob and Setzer (2013)).