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: \(y = a \cdot exp (x/b)\) with \(a > 0\)

• Model 3: \(y = a \cdot exp ( ± (x/b)^{d})\) with \(a > 0, b > 0, d ≥ 1\)

• Model 4: \(y = a [c-(c-1) exp ( - x/b)]\) with \(a > 0, b > 0, c > 0\)

• Model 5: \(y -(c-1) exp ( - (x/b)^{d})]\) with \(a > 0, b > 0, c > 0, d ≥ 1\)

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)).