The full specification of a statistical dose-response (regression) model involves specifying how the mean is described by a parametric function of dose as well as specifying assumptions about the distribution of the response.

We will focus on ways to model the mean trends through mostly s-shaped or related biphasic functions because these functions have in common that they reflect a basic understanding about the causal relationship between the dose and the response, e.g., when a dose increases the response monotonically decreases or increases one way or another towards minimum or maximum response limits, respectively. Consequently, these functions have turned out to be extremely versatile for describing various biological mechanisms involving model parameters that allow the interpretation of observed effects within a biologically plausible framework.

So we define dose-response models to be a collection of statistical models having a certain mean structure in common; this is not a strict mathematical definition, but rather a definition driven by applications. Consequently, dose-response models encompass a range of statistical models from nonlinear regression, generalized (non)linear regression, and parametric survival analysis.

Let y denote an observed response value, possibly aggregated in some way, corresponding to a dose value x ≥ 0. The values of y are often positive but may take arbitrary positive or negative values. Furthermore, we will assume that observation of y is subject to sampling variation, necessiating the specification of a statistical model describing the random variation. Specifically, we will focus on characterizing the mean of y (denoted E(y) below) in terms of a model function f that depends on the dose x: (1) So, for a given dose x, the corresponding observed response values will be distributed around f(x, β). The function f is completely known as it reflects the assumed relationship between x and y, except for the values of the model parameters β = (β1, …, βp), which will be estimated from the data to obtain the best fitting function. The remaining distributional assumptions on y will depend on the type of response. For instance, for a continuous response the normal distribution is commonly assumed whereas for a binary or quantal response the binomial distribution is commonly assumed.

A large number of more or less well-known model functions are built-in in drc (see Table 1). These models are parameterized using a unified structure with a coefficient b denoting the steepness of the dose-response curve, c, d the lower and upper asymptotes or limits of the response, and, for some models, e the effective dose ED50.

https://doi.org/10.1371/journal.pone.0146021.t001

By far the log-logistic models are the most used dose-response models [9]. The four-parameter log-logistic model corresponds to the model function: (2) Two slightly different parameterizations are available: one where ED50 is a model parameter, that is e in Eq (2), and another where the logarithm of ED50 denoted by , say, is a model parameter as in the following Eq (3): (3) This second version of the four-parameter log-logistic model may be preferred for dose-response analysis involving very small datasets (