beta_model {OptimModel}

Beta hook-effect model, gradient, starting values, and back-calculation functions

Description

Five-parameter hook-effect model for dose-response curve fitting

Usage

 
       beta_model(theta, x)

Arguments

theta

Vector of five parameters: (e_{\min}, e_{\max}, \log(\delta_1), \log(\delta_2), \log(\delta_3)) . See details.

x

Vector of concentrations for the Beta model.

Details

The five-parameter Beta model is given by:

y = e_{\min} + e_{\max} \times \exp( log( \beta(\delta_1, \delta_2) ) + \delta_1 \times\log(x) + \delta_2*\log(\text{sc}-x) - (\delta_1+\delta_2)\times\log(\text{sc})

where

\beta(\delta_1, \delta_2) = (\delta_1+\delta_2)^(\delta_1+\delta_2) /(\delta_1^{\delta_1} \times \delta_2^{\delta_2})

and

\text{sc} = \max(x) + \delta_3.

Note that the Beta model depends on the maximum x value. For a particular data set, this may be set by

attr(theta), "maxX") = max(x).

Value

Let N = length(x). Then

• beta_model(theta, x) returns a numeric vector of length N.

• attr(beta_model, "gradient")(theta, x) returns an N x 5 matrix.

• attr(beta_model, "start")(x, y) returns a numeric vector of length 5 with starting values for

  (e_{\min}, e_{\max}, \log(\delta_1), \log(\delta_2), \log(\delta_3)).

• attr(beta_model, "backsolve")(theta, y) returns a numeric vector of length=length(y) with the first x such that beta_model(theta, x)=y.

Author(s)

Steven Novick

See Also

optim_fit, rout_fitter

Examples

set.seed(123L)
x = rep( c(0, 2^(-4:4)), each=4 )
theta = c(emin=0, emax=115, ldelta1=-1.5, ldelta2=9, ldelta3=11.5)
y = beta_model(theta, x)  + rnorm( length(x), mean=0, sd=1 )

beta_model(theta, x)
attr(beta_model, "gradient")(theta, x)
attr(beta_model, "start")(x, y)

attr(theta, "maxX") = max(x)
attr(beta_model, "backsolve")(theta, 50)