OptimModel (version 2.0-1)

hill_quad_model: Five-parameter Hill model with quadratic component, gradient, starting values, and back-calculation functions

Description

Five-parameter Hill model with quadratic component, gradient, starting values, and back-calculation functions.

Usage

hill_quad_model(theta, x)

Value

Let N = length(x). Then

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

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

• attr(hill_quad_model, "start")(x, y) returns a numeric vector of length 5 with starting values for (A, B, a, b, c).

  If the quadratic roots are real-valued, attr(hill_quad_model, "backsolve")(theta, y) returns a numeric vector of length=2.

Arguments

theta

Vector of five parameters: (A, B, a, b, c). See details.

x

Vector of concentrations for the five-parameter Hill model with quadratic component.

Author

Steven Novick

Details

The five parameter Hill model with quadratic component is given by:

$$y = A + \frac{B-A}{( 1 + \exp( -(a + bz + cz^2) ) )}\text{, where }z = \log(x)$$

\(A =\min y\) ( minimum y value), \(B = \max y\) (maximum y value), (a, b, c) are quadratic parameters for \(\log(x)\).

Notes:

1. If \(c = 0\), this model is equivalent to the four-parameter Hill model (hill.model).

2. The ic50 is defined such that \(a + bz + cz^2 = 0\). If the roots of the quadratic equation are real, then the ic50 is given by \(\tfrac{-b \pm\sqrt{b^2 - 4ac }}{2a}\).

See Also

optim_fit, rout_fitter

Examples

set.seed(123L)
x = rep( c(0, 2^(-4:4)), each=3 )      ## Dose
theta = c(0, 100, 2, 1, -0.5)          ## Model parameters
y = hill_quad_model(theta, x) + rnorm( length(x), mean=0, sd=5 )

## Generate data
hill_quad_model(theta, x)
attr(hill_quad_model, "gradient")(theta, x)
attr(hill_quad_model, "start")(x, y)
attr(hill_quad_model, "backsolve")(theta, 50)