Second-Order Inverse Polynomial Model

These functions provide Second-Order Inverse Polynomial Model equation (invpoly2.fun()), as well as the self-starters for the nls() (SSinvpoly2()) and drc::drm() functions (DRCinvpoly2()).

Usage

invpoly2.fun(x, a, b, c)

SSinvpoly2(x, a, b, c)

DRCinvpoly2()

Arguments

x

A numeric vector of non-zero positive values (e.g., dose, time).

a

The coefficient of the \(1/x\) term in the linearised form. Controls the initial rise of the curve near the origin.

b

The coefficient of the linear term in the denominator. Along with c, determines the location and height of the response peak.

c

The coefficient of the quadratic term in the denominator. A positive value causes the curve to decline after reaching a maximum, producing a unimodal response.

Value

invpoly2.fun() and SSinvpoly2() return a numeric value, while DRCinvpoly2() returns a list containing the nonlinear function and the self starter function.

Details

The model is defined as:

$$y = \frac{x}{a + b x + c x^2}$$

where a controls the initial rise near the origin, b is the linear term in the denominator, and c is the quadratic term. A positive c causes the curve to reach a maximum and subsequently decline, producing a unimodal response.

Unlike the first-order model SSinvpoly1(), the quadratic term in the denominator allows the curve to reach a maximum and subsequently decline, making it suitable for unimodal dose-response relationships.

See also

Other non-linear functions, self-starters: chaprich.fun(), exp3.fun(), expneg.fun(), expneg2.fun(), invpoly1.fun(), invpoly3.fun(), kostmod.fun(), varexp.fun(), vargauss.fun(), varsph.fun()

Examples

x <- seq(1, 20, length.out = 50)
y <- invpoly2.fun(x, a = 1, b = 0.5, c = 0.02) + rnorm(50, sd = 0.05)
df <- data.frame(x = x, y = y)
mod = nls(y ~ SSinvpoly2(x, a, b, c), data = df)
summary(mod)
#> 
#> Formula: y ~ SSinvpoly2(x, a, b, c)
#> 
#> Parameters:
#>   Estimate Std. Error t value Pr(>|t|)    
#> a 0.886425   0.073827   12.01 6.36e-16 ***
#> b 0.527991   0.025684   20.56  < 2e-16 ***
#> c 0.017986   0.001615   11.14 8.79e-15 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Residual standard error: 0.05142 on 47 degrees of freedom
#> 
#> Number of iterations to convergence: 2 
#> Achieved convergence tolerance: 4.586e-06
#> 
plot(x, y, cex = 0.8)
lines(x, predict(mod), col = 'blue')


if (FALSE) { # \dontrun{
mod = drc::drm(y ~ x, data = df, fct = DRCinvpoly2())
summary(mod)
plot(mod, log = "")
} # }