Third-Order Inverse Polynomial Model

These functions provide Third-Order Inverse Polynomial Model equation (invpoly3.fun()), as well as the self-starters for the nls() (SSinvpoly3()) and drc::drm() functions (DRCinvpoly3()).

Usage

invpoly3.fun(x, a, b, c, d)

SSinvpoly3(x, a, b, c, d)

DRCinvpoly3()

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 and d, determines the shape and location of any peaks or troughs in the response.

c

The coefficient of the quadratic term in the denominator. Contributes to the curvature and can produce a unimodal response.

d

The coefficient of the cubic term in the denominator. Allows for additional inflection points, enabling bimodal or more complex response shapes.

Value

invpoly3.fun() and SSinvpoly3() return a numeric value, while DRCinvpoly3() returns a list containing the nonlinear function and the self starter function.

Details

The model is defined as:

$$y = \frac{x}{a + b x + c x^2 + d x^3}$$

where a controls the initial rise near the origin, b is the linear term, c is the quadratic term, and d is the cubic term in the denominator. The cubic term extends it by allowing for additional inflection points, enabling bimodal or more complex response shapes.

See also

Other non-linear functions, self-starters: chaprich.fun(), exp3.fun(), expneg.fun(), expneg2.fun(), invpoly1.fun(), invpoly2.fun(), kostmod.fun(), varexp.fun(), vargauss.fun(), varsph.fun()

Examples

x <- seq(1, 20, length.out = 50)
y <- invpoly3.fun(x, a = 1, b = 0.5, c = 0.02, d = 0.001) + rnorm(50, sd = 0.05)
df <- data.frame(x = x, y = y)
mod = nls(y ~ SSinvpoly3(x, a, b, c, d), data = df)
summary(mod)
#> 
#> Formula: y ~ SSinvpoly3(x, a, b, c, d)
#> 
#> Parameters:
#>    Estimate Std. Error t value Pr(>|t|)    
#> a 1.1287488  0.1374812   8.210 1.45e-10 ***
#> b 0.4109026  0.0784659   5.237 3.96e-06 ***
#> c 0.0321860  0.0117326   2.743  0.00864 ** 
#> d 0.0005805  0.0004764   1.219  0.22919    
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Residual standard error: 0.05274 on 46 degrees of freedom
#> 
#> Number of iterations to convergence: 2 
#> Achieved convergence tolerance: 6.331e-06
#> 
plot(x, y, cex = 0.8)
lines(x, predict(mod), col = 'blue')


if (FALSE) { # \dontrun{
mod = drc::drm(y ~ x, data = df, fct = DRCinvpoly3())
summary(mod)
plot(mod, log = "")
} # }