ursa: Model function for the universal response surface approach...
In MaximeBSanofi/drc2:

Description Usage Arguments Details Value Author(s) References See Also Examples

URSA provides a parametric approach for modelling the joint action of several agents. The model allows quantification of synergistic effects through a single parameter.

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  ursa(fixed = rep(NA, 7), names = c("b1", "b2", "c", "d", "e1", "e1", "f"), 
  ssfct = NULL)

`fixed`	numeric vector. Specifies which parameters are fixed and at what value they are fixed. NAs for parameter that are not fixed.
`names`	a vector of character strings giving the names of the parameters. The default is reasonable.
`ssfct`	a self starter function to be used (optional).

The model function is defined implicitly through an appropriate equation. More details are provided by Greco et al (1990, 1995).

A list containing the nonlinear function, the self starter function, and the parameter names.

Christian Ritz after an idea by Hugo Ceulemans.

Greco, W. R. and Park H. S. and Rustum, Y. M. (1990) Application of a New Approach for the Quantitation of Drug Synergism to the Combination of cis-Diamminedichloroplatinum and 1-beta-D-Arabinofuranosylcytosine, Cancer Research, 50, 5318–5327.

Greco, W. R. Bravo, G. and Parsons, J. C. (1995) The Search for Synergy: A Critical Review from a Response Surface Perspective, Pharmacological Reviews, 47, Issue 2, 331–385.

Other models for fitting mixture data are the Hewlett and Voelund models mixture.

## Here is the complete statistical analysis of the data 
##  from Greco et al. (1995) by means of the URSA model
if (FALSE)
{
d1 <- c(0, 0, 0, 0, 0, 0, 0, 0, 2, 5, 10, 20, 50, 2, 2, 2, 
2, 2, 5, 5, 5, 5, 5, 10, 10, 10, 10, 10, 20, 20, 20, 20, 
20, 50, 50, 50, 50, 50)

d2 <- c(0, 0, 0, 0.2, 0.5, 1, 2, 5, 0, 0, 0, 0, 0, 0.2, 
0.5, 1, 2, 5, 0.2, 0.5,  1, 2, 5, 0.2, 0.5, 1, 2, 5, 0.2, 
0.5, 1, 2, 5, 0.2, 0.5, 1, 2, 5)

effect <- c(106.00, 99.20, 115.00, 79.20, 70.10, 49.00, 
21.00, 3.83, 74.20, 71.50,48.10, 30.90, 16.30, 76.30, 
48.80, 44.50, 15.50, 3.21, 56.70, 47.50, 26.80, 16.90, 
3.25, 46.70, 35.60, 21.50, 11.10, 2.94, 24.80, 21.60, 
17.30, 7.78, 1.84, 13.60, 11.10, 6.43, 3.34, 0.89)

greco <- data.frame(d1, d2, effect)

greco.m1 <- drm(effect ~ d1 + d2, data = greco, fct = ursa(fixed = c(NA, NA, 0, NA, NA, NA, NA)))

plot(fitted(greco.m1), residuals(greco.m1)) # wedge-shaped

summary(greco.m1)
 
## Transform-both-sides approach using a logarithm transformation
greco.m2 <- drm(effect ~ d1 + d2, data = greco, fct = ursa(fixed = c(NA, NA, 0, NA, NA, NA, NA)), 
bcVal = 0, control = drmc(relTol = 1e-12))

plot(fitted(greco.m2), residuals(greco.m2))  # looks okay

summary(greco.m2)
# close to the estimates reported by Greco et al. (1995)
}