DRModels {DoseFinding}  R Documentation 
Doseresponse model functions and gradients.
Below are the definitions of the model functions:
Emax model
f(d,\theta)=E_0+E_{max}\frac{d}{ED_{50}+d}
Sigmoid Emax Model
f(d,\theta)=E_0+E_{max}\frac{d^h}{ED^h_{50}+d^h}
Exponential Model
f(d,\theta)=E_0+E_1(\exp(d/\delta)1)
Beta model
f(d,\theta)=E_0+E_{max}B(\delta_1,\delta_2)(d/scal)^{\delta_1}(1d/scal)^{\delta_2}
here
B(\delta_1,\delta_2)=(\delta_1+\delta_2)^{\delta_1+\delta_2}/(\delta_1^{\delta_1}
\delta_2^{\delta_2})
and scal
is a fixed dose scaling parameter.
Linear Model
f(d,\theta)=E_0+\delta d
Linear in log Model
f(d,\theta)=E_0+\delta \log(d + off)
here off
is a fixed offset parameter.
Logistic Model
f(d, \theta) = E_0 + E_{\max}/\left\{1 + \exp\left[ \left(ED_{50}  d
\right)/\delta \right] \right\}
Quadratic Model
f(d,\theta)=E_0+\beta_1d+\beta_2d^2
Linear Interpolation model
The linInt model provides linear interpolation at the values
defined by the nodes vector. In virtually all situations the nodes
vector is equal to the doses used in the analysis. For example the
Mods
and the fitMod
function automatically
use the doses that are used in the context of the function call as
nodes. The guesstimates specified in the Mods
function
need to be the treatment effects at the active doses standardized to
the interval [0,1] (see the examples in the Mods
function).
emax(dose, e0, eMax, ed50)
emaxGrad(dose, eMax, ed50, ...)
sigEmax(dose, e0, eMax, ed50, h)
sigEmaxGrad(dose, eMax, ed50, h, ...)
exponential(dose, e0, e1, delta)
exponentialGrad(dose, e1, delta, ...)
quadratic(dose, e0, b1, b2)
quadraticGrad(dose, ...)
betaMod(dose, e0, eMax, delta1, delta2, scal)
betaModGrad(dose, eMax, delta1, delta2, scal, ...)
linear(dose, e0, delta)
linearGrad(dose, ...)
linlog(dose, e0, delta, off = 1)
linlogGrad(dose, off, ...)
logistic(dose, e0, eMax, ed50, delta)
logisticGrad(dose, eMax, ed50, delta, ...)
linInt(dose, resp, nodes)
linIntGrad(dose, resp, nodes, ...)
dose 
Dose variable 
e0 
For most models placebo effect. For logistic model leftasymptote parameter, corresponding to a basal effect level (not the placebo effect) 
eMax 
Beta Model: Maximum effect within doserange 
ed50 
Dose giving half of the asymptotic maximum effect 
h 
Hill parameter, determining the steepness of the model at the ED50 
e1 
Slope parameter for exponential model 
delta 
Exponential model: Parameter, controlling the convexity
of the model. 
delta1 
delta1 parameter for beta model 
delta2 
delta2 parameter for beta model 
b1 
first parameter of quadratic model 
b2 
second parameter of quadratic model (controls, whether model is convex or concave) 
resp 
Response values at the nodes for the linInt model 
off 
Offset value to avoid problems with dose=0 (treated as a fixed value, not estimated) 
scal 
Scale parameter (treated as a fixed value, not estimated) 
nodes 
Interpolation nodes for the linear interpolation for the linInt model (treated as a fixed value, not estimated) 
... 
Just included for convenience in the gradient functions, so that for example

The Emax model is used to represent monotone, concave doseresponse shapes. To distinguish it from the more general sigmoid emax model it is sometimes also called hyperbolic emax model.
The sigmoid Emax model is an extension of the (hyperbolic) Emax model by introducing an additional parameter h, that determines the steepness of the curve at the ed50 value. The sigmoid Emax model describes monotonic, sigmoid doseresponse relationships. In the toxicology literature this model is also called fourparameter loglogistic (4pLL) model.
The quadratic model is intended to capture a possible nonmonotonic doseresponse relationship.
The exponential model is intended to capture a possible sublinear or a convex doseresponse relationship.
The beta model is intended to capture nonmonotone doseresponse relationships and is more flexible than the quadratic model. The kernel of the beta model function consists of the kernel of the density function of a beta distribution on the interval [0,scal]. The parameter scal is not estimated but needs to be set to a value larger than the maximum dose. It can be set in most functions (‘fitMod’, ‘Mods’) via the ‘addArgs’ argument, when omitted a value of ‘1.2*(maximum dose)’ is used as default, where the maximum dose is inferred from other input to the respective function.
The linear in logdose model is intended to capture concave
shapes. The parameter off
is not estimated in the code but set to
a prespecified value. It can be set in most functions (‘fitMod’,
‘Mods’) via the ‘addArgs’ argument, when omitted a value of
‘0.01*(maximum dose)’ is used as default, where the maximum dose is
inferred from other input to the respective function.
The logistic model is intended to capture general monotone, sigmoid doseresponse relationships. The logistic model and the sigmoid Emax model are closely related: The sigmoid Emax model is a logistic model in log(dose).
The linInt model provids linear interpolation of the means at the doses. This can be used as a "nonparametric" estimate of the doseresponse curve, but is probably most interesting for specifying a "nonparametric" truth during planning and assess how well parametric models work under a nonparametric truth. For the function ‘Mods’ and ‘fitMod’ the interpolation ‘nodes’ are selected equal to the doselevels specified.
Response value for model functions or matrix containing the gradient evaluations.
MacDougall, J. (2006). Analysis of doseresponse studies  Emax model,in N. Ting (ed.), Dose Finding in Drug Development, Springer, New York, pp. 127–145
Pinheiro, J. C., Bretz, F. and Branson, M. (2006). Analysis of doseresponse studies  modeling approaches, in N. Ting (ed.). Dose Finding in Drug Development, Springer, New York, pp. 146–171
## some quadratic example shapes
quadModList < Mods(quadratic = c(0.5, 0.75, 0.85, 1), doses = c(0,1))
plot(quadModList)
## some emax example shapes
emaxModList < Mods(emax = c(0.02,0.1,0.5,1), doses = c(0,1))
plot(emaxModList)
## example for gradient
emaxGrad(dose = (0:4)/4, eMax = 1, ed50 = 0.5)
## some sigmoid emax example shapes
sigEmaxModList < Mods(sigEmax = rbind(c(0.05,1), c(0.15,3), c(0.4,8),
c(0.7,8)), doses = c(0,1))
plot(sigEmaxModList)
sigEmaxGrad(dose = (0:4)/4, eMax = 1, ed50 = 0.5, h = 8)
## some exponential example shapes
expoModList < Mods(exponential = c(0.1,0.25,0.5,2), doses=c(0,1))
plot(expoModList)
exponentialGrad(dose = (0:4)/4, e1 = 1, delta = 2)
## some beta model example shapes
betaModList < Mods(betaMod = rbind(c(1,1), c(1.5,0.75), c(0.8,2.5),
c(0.4,0.9)), doses=c(0,1), addArgs=list(scal = 1.2))
plot(betaModList)
betaModGrad(dose = (0:4)/4, eMax = 1, delta1 = 1, delta2 = 1, scal = 5)
## some logistic model example shapes
logistModList < Mods(logistic = rbind(c(0.5,0.05), c(0.5,0.15),
c(0.2,0.05), c(0.2,0.15)), doses=c(0,1))
plot(logistModList)
logisticGrad(dose = (0:4)/4, eMax = 1, ed50 = 0.5, delta = 0.05)
## some linInt shapes
genModList < Mods(linInt = rbind(c(0.5,1,1),
c(0,1,1), c(0,0,1)), doses=c(0,0.5,1,1.5))
plot(genModList)
linIntGrad(dose = (0:4)/4, resp=c(0,0.5,1,1,1), nodes=(0:4)/4)