| Models {tcpl} | R Documentation |
Model objective functions
Description
These functions take in the dose-response data and the model parameters, and
return a likelihood value. They are intended to be optimized using
constrOptim in the tcplFit function.
Usage
tcplObjCnst(p, resp)
tcplObjGnls(p, lconc, resp)
tcplObjHill(p, lconc, resp)
tcplObjCnst(p, resp)
tcplObjGnls(p, lconc, resp)
tcplObjHill(p, lconc, resp)
Arguments
p |
Numeric, the parameter values. See details for more information. |
resp |
Numeric, the response values |
lconc |
Numeric, the log10 concentration values |
Details
These functions produce an estimated value based on the model and given parameters for each observation. Those estimated values are then used with the observed values and a scale term to calculate the log-likelihood.
Let t(z,\nu) be the Student's t-distribution with \nu degrees of
freedom, y_{i} be the observed response at the i^{th}
observation, and \mu_{i} be the estimated response at the i^{th}
observation. We calculate z_{i} as:
z_{i} = \frac{y_{i} - \mu_{i}}{e^\sigma}
where \sigma is the scale term. Then the log-likelihood is:
\sum_{i=1}^{n} [ln(t(z_{i}, 4)) - \sigma]
Where n is the number of observations.
Value
The log-likelihood.
Constant Model (cnst)
tcplObjCnst calculates the likelyhood for a constant model at 0. The
only parameter passed to tcplObjCnst by p is the scale term
\sigma. The constant model value \mu_{i} for the
i^{th} observation is given by:
\mu_{i} = 0
tcplObjCnst calculates the likelyhood for a constant model at 0. The
only parameter passed to tcplObjCnst by p is the scale term
\sigma. The constant model value \mu_{i} for the
i^{th} observation is given by:
\mu_{i} = 0
Gain-Loss Model (gnls)
tcplObjGnls calculates the likelyhood for a 5 parameter model as the
product of two Hill models with the same top and both bottoms equal to 0.
The parameters passed to tcplObjGnls by p are (in order) top
(\mathit{tp}), gain log AC50 (\mathit{ga}), gain hill coefficient (gw),
loss log AC50 \mathit{la}, loss hill coefficient \mathit{lw}, and the scale
term (\sigma). The gain-loss model value \mu_{i} for the
i^{th} observation is given by:
g_{i} = \frac{1}{1 + 10^{(\mathit{ga} - x_{i})\mathit{gw}}}
l_{i} = \frac{1}{1 + 10^{(x_{i} - \mathit{la})\mathit{lw}}}
\mu_{i} = \mathit{tp}(g_{i})(l_{i})
where x_{i} is the log concentration for the i^{th}
observation.
tcplObjGnls calculates the likelyhood for a 5 parameter model as the
product of two Hill models with the same top and both bottoms equal to 0.
The parameters passed to tcplObjGnls by p are (in order) top
(\mathit{tp}), gain log AC50 (\mathit{ga}), gain hill coefficient (gw),
loss log AC50 \mathit{la}, loss hill coefficient \mathit{lw}, and the scale
term (\sigma). The gain-loss model value \mu_{i} for the
i^{th} observation is given by:
g_{i} = \frac{1}{1 + 10^{(\mathit{ga} - x_{i})\mathit{gw}}}
l_{i} = \frac{1}{1 + 10^{(x_{i} - \mathit{la})\mathit{lw}}}
\mu_{i} = \mathit{tp}(g_{i})(l_{i})
where x_{i} is the log concentration for the i^{th}
observation.
Hill Model (hill)
tcplObjHill calculates the likelyhood for a 3 parameter Hill model
with the bottom equal to 0. The parameters passed to tcplObjHill by
p are (in order) top (\mathit{tp}), log AC50 (\mathit{ga}), hill
coefficient (\mathit{gw}), and the scale term (\sigma). The hill model
value \mu_{i} for the i^{th} observation is given
by:
\mu_{i} = \frac{tp}{1 + 10^{(\mathit{ga} - x_{i})\mathit{gw}}}
where x_{i} is the log concentration for the i^{th}
observation.
tcplObjHill calculates the likelyhood for a 3 parameter Hill model
with the bottom equal to 0. The parameters passed to tcplObjHill by
p are (in order) top (\mathit{tp}), log AC50 (\mathit{ga}), hill
coefficient (\mathit{gw}), and the scale term (\sigma). The hill model
value \mu_{i} for the i^{th} observation is given
by:
\mu_{i} = \frac{tp}{1 + 10^{(\mathit{ga} - x_{i})\mathit{gw}}}
where x_{i} is the log concentration for the i^{th}
observation.