constrOptim in the tcplFit function.
tcplObjCnst(p, resp)
tcplObjGnls(p, lconc, resp)
tcplObjHill(p, lconc, resp)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
$ith$ observation is given by:
$$\mu_{i} = 0$$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
$ith$ 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 $ith$
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 $ith$ 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 $ith$
observation.Let $t(z,\nu)$ be the Student's t-ditribution with $\nu$ degrees of freedom, $y[i]$ be the observed response at the $ith$ observation, and $\mu[i]$ be the estimated response at the $ith$ observation. We calculate $z[i]$ as: $$ z_{i} = \frac{y_{i} - \mu_{i}}{e^\sigma} $$ where $\sigma$ is the scale term. Then the log-likelyhood is: $$ \sum_{i=1}^{n} [ln(t(z_{i}, 4)) - \sigma] $$ Where $n$ is the number of observations.