# Models: Model objective functions In USEPA/CompTox-ToxCast-tcpl: ToxCast Data Analysis Pipeline

## 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

 ``` 1 2 3 4 5 6 7 8 9 10 11``` ```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,ν) be the Student's t-distribution with ν degrees of freedom, y[i] be the observed response at the ith observation, and μ[i] be the estimated response at the ith observation. We calculate z[i] as:

z[i] = (y[i] - μ[i])/e^σ

where σ is the scale term. Then the log-likelihood is:

sum_{i=1}^{n} [ln(t(z[i], 4)) - σ]

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 σ. The constant model value μ[i] for the ith observation is given by:

μ[i] = 0

`tcplObjCnst` calculates the likelyhood for a constant model at 0. The only parameter passed to `tcplObjCnst` by `p` is the scale term σ. The constant model value μ[i] for the ith observation is given by:

μ[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 (σ). The gain-loss model value μ[i] for the ith observation is given by:

g[i] = 1/(1 + 10^(ga - x[i])*gw)

l[i] = 1/(1 + 10^(x[i] - la)*lw)

μ[i] = tp*g[i]*l[i]

where x[i] is the log concentration for the ith 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 (σ). The gain-loss model value μ[i] for the ith observation is given by:

g[i] = 1/(1 + 10^(ga - x[i])*gw)

l[i] = 1/(1 + 10^(x[i] - la)*lw)

μ[i] = tp*g[i]*l[i]

where x[i] is the log concentration for the ith 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 (σ). The hill model value μ[i] for the ith observation is given by:

μ[i] = tp/(1 + 10^(ga - x[i])*gw)

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 (σ). The hill model value μ[i] for the ith observation is given by:

μ[i] = tp/(1 + 10^(ga - x[i])*gw)

where x[i] is the log concentration for the ith observation.

USEPA/CompTox-ToxCast-tcpl documentation built on May 5, 2019, 4:48 p.m.