`bayesnec`

There are a range of models available in `bayesnec`

and
the working `bnec`

function supports individual model
fitting, as well as multi-model fitting with Bayesian model
averaging.

The argument `model`

in a `bayesnecformula`

is
a character string indicating the name(s) of the desired model (see
`?models`

for more details, and the list of models
available). If a recognised model name is provided, a single model of
the specified type is fit, and `bnec`

returns a model object
of class `bayesnecfit`

. If a vector of two or more of the
available models are supplied, `bnec`

returns a model object
of class `bayesmanecfit`

containing Bayesian model averaged
predictions for the supplied models, providing they were successfully
fitted.

Model averaging is achieved through a weighted sample of each fitted
models’ posterior predictions, with weights derived using the
`loo_model_weights`

function from `loo`

(Vehtari et al. 2020; Vehtari, Gelman, and Gabry
2017). Individual `brms`

model fits can be extracted
from the `mod_fits`

element and can be examined
individually.

The `model`

may also be one of `"all"`

, meaning
all of the available models will be fit; `"ecx"`

meaning only
models excluding the
$\eta = \text{NEC}$
step parameter will be fit; `"nec"`

meaning only models with
a specific
$\eta = \text{NEC}$
step parameter will be fit; `"bot_free"`

meaning only models
without a `"bot"`

parameter (without a bottom plateau) will
be fit; `"zero_bounded"`

are models that are bounded to be
zero; or `"decline"`

excludes all hormesis models, i.e., only
allows a strict decline in response across the whole predictor range
(see below **Parameter definitions**). There are a range of
other pre-defined model groups available. The full list of currently
implemented model groups can be seen using:

```
library(bayesnec)
models()
#> $nec
#> [1] "nec3param" "nec4param" "nechorme" "nechorme4" "necsigm" "neclin" "neclinhorme"
#> [8] "nechormepwr" "nechorme4pwr" "nechormepwr01"
#>
#> $ecx
#> [1] "ecx4param" "ecxlin" "ecxexp" "ecxsigm" "ecxwb1" "ecxwb2" "ecxwb1p3" "ecxwb2p3"
#> [9] "ecxll5" "ecxll4" "ecxll3" "ecxhormebc4" "ecxhormebc5"
#>
#> $all
#> [1] "nec3param" "nec4param" "nechorme" "nechorme4" "necsigm" "neclin" "neclinhorme"
#> [8] "nechormepwr" "nechorme4pwr" "nechormepwr01" "ecxlin" "ecxexp" "ecxsigm" "ecx4param"
#> [15] "ecxwb1" "ecxwb2" "ecxwb1p3" "ecxwb2p3" "ecxll5" "ecxll4" "ecxll3"
#> [22] "ecxhormebc4" "ecxhormebc5"
#>
#> $bot_free
#> [1] "nec3param" "nechorme" "necsigm" "neclin" "neclinhorme" "nechormepwr" "ecxlin"
#> [8] "ecxexp" "ecxsigm" "ecxwb1p3" "ecxwb2p3" "ecxll3" "ecxhormebc4" "nechormepwr01"
#>
#> $zero_bounded
#> [1] "nec3param" "nechorme" "necsigm" "nechormepwr" "nechormepwr01" "ecxexp" "ecxsigm"
#> [8] "ecxwb1p3" "ecxwb2p3" "ecxll3" "ecxhormebc4"
#>
#> $decline
#> [1] "nec3param" "nec4param" "neclin" "ecxlin" "ecxexp" "ecxsigm" "ecx4param" "ecxwb1" "ecxwb2"
#> [10] "ecxwb1p3" "ecxwb2p3" "ecxll5" "ecxll4" "ecxll3"
#>
#> $hormesis
#> [1] "nechorme" "nechorme4" "neclinhorme" "nechormepwr" "nechorme4pwr" "nechormepwr01" "ecxhormebc4"
#> [8] "ecxhormebc5"
```

Where possible we have aimed for consistency in the interpretable
meaning of the individual parameters across models. Across the currently
implemented model set, models contain from two (basic linear or
exponential decay, see **ecxlin** or
**ecxexp**) to five possible parameters
(**nechorme4**), including:

$\tau = \text{top}$, usually interpretable as either the y-intercept or the upper plateau representing the mean concentration of the response at zero concentration;

$\eta = \text{NEC}$,
the No-Effect-Concentration value (the x concentration value where the
breakpoint in the regression is estimated at, see **Model types
for NEC and EC_{x} estimation** and
(Fox 2010) for more details on parameter
based

$\beta = \text{beta}$,
generally the exponential decay rate of response, either from 0
concentration or from the estimated
$\eta$
value, with the exception of the **neclinhorme** model
where it represents a linear decay from
$\eta$
because slope
($\alpha$)
is required for the linear increase;

$\delta = \text{bottom}$, representing the lower plateau for the response at infinite concentration;

$\alpha = \text{slope}$,
the linear decay rate in the models **neclin** and
**ecxlin**, or the linear increase rate prior to
$\eta$
for all hormesis models;

$\omega$ = $\text{EC\textsubscript{50}}$ notionally the 50% effect concentration but may be influenced by scaling and should therefore not be strictly interpreted, and

$\epsilon = \text{d}$,
the exponent in the **ecxsigm** and
**necisgm** models.

$\phi = \text{f}$
A scaling exponent exclusive to model **ecxll5**.

In addition to the model parameters, all
**nec**-containing models have a step function used to
define the breakpoint in the regression, which can be defined as

$f(x_i, \eta) = \begin{cases} 0, & x_i - \eta < 0 \\ 1, & x_i - \eta \geq 0 \\ \end{cases}$

In principle all models provide an estimate for “no-effect” toxicity
concentration. As seen above, for model strings with
**nec** as a prefix, the NEC is directly estimated as
parameter
$\eta = \text{NEC}$
in the model, as per Fox (2010). On the
other hand, model strings with **ecx** as a prefix are
continuous curve models with no threshold, typically used for extracting
ECx values from concentration-response data. In this instance, the NEC
reported is actually the No-Significant-Effect-Concentration (NSEC, see details in Fisher and Fox 2023),
defined as the concentration at which there is a user supplied certainty
(based on the Bayesian posterior estimate) that the response falls below
the estimated value of the upper asymptote
($\tau = \text{top}$)
of the response (i.e., the response value is significantly lower than
that expected in the case of no exposure). The default value for this
NSEC proportion is 0.01, which corresponds to an alpha value (Type-I
error rate) of 0.01 for a one-sided test of significance. The NSEC
concept has been recently explored using simulation studies and case
study examples, and when combined with the NEC estimates of threshold
models within a model‐ averaging approach, can yield robust estimates of
N(S)EC and of their uncertainty within a single analysis framework (Fisher et al. 2023). Both NEC and NSEC can be
calculated from fitted models using the functions and . The model
averaged N(S)EC is automatically returned as part of the fitted model
for any that contains a combination of both and models. The significance
level used can be adjusted from the default value using .

*EC _{x}* estimates can be equally obtained from both

`"nec"`

and `"ecx"`

models.
`"ecx"`

models fitted to the same data as
`"nec"`

models (see the Comparing
posterior predictions) vignette for an example. However, we
recommend using `"all"`

models where `"nec"`

models can fit some
datasets better than `"ecx"`

models and the model averaging
approach will place the greatest weight for the outcome that best fits
the supplied data. This approach will yield There is ambiguity in the definition of *EC _{x}*
estimates from hormesis models—these allow an initial increase in the
response (see Mattson 2008) and include
models with the character string

`horme`

in their name—as
well as those that have no natural lower bound on the scale of the
response (models with the string `ecx`

function has arguments `hormesis_def`

and
`type`

, both character vectors indicating the desired
behaviour. For `hormesis_def = "max"`

,
`hormesis_def = "control"`

(the default) indicates that
`type = "relative"`

`type = "absolute"`

(the default)
`type = "direct"`

, a direct interpolation of the
response on the predictor is obtained.Models that have an exponential decay (most models with parameter
$\beta = \text{beta}$)
with no
$\delta = \text{bottom}$
parameter are 0-bounded and are not suitable for the Gaussian family, or
any family modelled using a `"logit"`

or `"log"`

link because they cannot generate predictions of negative response
values. Conversely, models with a linear decay (containing the string
**lin** in their name) are not suitable for modelling
families that are 0-bounded (Gamma, Poisson, Negative Binomial, Beta,
Binomial, Beta-Binomial) using an `"identity"`

link. These
restrictions do not need to be controlled by the user, as a call to
`bnec`

with `models = "all"`

in the formula will
simply exclude inappropriate models, albeit with a message.

Strictly speaking, models with a linear hormesis increase are not
suitable for modelling responses that are 0, 1-bounded (Binomial-, Beta-
and Beta-Binomial-distributed), however they are currently allowed in
`bayesnec`

, with a reasonable fit achieved through a
combination of the appropriate distribution being applied to the
response, and `bayesnec`

’s `make_inits`

function
which ensures initial values passed to `brms`

yield response
values within the range of the user-supplied response data.

The **ecxlin** model is a basic linear decay model,
given by the equation:
$y_i = \tau - e^{\alpha} x_i$
with the respective `brmsformula`

being

```
#> y ~ top - exp(slope) * x
#> top ~ 1
#> slope ~ 1
```

Because the model contains linear predictors it is not suitable for
0, 1-bounded data (i.e. Binomial and Beta families with an
`"identity"`

link function). As the model includes a linear
decline with concentration, it is also not suitable for 0,
`Inf`

bounded data (Gamma, Poisson, Negative Binomial with an
`"identity"`

link).

The **ecxexp** model is a basic exponential decay model,
given by the equation:
$y_i = \tau e^{-e^{\beta} x_i}$
with the respective `brmsformula`

being

```
#> y ~ top * exp(-exp(beta) * x)
#> top ~ 1
#> beta ~ 1
```

The model is 0-bounded, thus not suitable for Gaussian response data
or the use of a `"logit"`

or `"log"`

link
function.

The **ecxsigm** model is a simple sigmoidal decay model,
given by the equation:
$y_i = \tau e^{-e^{\beta} x_i^{e^\epsilon}}$
with the respective `brmsformula`

being

```
#> y ~ top * exp(-exp(beta) * x^exp(d))
#> d ~ 1
#> top ~ 1
#> beta ~ 1
```

The model is 0-bounded, thus not suitable for Gaussian response data
or the use of a `"logit"`

or `"log"`

link
function.

The **ecx4param** model is a 4-parameter sigmoidal decay
model, given by the equation:
$y_i = \tau + (\delta - \tau)/(1 + e^{e^{\beta} (\omega - x_i)})$
with the respective `brmsformula`

being

```
#> y ~ top + (bot - top)/(1 + exp((ec50 - x) * exp(beta)))
#> bot ~ 1
#> ec50 ~ 1
#> top ~ 1
#> beta ~ 1
```

The **ecxwb1** model is a 4-parameter sigmoidal decay
model which is a slight reformulation of the Weibull1 model of Ritz et al. (2016), given by the equation:
$y_i = \delta + (\tau - \delta) e^{-e^{e^{\beta} (x_i - \omega)}}$
with the respective `brmsformula`

being

```
#> y ~ bot + (top - bot) * exp(-exp(exp(beta) * (x - ec50)))
#> bot ~ 1
#> ec50 ~ 1
#> top ~ 1
#> beta ~ 1
```

The **ecxwb1p3** model is a 3-parameter sigmoidal decay
model which is a slight reformulation of the Weibull1 model of Ritz et al. (2016), given by the equation:
$y_i = {0} + (\tau - {0}) e^{-e^{e^{\beta} (x_i - \omega)}}$
with the respective `brmsformula`

being

```
#> y ~ 0 + (top - 0) * exp(-exp(exp(beta) * (x - ec50)))
#> ec50 ~ 1
#> top ~ 1
#> beta ~ 1
```

The model is 0-bounded, thus not suitable for Gaussian response data
or the use of a `"logit"`

or `"log"`

link
function.

The **ecxwb2** model is a 4-parameter sigmoidal decay
model which is a slight reformulation of the Weibull2 model of Ritz et al. (2016), given by the equation:
$y_i = \delta + (\tau - \delta) (1 - e^{-e^{e^{\beta} (x_i - \omega)}})$
with the respective `brmsformula`

being

```
#> y ~ bot + (top - bot) * (1 - exp(-exp(-exp(beta) * (x - ec50))))
#> bot ~ 1
#> ec50 ~ 1
#> top ~ 1
#> beta ~ 1
```

While very similar to the **ecxwb1** (according to Ritz et al. 2016), fitted
**ecxwb1** and **ecxwb2** models can differ
slightly.

The **ecxwb2p3** model is a 3-parameter sigmoidal decay
model, which is a slight reformulation of the Weibull2 model of Ritz et al. (2016), given by the equation:
$y_i = {0} + (\tau -{0}) (1 - e^{-e^{e^{\beta} (x_i - \omega)}})$
with the respective `brmsformula`

being

```
#> y ~ 0 + (top - 0) * (1 - exp(-exp(-exp(beta) * (x - ec50))))
#> ec50 ~ 1
#> top ~ 1
#> beta ~ 1
```

While very similar to the **ecxwb1p3** (according to Ritz et al. 2016), fitted
**ecxwb1p3** and **ecxwb2p3** models can
differ slightly. The model is 0-bounded, thus not suitable for Gaussian
response data or the use of a logit or log link function.

The **ecxll5** model is a 5-parameter sigmoidal
log-logistic decay model, which is a slight reformulation of the LL.5
model of Ritz et al. (2016), given by the
equation:
$y_i = \delta + (\tau - \delta) / (1 + e^{-e^{\beta} (x_i - \omega)})^{e^\phi}$
with the respective `brmsformula`

being

```
#> y ~ bot + (top - bot)/(1 + exp(exp(beta) * (x - ec50)))^exp(f)
#> bot ~ 1
#> top ~ 1
#> beta ~ 1
#> ec50 ~ 1
#> f ~ 1
```

The **ecxll4** model is a 4-parameter sigmoidal
log-logistic decay model which is a slight reformulation of the LL.4
model of Ritz et al. (2016), given by the
equation:
$y_i = \delta + (\tau - \delta)/ (1 + e^{e^{\beta} (x_i - \omega)})$
with the respective `brmsformula`

being

```
#> y ~ bot + (top - bot)/(1 + exp(exp(beta) * (x - ec50)))
#> bot ~ 1
#> top ~ 1
#> beta ~ 1
#> ec50 ~ 1
```

The **ecxll3** model is a 3-parameter sigmoidal
log-logistic decay model, which is a slight reformulation of the LL.3
model of Ritz et al. (2016), given by the
equation:
$y_i = 0 + (\tau - 0)/ (1 + e^{e^{\beta} (x_i - \omega)})$
with the respective `brmsformula`

being

```
#> y ~ 0 + (top - 0)/(1 + exp(exp(beta) * (x - ec50)))
#> top ~ 1
#> beta ~ 1
#> ec50 ~ 1
```

`"logit"`

or `"log"`

link
function.

The **ecxhormebc5** model is a 5 parameter log-logistic
model modified to accommodate a non-linear hormesis at low
concentrations. It has been modified from to the “Brain-Cousens” (BC.5)
model of Ritz et al. (2016), given by the
equation:
$y_i = \delta + (\tau - \delta + e^{\alpha} x)/ (1 + e^{e^{\beta} (x_i - \omega)})$
with the respective `brmsformula`

being

```
#> y ~ bot + (top - bot + exp(slope) * x)/(1 + exp(exp(beta) * (x - ec50)))
#> bot ~ 1
#> top ~ 1
#> beta ~ 1
#> ec50 ~ 1
#> slope ~ 1
```

The **ecxhormebc4** model is a 5-parameter log-logistic
model similar to the **exchormebc5** model but with a lower
bound of 0, given by the equation:
$y_i = 0 + (\tau - 0 + e^{\alpha} x)/ (1 + e^{e^{\beta} (x_i - \omega)})$
with the respective `brmsformula`

being

```
#> y ~ 0 + (top - 0 + exp(slope) * x)/(1 + exp(exp(beta) * (x - ec50)))
#> top ~ 1
#> beta ~ 1
#> ec50 ~ 1
#> slope ~ 1
```

`"logit"`

or `"log"`

link
function.

The **neclin** model is a basic linear decay model
equivalent to **ecxlin** with the addition of the
*NEC* step function, given by the equation:
$y_i = \tau - e^{\alpha} \left(x_i - \eta \right) f(x_i, \eta)$
with the respective `brmsformula`

being

```
#> y ~ top - exp(slope) * (x - nec) * step(x - nec)
#> top ~ 1
#> slope ~ 1
#> nec ~ 1
```

Because the model contains linear predictors it is not suitable for
0, 1-bounded data (Binomial and Beta distributions with
`"identity"`

link). As the model includes a linear decline
with concentration, it is also not suitable for 0, `Inf`

bounded data (Gamma, Poisson, Negative Binomial with
`"identity"`

link).

The **nec3param** model is a basic exponential decay
model equivalent to **ecxexp** with the addition of the
*NEC* step function, given by the equation:
$y_i = \tau e^{-e^{\beta} \left(x_i - \eta \right) f(x_i, \eta)}$
with the respective `brmsformula`

being

```
#> y ~ top * exp(-exp(beta) * (x - nec) * step(x - nec))
#> top ~ 1
#> beta ~ 1
#> nec ~ 1
```

For Binomial-distributed response data in the case of
`"identity"`

link this model is equivalent to that in Fox (2010). The model is 0-bounded, thus not
suitable for Gaussian response data or the use of a `"logit"`

or `"log"`

link function.

The **nec4param** model is a 3-parameter decay model
with the addition of the *NEC* step function, given by the
equation:
$y_i = \delta + (\tau - \delta) e^{-e^{\beta} \left(x_i - \eta \right) f(x_i, \eta)}$
with the respective `brmsformula`

being

```
#> y ~ bot + (top - bot) * exp(-exp(beta) * (x - nec) * step(x - nec))
#> bot ~ 1
#> top ~ 1
#> beta ~ 1
#> nec ~ 1
```

The **nechorme** model is a basic exponential decay
model with an *NEC* step function equivalent to
**nec3param**, with the addition of a linear increase prior
to
$\eta$,
given by the equation
$y_i = (\tau + e^{\alpha} x_i) e^{-e^{\beta} \left(x_i - \eta \right) f(x_i, \eta)}$
with the respective `brmsformula`

being

```
#> y ~ (top + exp(slope) * x) * exp(-exp(beta) * (x - nec) * step(x - nec))
#> top ~ 1
#> beta ~ 1
#> nec ~ 1
#> slope ~ 1
```

The **nechorme** model is a *hormesis* model
(Mattson 2008), allowing an initial
increase in the response variable at concentrations below
$\eta$.
The model is 0-bounded, thus not suitable for Gaussian response data or
the use of a `"logit"`

or `"log"`

link function.
In this case the linear version (**neclinhorme**) should be
used.

The **nechormepwr** model is a basic exponential decay
model with an *NEC* step function equivalent to
**nec3param**, with the addition of a power increase prior
to
$\eta$,
given by the equation:
$y_i = (\tau + x_i^{1/(1+e^{\alpha})}) e^{-e^{\beta} \left(x_i - \eta \right) f(x_i, \eta)}$
with the respective `brmsformula`

being

```
#> y ~ (top + x^(1/(1 + exp(slope)))) * exp(-exp(beta) * (x - nec) * step(x - nec))
#> top ~ 1
#> beta ~ 1
#> nec ~ 1
#> slope ~ 1
```

The **nechormepwr** model is a *hormesis* model
(Mattson 2008), allowing an initial
increase in the response variable at concentrations below
$\eta$.
The model is 0-bounded, thus not suitable for Gaussian response data or
the use of a `"logit"`

or `"log"`

link function.
Because the model can generate predictions > 1 it should not be used
for Binomial and Beta distributions with `"identity"`

link.
In this case the **nechromepwr01** model should be
used.

The **neclinhorme** model is a basic linear decay model
with an *NEC* step function equivalent to
**neclin**, with the addition of a linear increase prior to
$\eta$,
given by the equation:
$y_i = (\tau + e^{\alpha} x_i) - e^{\beta} \left(x_i - \eta \right) f(x_i, \eta)$
with the respective `brmsformula`

being.

```
#> y ~ (top + exp(slope) * x) - exp(beta) * (x - nec) * step(x - nec)
#> top ~ 1
#> beta ~ 1
#> nec ~ 1
#> slope ~ 1
```

The **neclinhorme** model is a *hormesis* model
(Mattson 2008), allowing an initial
increase in the response variable at concentrations below
$\eta$.
This model contains linear predictors and is not suitable for 0,
1-bounded data (Binomial and Beta distributions with
`"identity"`

link). As the model includes a linear decline
with concentration, it is also not suitable for 0, `Inf`

bounded data (Gamma, Poisson, Negative Binomial with
`"identity"`

link).

The **nechorme4** model is 4 parameter decay model with
an *NEC* step function equivalent to **nec4param**
with the addition of a linear increase prior to
$\eta$,
given by the equation:
$y_i = \delta + ((\tau + e^{\alpha} x_i) - \delta ) e^{-e^{\beta} \left(x_i - \eta \right) f(x_i, \eta)}$
with the respective `brmsformula`

being

```
#> y ~ bot + ((top + exp(slope) * x) - bot) * exp(-exp(beta) * (x - nec) * step(x - nec))
#> bot ~ 1
#> top ~ 1
#> beta ~ 1
#> nec ~ 1
#> slope ~ 1
```

The **nechorme4** model is a *hormesis* model
(Mattson 2008), allowing an initial
increase in the response variable at concentrations below
$\eta$.

The **nechorme4pwr** model is 4 parameter decay model
with an *NEC* step function equivalent to
**nec4param** with the addition of a power increase prior
to
$\eta$,
given by the equation:
$y_i = \delta + ((\tau + x_i^{1/(1+e^{\alpha})}) - \delta) e^{-e^{\beta} \left(x_i - \eta \right) f(x_i, \eta)}$
with the respective `brmsformula`

being

```
#> y ~ bot + ((top + x^(1/(1 + exp(slope)))) - bot) * exp(-exp(beta) * (x - nec) * step(x - nec))
#> bot ~ 1
#> top ~ 1
#> beta ~ 1
#> nec ~ 1
#> slope ~ 1
```

The **nechorme4pwr** model is a *hormesis* model
(Mattson 2008), allowing an initial power
increase in the response variable at concentrations below
$\eta$.
Because the model can generate predictions > 1 it should not be used
for Binomial and Beta distributions with `"identity"`

link.
In this case the **nechromepwr01** model should be
used.

The **nechormepwr01** model is a basic exponential decay
model with an *NEC* step function equivalent to
**nec3param**, with the addition of a power increase prior
to
$\eta$,
given by the equation:
$y_i = \left(\frac{1}{(1 + ((1/\tau)-1) e^{-e^{\alpha}x_i}}\right) e^{-e^{\beta} \left(x_i - \eta \right) f(x_i, \eta)}$
with the respective `brmsformula`

being

```
#> y ~ (1/(1 + ((1/top) - 1) * exp(-exp(slope) * x))) * exp(-exp(beta) * (x - nec) * step(x - nec))
#> top ~ 1
#> beta ~ 1
#> nec ~ 1
#> slope ~ 1
```

The **nechormepwr01** model is a *hormesis* model
(Mattson 2008), allowing an initial
increase in the response variable at concentrations below
$\eta$.
The model is 0-bounded, thus not suitable for Gaussian response data or
the use of a `"logit"`

or `"log"`

link function.
In this case the linear version (**neclinhorme**) should be
used.

The **necsigm** model is a basic exponential decay model
equivalent to **ecxlin** with the addition of the
*NEC* step function, given by the equation:
$y_i = \tau e^{-e^{\beta} ((x_i - \eta) f(x_i, \eta))^{e^\epsilon}f(x_i, \eta)}$
with the respective `brmsformula`

being

```
#> y ~ top * exp(-exp(beta) * (step(x - nec) * (x - nec))^exp(d) * step(x - nec))
#> top ~ 1
#> beta ~ 1
#> nec ~ 1
#> d ~ 1
```

The model is 0-bounded, thus not suitable for Gaussian response data
or the use of a `"logit"`

or `"log"`

link
function. In addition, there may be theoretical issues with combining a
sigmoidal decay model with an *NEC* step function because where
there is an upper plateau in the data the location of
$\eta$
may become ambiguous. Estimation of No-Effect-Concentrations using this
model are not currently recommended without further testing.

Fisher, Rebecca, and David R Fox. 2023. “Introducing the No
Significant Effect Concentration (NSEC).” Journal
Article. *Environmental Toxicology and Chemistry* 42 (9):
2019–28. https://doi.org/https://doi.org/10.1002/etc.5610.

Fisher, Rebecca, David R. Fox, Andrew P. Negri, Joost van Dam, Florita
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