Model definitions
ECx models
ecxlin
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 ~ 1Because 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).
ecxexp
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 ~ 1The model is 0-bounded, thus not suitable for Gaussian response data
or the use of a "logit" or "log" link
function.
ecxsigm
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 ~ 1The model is 0-bounded, thus not suitable for Gaussian response data
or the use of a "logit" or "log" link
function.
ecx4param
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 ~ 1ecxwb1
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 ~ 1ecxwb1p3
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 ~ 1The model is 0-bounded, thus not suitable for Gaussian response data
or the use of a "logit" or "log" link
function.
ecxwb2
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 ~ 1While very similar to the ecxwb1 (according to Ritz et al. 2016), fitted ecxwb1 and ecxwb2 models can differ slightly.
ecxwb2p3
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 ~ 1While 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.
ecxll5
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 ~ 1ecxll4
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 ~ 1ecxll3
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 ~ 1The model is 0-bounded, thus not suitable for Gaussian response data
or the use of a "logit" or "log" link
function.
ecxhormebc5
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 ~ 1ecxhormebc4
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 ~ 1The model is 0-bounded, thus not suitable for Gaussian response data
or the use of a "logit" or "log" link
function.
NEC models
neclin
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 ~ 1Because 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).
nec3param
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 ~ 1For 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.
nec4param
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 ~ 1nechorme
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 ~ 1The 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.
nechormepwr
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 ~ 1The 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.
neclinhorme
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 ~ 1The 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).
nechorme4
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 ~ 1The nechorme4 model is a hormesis model (Mattson 2008), allowing an initial increase in the response variable at concentrations below \(\eta\).
nechorme4pwr
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 ~ 1The 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.
nechormepwr01
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 ~ 1The 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.
necsigm
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 ~ 1The 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.