This Howto provides instructions on how to address certain modeling
challenges and offers additional details and context for certain
features of the package. A final section provides a complete worked out
example. For a more general overview, please refer to the manual.
How to access scenario properties
The package provides a number of set*() functions to
modify scenario properties such as set_param() and
set_times(). Analogous set* functions are not
provided because, generally, it should not be necessary to retrieve data
from scenarios. However, it is possible (but not recommended) to access
all scenario settings via objects slots. A slot is the
name for an object attribute in R. The slots of an object can
be accessed by using the @ operator. It behaves similar to
the $ operator on named lists:
# Create a new scenario object
myscenario <- Lemna_Schmitt()
# Get model name
get_model(myscenario)
#> [1] "Lemna_Schmitt"
# Set a custom tag to identify the scenario
myscenario %>%
set_tag("Lab experiment #1") -> myscenario
# Get custom tag
get_tag(myscenario)
#> [1] "Lab experiment #1"
# The tag is also displayed when printing scenario properties
myscenario
#> 'Lemna_Schmitt' scenario
#> tag : Lab experiment #1
#> param: Emax=1, AperBM=1000, Kbm=1, P_Temp=0, MolWeight=390.4, k_phot_fix=1, k_phot_max=0.47, k_resp=0.05, k_loss=0, Tmin=8, Tmax=40.5, Topt=26.7, t_ref=25, Q10=2, k_0=3, a_k=5e-05, C_P=0.3, CP50=0.0043, a_P=1, KiP=101, C_N=0.6, CN50=0.034, a_N=1, KiN=604, BM50=176, mass_per_frond=1e-04, BMw2BMd=16.7
#> init : BM=0, E=1, M_int=0
#> endpt: BM, r
#> times: none
#> forcs: none
#> expsr: none
#> >> exposure series is empty
# Accessing scenario slots and their default values
myscenario@forcings.req # forcings required for effect calculations
#> [1] "temp" "rad"
myscenario@endpoints # available effect endpoints
#> [1] "BM" "r"
myscenario@control.req # are control runs required for effect calculation?
#> [1] TRUE
The previous example displays some of the default values of a
Lemna_Schmitt scenario. The set of available slots depends on
the model type and is documented in the package help. For instance,
scenario properties shared by all models are documented in the effect
scenario class:
# Call the help page of effect scenarios class
?scenarios
A scenario class inherits all slots from its ancestors. A notable
class which modifies simulation behavior and provides additional
scenario properties is Transferable: it provides
capabilities to consider biomass transfers at defined time points during
simulation. Details about its class slots and functionality are
described in the help pages.
# Call the help page of the biomass transfer class
?Transferable
Moving exposure windows
A moving time window is a computing technique used in data analysis.
Data is systematically analysed within a window of fixed length that
moves or slides through the data set, in our case an exposure
time-series. The window captures a subset of successive data points, and
as it moves through the data; simulations and effect calculations are
performed for each window.
effect() can report effect levels for all evaluated
exposure windows on demand:
# Derive effect levels of all exposure windows
metsulfuron %>%
set_window(length=7, interval=1) %>%
effect(max_only=FALSE)
#> # A tibble: 8 × 5
#> scenario BM r dat.start dat.end
#> <list> <dbl> <dbl> <dbl> <dbl>
#> 1 <LmnSchmS> 2.68e- 1 9.06e- 1 0 7
#> 2 <LmnSchmS> 2.65e- 1 8.95e- 1 1 8
#> 3 <LmnSchmS> 2.48e- 1 8.29e- 1 2 9
#> 4 <LmnSchmS> 2.05e- 1 6.69e- 1 3 10
#> 5 <LmnSchmS> 1.47e- 1 4.63e- 1 4 11
#> 6 <LmnSchmS> 7.39e- 2 2.23e- 1 5 12
#> 7 <LmnSchmS> 3.15e- 3 9.18e- 3 6 13
#> 8 <LmnSchmS> -6.66e-16 -1.78e-15 7 14
The resulting table describes how effect levels change when the
exposure window moves along the exposure series. It is also possible to
specify the marginal effect threshold of reported results (this prevents
overinterpretation of spurious effect levels originating from marginal
numerical errors introduced during simulation), as in the following
example:
# Only report effect levels larger than 1e-5 = 0.001%
metsulfuron %>%
set_window(length=7, interval=1) %>%
effect(max_only=FALSE, marginal_effect=1e-5)
#> # A tibble: 8 × 5
#> scenario BM r dat.start dat.end
#> <list> <dbl> <dbl> <dbl> <dbl>
#> 1 <LmnSchmS> 0.268 0.906 0 7
#> 2 <LmnSchmS> 0.265 0.895 1 8
#> 3 <LmnSchmS> 0.248 0.829 2 9
#> 4 <LmnSchmS> 0.205 0.669 3 10
#> 5 <LmnSchmS> 0.147 0.463 4 11
#> 6 <LmnSchmS> 0.0739 0.223 5 12
#> 7 <LmnSchmS> 0.00315 0.00918 6 13
#> 8 <LmnSchmS> 0 0 7 14
The effect over all moving windows can be visualized using
ggplot:
set.seed(123)
# Generate a random exposure series spanning 14 days
ts <- data.frame(time = 0:14,
conc = runif(15, min=0, max=0.1))
# Run EPx calculations for a window length of 7 days and a step size of 1 day
metsulfuron %>%
set_exposure(ts) %>%
set_window(length=7, interval=1) %>%
effect(max_only=FALSE) -> results
results
#> # A tibble: 8 × 5
#> scenario BM r dat.start dat.end
#> <list> <dbl> <dbl> <dbl> <dbl>
#> 1 <LmnSchmS> 0.000113 0.000327 0 7
#> 2 <LmnSchmS> 0.000115 0.000333 1 8
#> 3 <LmnSchmS> 0.000109 0.000317 2 9
#> 4 <LmnSchmS> 0.0000971 0.000283 3 10
#> 5 <LmnSchmS> 0.000104 0.000302 4 11
#> 6 <LmnSchmS> 0.000119 0.000346 5 12
#> 7 <LmnSchmS> 0.000136 0.000394 6 13
#> 8 <LmnSchmS> 0.000116 0.000337 7 14
# Create a plot of effects over time
library(ggplot2)
ggplot(results) +
geom_point(aes(dat.start, BM*100)) +
labs(x="Start of window (day)", y="Effect on biomass (%)")
The
effect plot shows the effect for the time point where each window
starts. Effects are not available, and therefore not plotted, for time
points where the window exceeds the simulated timeframe.
Fitting model parameters
Parameters of a model can be fitted (calibrated) against observed
effect data. To fit a scenario to observed effect data:
- Combine the necessary inputs (model, exposure time-series, effect
data).
- Fit the scenario for the given parameters and observed effect
data.
For the first step, two options are available:
When only one exposure level (e.g., one experimental treatment)
and corresponding effect dataset (effect data only contain observations
corresponding to one exposure scenario) are of interest, then a scenario
consisting of model parameters and exposure time-series can directly be
fitted to effect data using the calibrate()
function.
As a more complex and versatile option, scenarios and effect data
are combined into one or more calibration sets. Each
calibration set has exactly one scenario that describes the
parameters of the experiment. If more than one calibration set
is defined, scenario parameters can differ between set, but don’t have
to, A numerical weight can be assigned to each calibration set
to control its impact on the calculated error term and derived fitting
procedure. All calibration sets are then passed to the
calibrate() function.
The following describes an example which fits selected parameters of
the Lemna model by Schmitt et al. (2013) to observed
frond numbers from experiments with the herbicide
metsulfuron:
Option 1: direct calibration with one scenario and one effect
dataset
library(dplyr)
# No exposure in control scenario
exposure <- data.frame(time=0:14, conc=0)
# set k_phot_fix, k_resp and Emax to the defaults recommended in Klein et al. (2022)
# for Tier 2C studies. Set initial biomass to 12.0 (original data is in fronds
# rather than biomass, but for sake of simplicity, no conversion was performed).
control <- metsulfuron %>%
set_param(c(k_phot_fix=TRUE, k_resp=0, Emax=1)) %>%
set_init(c(BM=12)) %>%
set_exposure(exposure)
# `metsulfuron` is an example scenario based on Schmitt et al. (2013) and therefore
# uses the historical, superseded nomenclature by Schmitt et al. We recommend using
# the newer SETAC Lemna model by Klein et al. (2022) for current applications,
# see `Lemna_SETAC()` for details.
# Get control data from Schmitt et al. (2013)
obs_control <- Schmitt2013 %>%
filter(ID == "T0") %>%
select(t, BM=obs)
# Fit parameter `k_phot_max` to observed data: `k_phot_max` is a physiological
# parameter which is typically fitted to the control data
fit1 <- calibrate(
x = control,
par = c(k_phot_max = 1),
data = obs_control,
output = "BM",
method="Brent", # Brent is recommended for one-dimensional optimization
lower=0, # lower parameter boundary
upper=0.5 # upper parameter boundary
)
#> Calibration based on 1 data set
#> Fitted parameter: k_phot_max
#> Output variable: BM
#> Error function: Sum of squared errors
#> Testing: k_phot_max=0.190983005625053, Error: 7773063.37078419
#> Testing: k_phot_max=0.309016994374947, Error: 3773478.40651822
#> Testing: k_phot_max=0.381966011250105, Error: 32463.0450211169
#> Testing: k_phot_max=0.427050983124842, Error: 5007518.67666603
#> Testing: k_phot_max=0.364216527777377, Error: 595120.655295876
#> Testing: k_phot_max=0.399186938124422, Error: 350364.561310557
#> Testing: k_phot_max=0.384141480406814, Error: 12409.3333143064
#> Testing: k_phot_max=0.385559150655713, Error: 7053.80696240271
#> Testing: k_phot_max=0.386097811329519, Error: 6707.77702709609
#> Testing: k_phot_max=0.386028896259684, Error: 6698.81329814015
#> Testing: k_phot_max=0.386023805114469, Error: 6698.7746084863
#> Testing: k_phot_max=0.386024054599214, Error: 6698.77450749263
#> Testing: k_phot_max=0.386024043879953, Error: 6698.77450760104
#> Testing: k_phot_max=0.386024065318474, Error: 6698.77450776412
#> Testing: k_phot_max=0.386024054599214, Error: 6698.77450749263
#> Best fit: k_phot_max=0.386024054599214
fit1$par
#> k_phot_max
#> 0.3860241
# Update the scenario with fitted parameter and simulate it
fitted_growth <- control %>%
set_param(fit1$par)
sim_mean <- fitted_growth %>%
simulate() %>%
mutate(trial="control")
# Exposure and observations in long format for plotting
treatments <- exposure %>%
mutate(trial="control")
obs_mean <- obs_control %>%
mutate(trial="control") %>%
select(time=t, data=BM, trial)
# Plot results
plot_sd(
model_base = fitted_growth,
treatments = treatments,
rs_mean = sim_mean,
obs_mean = obs_mean
)
Option 2: Create a list of calibration sets and then fit
TK/TD model parameters on all datasets and exposure levels at the same
time:
# get all the trials and exposure series from Schmitt et al. (2013) and create
# a list of calibration sets
Schmitt2013 %>%
group_by(ID) %>%
group_map(function(data, key) {
exp <- data %>% select(t, conc)
obs <- data %>% select(t, BM=obs)
sc <- fitted_growth %>% set_exposure(exp)
caliset(sc, obs)
}) -> cs
# Fit parameters on results of all trials at once
fit2 <- calibrate(
cs,
par=c(EC50=0.3, b=4.16, P_up=0.005),
output="BM",
method="L-BFGS-B",
lower=c(0, 0.1, 0),
upper=c(1000, 10, 0.1),
verbose=FALSE
)
fit2$par
#> EC50 b P_up
#> 0.50704017 4.16140566 0.01367997
# Update the scenario with fitted parameter and simulate all trials
fitted_tktd <- fitted_growth %>%
set_param(fit2$par)
treatments <- Schmitt2013 %>% select(time=t, conc, trial=ID)
rs_mean <- simulate_batch(
model_base = fitted_tktd,
treatments = treatments
)
# Observations in long format for plotting
obs_mean <- Schmitt2013 %>%
select(time=t, data=obs, trial=ID)
# Plot results
plot_sd(
model_base = fitted_tktd,
treatments = treatments,
rs_mean = rs_mean,
obs_mean = obs_mean
)
The resulting scenario with fitted parameters shows a very good fit
with the observed effects from experiments.
After model parameters have been calibrated during model fitting
against observational data, the parameter estimates obtained can be
further evaluated by assessing the 95% confidence intervals for each
parameter through likelihood profiling.
First, the profiling is conducted using the
lik_profile() function. Then, the profiles can be
visualized using plot_lik_profile().
# using the calibration set and calibrated parameters from the previous Lemna
# Schmitt example, a likelihood profiling is done
# We set parameter boundaries to constrain the likelihood profiling within these
# boundaries (this is optional)
# conduct profiling
res <- lik_profile(x = cs, # the calibration set
par = fit2$par, # the parameter values after calibration
output = "BM", # the observational output against which to
# compare the model fit
bounds = list(EC50_int = list(0.1, 4),
b = list(1, 5),
P = list(0.0001, 0.2)))
#> Profiling: EC50
#> start param value: 0.507LL:-244.118
#> hit 95% Conf Region, changing direction
#> hit 95% Conf Region, end of profiling
#> Profiling: b
#> start param value: 4.161LL:-244.118
#> hit 95% Conf Region, changing direction
#> hit 95% Conf Region, end of profiling
#> Profiling: P_up
#> start param value: 0.014LL:-244.118
#> hit 95% Conf Region, changing direction
#> hit 95% Conf Region, end of profiling
# visualise profiling results
plot_lik_profile(res)
# access 95% confidence intervals of profiled parameters
res$EC50$confidence_interval
#> [1] 0.4970885 0.5156808
res$b$confidence_interval
#> [1] 3.541113 4.730779
Finally, relations between estimates of the calibrated parameters can
be visualized using a parameter space explorer. This supports
identifying potential parameter correlations and identifiability
issues.
The parameter space explorer draws random parameter sets from the 95%
confidence intervals obtained for each parameter during likelihood
profiling, and evaluates the fit (likelihood) of each of these parameter
sets by comparing the model with the original parameter set against the
model with the new, randomly draw set.
First, the parameter space exploration is conducted using the
explore_space() function. Then, the space can be visualized
using plot_lik_profile().
# Call the help page for more information about the parameter space explorer
?explore_space
# conduct space exploration
res_space <- explore_space(
x = cs,
par = fit2$par,
res = res, # output of the likelihood profiling function
output = "BM")
#> better optimum found in space explorer
# visualize the parameter space
plot_param_space(res_space)
#> Registered S3 method overwritten by 'GGally':
#> method from
#> +.gg ggplot2
Decrease assessment runtime
There are multiple ways to decrease the runtime of a simulation. One
option is to increase the maximum step length of the solver,
hmax. It is an optional argument which can be passed to
either simulate(), effect(), or
epx(). The larger hmax, the faster the
simulation generally completes. However, be careful: The larger
hmax, the greater the risk that simulation results will be
inaccurate and simulations may even fail due to numerical issues.
# Simulations with a maximum solver step length of hmax=0.01
metsulfuron %>%
set_times(0:7) %>%
simulate(hmax=0.1)
#> time BM E M_int C_int FrondNo
#> 1 0 50.00000 1 0.0000 0.0000000 500000.0
#> 2 1 52.15858 1 223.0745 0.2560989 521585.8
#> 3 2 52.43495 1 369.6344 0.4221192 524349.5
#> 4 3 52.33917 1 463.0261 0.5297392 523391.7
#> 5 4 52.17923 1 521.8927 0.5989176 521792.3
#> 6 5 52.00016 1 558.6268 0.6432808 520001.6
#> 7 6 51.81403 1 581.2187 0.6717006 518140.3
#> 8 7 51.63740 1 474.0716 0.5497473 516374.0
The calculation of EPx values may take a lot of time if one
or more of the following conditions apply:
- a large number of scenarios is assessed
- the simulated exposure time-series are very long
- moving exposure windows are considered which are very small in
comparison to the length of the exposure series
The epx() function is implemented in a way that it can
parallelize calculations by using more than one CPU. To enable parallel
processing, a command like the following needs to preceed the call to
epx():
future::plan(future::multisession)
If the computer running the calculation has n physical
CPU cores, then the time necessary to calculate EPx values will
(in the best case) decrease by the same factor. For more information on
how to make use of parallelization in R, please refer to the
future package:
vignette("future-1-overview", package="future")
Implementing custom models
The set of supported models can be extended by users as needed.
Experimental or one-time use models can be implemented in a user’s
script and inserted into the package’s workflow.
The starting point to add a new model is an implementation of the
model’s rules and dynamics in R code. In most cases, this will
be a code snippet which describes the model’s Ordinary Differential
Equations (ODE):
## An exemplary implementation of the GUTS-RED-SD TKTD model ##
# Model ODEs following the deSolve specification
sd_ode <- function(t, state, params) {
with(as.list(c(state, params)), {
dDw <- kd * (Cw(t) - Dw) # dDw/dt, scaled damage
dH <- kk * max(0, Dw - z) # dH/dt, cumulative hazard
list(c(dDw, dH)) # return derivatives
})
}
The previous example implements the model equations in a way that can
be processed by the deSolve package for numerical
integration. For a detailed description on how to use
deSolve, please refer to its vignette:
vignette("deSolve", package="deSolve")
With some additional scenario data such as initial state, output time
points, and model parameters, we are able to simulate our custom TKTD
model:
## Properties of a sample scenario ##
init <- c(Dw=0, H=0) # initial state
times <- 0:5 # output time points [0,5]
param <- c(kd=22, hb=0.01, z=0.5, kk=0.08) # model parameters
exp <- data.frame(time=0:5, # exposure time-series
conc=c(0,1,1,0.5,0.6,0.2))
# Create a linear interpolation of the exposure time-series
expf <- approxfun(x=exp$time, y=exp$conc, method="linear", rule=2)
# Extend parameter set by interpolated exposure series
paramx <- as.list(c(param, Cw=expf))
# Numerically solve the ODE
deSolve::ode(y=init, times=times, parms=paramx, func=sd_ode)
#> time Dw H
#> 1 0 0.0000000 0.000000000
#> 2 1 0.9545447 0.008266888
#> 3 2 0.9999995 0.048101596
#> 4 3 0.5227279 0.069837063
#> 5 4 0.5954544 0.073573877
#> 6 5 0.2181819 0.074888021
When we have made sure that our custom model can be simulated and
works as expected, we can continue with integrating it into the
package’s workflow. First, we will define a new scenario class that
identifies our custom model by a unique name. Next, we have to tell the
package how the models of this class can be simulated. In a final step,
we may have to describe how effects are calculated for these models.
We start by defining a new scenario class that derives from a
suitable ancestor. In general, we can inherit from the base class
EffectScenario which provides the general scenario
capabilities. However, if the custom model is just a variant of an
existing model category, then it will be easier to derive from
specialized scenario class that already provides certain features such
as effect calculation. The following class tree gives an overview of the
scenario classes in use:
EffectScenario
|
| Transferable
|___|__ Lemna
| | |__ LemnaSchmittScenario
| | |__ LemnaSetacScenario
| |
| |__ Myriophyllum
| | |__ MyrioExpScenario
| | |__ MyrioLogScenario
| |
| |__ Algae
| |__ AlgaeWeberScenario
| |__ AlgaeTKTDScenario
| |__ AlgaeSimpleScenario
|
|__ GutsRedSd
|__ GutsRedIt
|
|__ DebScenario
|__ DebAbj
|__ DebTox
To give an example, to implement a variant of a Lemna model,
it would be advisable to derive from the class Lemna to
benefit from already implemented features such as effect endpoint
calculation and simulation of biomass transfers. For the custom
GUTS-RED-SD model, the ideal choice would be to derive from
GutsRedSd to minimize the implementation overhead. However,
we will derive from the general EffectScenario class for
the sake of our example and create a scenario object:
## Integrate a new model class into the package workflow ##
# Create a unique class that derives from 'EffectScenario'
setClass("MyGuts", contains="EffectScenario")
# Create an object of the new class and assign scenario properties
new("MyGuts", name="My custom model") %>%
set_init(init) %>%
set_times(times) %>%
set_param(param) %>%
set_exposure(exp, reset_times=FALSE) %>%
set_endpoints("L") -> myscenario
myscenario
#> 'My custom model' scenario
#> param: kd=22, hb=0.01, z=0.5, kk=0.08
#> init : Dw=0, H=0
#> endpt: L
#> times: [0,5] n=6, regular
#> forcs: none
#> expsr: none
#> time conc
#> 1 0 0.0
#> 2 1 1.0
#> 3 2 1.0
#> 4 3 0.5
#> 5 4 0.6
#> 6 5 0.2
The object myscenario now carries all the settings and
data which we also used for our test simulation. In the next step, the
solver() function will be overloaded to handle objects of
the newly defined scenario class MyGuts. The adapted
solver() function will collect the properties of a given
scenario object, call the ODE solver, and return simulation results:
# the actual function calling deSolve can have a different signature
solver_myguts <- function(scenario, times, ...) {
# overriding output times by function argument must be possible
if(missing(times))
times <- scenario@times
# get relevant data from scenario
init <- scenario@init
param <- scenario@param
exp <- scenario@exposure@series
if(nrow(exp) == 1) { # extend exposure series to have at least two rows
row2 <- exp[1,]
row2[[1]] <- row2[[1]]+1
exp <- rbind(exp, row2)
}
# Create a linear interpolation of the exposure time-series
expf <- approxfun(x=exp[,1], y=exp[,2], method="linear", rule=2)
# Extend parameter set by interpolated exposure series
paramx <- as.list(c(param, Cw=expf))
as.data.frame(deSolve::ode(y=init, times=times, parms=paramx, func=sd_ode, ...))
}
## Overload the solver() function ##
# The functions signature, i.e. the number and names of its arguments, must stay as is
setMethod("solver", "MyGuts", function(scenario, times, ...) solver_myguts(scenario, times, ...))
Overloading an S4 function is done using
setMethod(). The first argument to setMethod()
identifies which function we want to overload, in this case it is
solver. The second argument, MyGuts, defines
which object type our overloaded function will accept. In this case, we
want to provide an implementation to simulate MyGuts
scenarios. R will decide during runtime which of the function
candidates to execute when solver() is called based on the
type of the first argument to solver(). The third argument,
function(object, ...) solver_myguts(scenario=object, ...)
forwards calls to an appropriate function which can handle
MyGuts objects. The function signature
function(object, ...) must stay exactly as it is and must
not be modified.
The code within the body of solver_myguts() is almost
identical to the original code used to simulate the prototyped model. We
have to deal with one corner case, though: some exposure time-series
will contain only a single row, representing constant exposure over
time, but this would raise an error in approxfun(). In
order to avoid this error, we will duplicate the first row and append it
to the series. Technically, this issue should be handled in the
prototyped code as well but was left out for reasons of brevity.
The parameterized scenario object myscenario of class
MyGuts can be passed to the overloaded
simulate() function:
myscenario %>% simulate()
#> time Dw H
#> 1 0 0.0000000 0.000000000
#> 2 1 0.9545447 0.008266888
#> 3 2 0.9999995 0.048101596
#> 4 3 0.5227279 0.069837063
#> 5 4 0.5954544 0.073573877
#> 6 5 0.2181819 0.074888021
The custom model can now be simulated using the framework but it is
still missing effect endpoints. If, on the other hand, we had decided to
inherit our scenario class from a suitable ancestor, such as
GutsRedSd, we could have stopped at this point as the
ancestor would provide routines to calculate effects.
By default, any state variable is available as an effect endpoint and
the calculated effect would reflect the change in the state variables’
value at the end of the simulation. However, we need to calculate the
survival endpoint in a different manner for GUTS-RED type
models. To implement this specialized endpoint, we need to overload the
function fx() which calculates effect endpoints:
## Overload effect endpoint calculation ##
# fx() is called by effect()
setMethod("fx", "MyGuts", function(scenario, ...) fx_myguts(scenario, ...))
# @param scenario Scenario object to assess
# @param window Start & end time of the moving exposure window
# @param ... any additional parameters
fx_myguts <- function(scenario, window, ...) {
# simulate the scenario (it is already clipped to the moving exposure window)
out <- simulate(scenario, ...)
# only use model state at the end of the simulation
out <- tail(out, 1)
# calculate survival according to EFSA Scientific Opinion on TKTD models
# p. 33, doi:10.2903/j.efsa.2018.5377
t <- unname(window[2] - window[1])
survival <- exp(-out$H) * exp(-scenario@param$hb * t)
return(c("L"=survival))
}
# Derive effect levels for our sample scenario
myscenario %>% effect()
#> # A tibble: 1 × 4
#> scenario L L.dat.start L.dat.end
#> <list> <dbl> <dbl> <dbl>
#> 1 <MyGuts> 0.0722 0 5
fx() is used by effect() to derive effect
endpoints for each model. By convention, any overload of the
fx() function must return a named numerical vector
containing the effect endpoints for the provided scenario and moving
exposure window. The scenario will already be parameterized to only
simulate the current exposure window. The return value of
fx() will then be used to calculate effect levels. In case
of models requiring a control scenario, the effect level will be
calculated as 1 - effect/control. For models not requiring
a control, the overloaded fx() must return the final effect
level, i.e. a value from the interval [0,1] (0% to 100%
effect).
