Pumas can use the observational data of a
Population to estimate the parameters in many types of models. This is done by two classes of methods. First, maximum likelihood methods find the parameters such that the observational data has the highest probability of occurring according to the chosen error distributions. Second, bayesian methods find a posterior probability distribution for the parameters to describe the chance that a parameter has a given value given the data. The following section describes how to fit an NLME model in Pumas via the two methods.
- Maximum likelihood methods find the parameters such that the observational data has the highest probability of occurring according to the chosen error distributions. - Bayesian methods find a posterior probability distribution for the parameters to describe the chance that a parameter has a given value given the data. - Maximum A Posteriori (MAP) can be considered as something in between maximum likelihood estimation and Bayesian inference. It incorporates priors of some or all variables and can be interpreted as penalized maximum likelihood or finding the mode of the posterior distribution in a Bayesian context.
The following sections describe how to fit a NLME model in Pumas via the two methods.
The observed data should be parsed using the same names as those found in the model. For example, if
subject.observations is a
NamedTuple with names
derived block or function in the
PumasModel should define distributions with matching names. Similarly, the
error block should define distrubtions with matching named if using a
PumasEMModel instead. If
dv is a scalar in the observation data, then
derived should also be a scalar. Likewise, if
dv is an array like a time series, then
dv should be a size-matching time series when returned from
derived. The likelihood of observing multiple dependent variables is calculated under the assumtion of independence between the two.
Maximum Likelihood Estimation (MLE) is performed using the
fit function. This function's signature is:
Distributions.fit(model::AbstractPumasModel, data::Population, param::NamedTuple, approx::LikelihoodApproximation; optimize_fn = DEFAULT_OPTIMIZE_FN, constantcoef::NamedTuple = NamedTuple(), omegas::Tuple = tuple(), ensemblealg::DiffEqBase.EnsembleAlgorithm = EnsembleSerial(), checkidentification=true, diffeq_options)
Fit the Pumas model
model to the dataset
data with starting values
param using the estimation method
approx. Currently supported values for the
approx argument are
BayesMCMC. See the bottom of this page for how to do Maximum A Posteriori (MAP) estimation based on these likelihood approximations. Example fit syntax
myfit = fit(mymodel, drug_x_population, drug_x_parameters, Pumas.FOCE())
Note that the
approx has to be qualified with
Pumas and has to end with the empty parenthesis.
optimize_fn is used for optimizing the objective function for all
approx methods except
BayesMCMC. The default optimization function uses the quasi-Newton routine
BFGS method from the
Optim package. Optimization specific arguments can be passed to
DefaultOptimizeFN, e.g. the optimization trace can be disabled by passing
Optim for more details. Example syntax
myfit = fit(mymodel, drug_x_population, drug_x_parameters, Pumas.FOCE(), optimize_fn=DefaultOptimizeFN(show_trace=false))
model is a
PumasModel, it is possible to fix It is possible to fix one or more parameters of the fit by passing a
NamedTuple as the
constantcoef argument with keys and values corresponding to the names and values of the fixed parameters, e.g.
constantcoef=(σ=0.1,). Example syntax
myfit = fit(mymodel, drug_x_population, drug_x_parameters, Pumas.FOCE(), constantcoef = (tvcl = 4, ), optimize_fn=DefaultOptimizeFN(show_trace=false))
When models include an
@random block and fitting with
NaivePooled is requested, it is required that the user supplies the names of the parameters of the random effects as the
omegas argument such that these can be ignored in the optimization, e.g.
omegas=(Ω,). Example syntax
myfit = fit(mymodel, drug_x_population, drug_x_parameters, Pumas.FOCEI(), omegas=(Ω,), optimize_fn=DefaultOptimizeFN(show_trace=false))
Parallelization of the optimization is supported for most estimation methods via the ensemble interface of DifferentialEquations.jl. The default is
EnsembleThreads(). This is currently the only supported parallelization for model fitting. It is also possible to ask for a non-parallelized fit by setting the keyword to
EnsembleSerial(). Example fit syntax for this is
myfit = fit(mymodel, drug_x_population, drug_x_parameters, Pumas.FOCEI(), ensemblealg = EnsembleSerial())
fit function will check if any gradients and throw an exception if any of the elements are exactly zero unless
checkidentification is set to
checkidentification currently requires using a
Keyword arguments that control the differential equation solver can be passed via the
diffeq_options argument. This accepts a
NamedTuple with arguments to the differential equations solver such as
reltol. The default values for these are
1e-8 respectively. See the DifferentialEquations.jl documentation for more details.
The return type of
fit is a
The following choices are available for the likelihood approximations:
FO(): first order approximation.
FOCE(): first order conditional estimation (with automatic detection for when to include the interaction terms).
LaplaceI(): second order Laplace approximation with interaction.
LaplceI() is supported when using a
The relevant fields of a
modelused in the estimation process.
Populationthat was estimated.
optim: the result returned by the optimizer
approx: the marginal likelihood approximation that was used.
param: the optimal parameters.
Bayesian parameter estimation is performed by using the
fit function as follows:
Distributions.fit( model::PumasModel, data::Population, param::NamedTuple, ::BayesMCMC; nadapts::Integer=2000, nsamples::Integer=10000, progress = Base.is_interactive, kwargs... )
We use a
NUTS sampler with the generalized no-U-turn termination criterion and multinomial sampling on Hamiltonian system
whose kinetic energy is specified with a diagonal metric (diagonal matrix with positive diagonal entries). For numerical intergation of the Hamiltonian system, we use the ordinary leapfrog integrator. The adaptation is done
using the Stan’s windowed adaptation routine with a target acceptance ratio of
The arguments are:
PumasModel, either defined by the
@modelDSL or the function-based interface.
param: a named tuple of parameters. Used as the initial condition for the sampler.
nsamplesdetermines the number of samples taken along each chain.
kwargsare passed on to the internal
simobscall and thus control the behavior of the differential equation solvers.
The result is a
The MCMC chain is stored in the
chain field of the returned
BayesMCMCResults. Additionally the result can be converted into a
Chains object from MCMCChains.jl, allowing utlilization of diagnostics and visualization tooling. This is discussed further in the Bayesian Estimation tutorial.
The following functions help with querying common results on the Bayesian posterior:
param_mean(br): returns a named tuple of parameters which represents the mean of each parameter's posterior distribution
param_var(br): returns a named tuple of parameters which represents the variance of each parameter's posterior distribution
param_std(br): returns a named tuple of parameters which represents the standard deviation of each parameter's posterior distribution
MAP estimation is carried out using the
fit function and by wrapping the chosen likelihood approximation in the
MAP type. The
fit signature is the same as for maximum likelihood. If we have a model (
data), and an initial parameter specification (
param) we can carry out the MAP estimation with the FOCE approximation of the log-likelihood using
result = fit(model, data, param, Pumas.MAP(Pumas.FOCE()))
The usual post-processing and diagnostics are available and are calculated according to the approximation wrapped in