# Estimating Parameters of Pumas Models

Pumas can use the observational data of a Subject or 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.

## Defining Data for Estimation

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 dv and resp, the 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 dv from 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

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 FO, FOCE, LaplaceI, NaivePooled, and 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.

The argument 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 DefaultOptimizeFN(show_trace=false). See Optim for more details. Example syntax

myfit = fit(mymodel, drug_x_population, drug_x_parameters, Pumas.FOCE(),
optimize_fn=DefaultOptimizeFN(show_trace=false))

If 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())

The fit function will check if any gradients and throw an exception if any of the elements are exactly zero unless checkidentification is set to false. checkidentification currently requires using a PumasModel.

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 alg, abstol, and reltol. The default values for these are AutoVern7(Rodas5()), 1e-12, and 1e-8 respectively. See the DifferentialEquations.jl documentation for more details.

The return type of fit is a FittedPumasModel.

### Marginal Likelihood Approximations

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.

Only LaplceI() is supported when using a PumasEMModel.

### FittedPumasModel

The relevant fields of a FittedPumasModel are:

• model: the model used in the estimation process.
• data: the Population that was estimated.
• optim: the result returned by the optimizer
• approx: the marginal likelihood approximation that was used.
• param: the optimal parameters.

## Bayesian Estimation

Bayesian parameter estimation is performed by using the fit function as follows:

Distributions.fit(
model::PumasModel,
data::Population,
param::NamedTuple,
::BayesMCMC;
nsamples::Integer=10000,
progress = Base.is_interactive,
kwargs...
)
Info

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 0.8.

The arguments are:

• model: a PumasModel, either defined by the @model DSL or the function-based interface.
• data: a Population.
• param: a named tuple of parameters. Used as the initial condition for the sampler.
• The approx must be BayesMCMC().
• nsamples determines the number of samples taken along each chain.
• Extra args and kwargs are passed on to the internal simobs call and thus control the behavior of the differential equation solvers.

The result is a BayesMCMCResults type.

### BayesMCMCResults

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

## Maximum A Posteriori (MAP) Estimation

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 (model), a Population (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 MAP.