# 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.

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 name 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 model should define distributions with matching names. 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::PumasModel,
data::Population,
param::NamedTuple,
approx::LikelihoodApproximation;
optimize_fn = DEFAULT_OPTIMIZE_FN,
constantcoef::NamedTuple = NamedTuple(),
omegas::Tuple = tuple(),
ensemblealg::DiffEqBase.EnsembleAlgorithm = EnsembleSerial(),
checkidentification=true,
kwargs...))

Fit the Pumas model model to the dataset population with starting values param using the estimation method approx. Currently supported values for the approx argument are FO, FOCE, FOCEI, LaplaceI, TwoStage, NaivePooled, and BayesMCMC. See the online documentation for more details about the different methods.

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 defails.

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,).

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,).

Parallelization of the optimization is supported for most estimation methods via the ensemble interface of DifferentialEquations.jl. The default is EnsembleSerial(). Currently, the only supported parallelization for model fitting is EnsembleThreads().

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.

Further keyword arguments can be passed via the kwargs... argument. This allows for passing 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.
• FOCEI(): first order conditional estimation with interaction.
• LaplaceI(): second order Laplace approximation with interaction.

### 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