# 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 distributions with matching names 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 assumption of independence between the two.

## Maximum Likelihood Estimation

Maximum Likelihood Estimation (MLE) is performed using the fit function:

StatsAPI.fitMethod
fit(
model::PumasModel,
population::Population,
param::NamedTuple,
approx::Union{LikelihoodApproximation, MAP};
optim_alg = Optim.BFGS(linesearch = Optim.LineSearches.BackTracking(), initial_stepnorm = 1.0),
optim_options::NamedTuple = NamedTuple(),
optimize_fn = nothing,
constantcoef::Tuple = (),
checkidentification = true,
diffeq_options = NamedTuple())

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, LaplaceI, NaivePooled, and BayesMCMC. See the online documentation for more details about the different methods.

To control the optimization procedure for finding population parameters (fixed effects), use the optim_alg and optim_options keyword arguments. In previous versions of Pumas the argument optimize_fn was used, but is now discouraged and will be removed in a later version of Pumas. These options control the optimization all approx methods except BayesMCMC. The default optimization function uses the quasi-Newton routine BFGS method from the Optim package. It can be changed by setting the optim_alg to an algorithm implemented in Optim.jl as long as it does not use second order derivatives. Optimization specific options can be passed in using the optim_options keyword and has to be a NamedTuple with names and values that match the Optim.Options type. For example, the optimization trace can be disabled and the algorithm can be changed to L-BFGS by passing optim_alg=Optim.LBFGS(), optim_options = ;(show_trace=false) to fit. See Optim for more defails.

It is possible to fix one or more parameters of the fit by passing a Tuple of Symbols as the constantcoef argument with elements corresponding to the names of the fixed parameters, e.g. constantcoef=(:σ,).

When models include an @random block and fitting with NaivePooled is requested, it is required that the user sets all varibility parameters to zero with constantcoef such that these can be ignored in the optimization, e.g. constantcoef = (:Ω,) while overwriting the corresponding values in param with (; init_params..., Ω = zeros(3, 3)).

Parallelization of the optimization is supported for most estimation methods via the ensemble interface of DifferentialEquations.jl. Currently, the only supported options are:

• EnsembleThreads(): the default. Accelerate fits by using multiple threads.
• EnsembleSerial(): fit using a single thread.
• EnsembleDistributed(): fit by using multiple worker processes.

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 diffeq_options 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(autodiff=true)), 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 LaplaceI() is supported when using a PumasEMModel:

StatsAPI.fitMethod
fit(
model::PumasEMModel,
population::Population,
param::NamedTuple,
approx::Union{SAEM,LaplaceI};
rng = default_rng()
)

Fit the PumasEMModel to the dataset population using the initial values specified in param. The supported methods for the approx argument are currently SAEM and LaplaceI. See the online documentation for more details about these two methods.

When fitting with SAEM the variance terms of the random effects (Ω) and the dispersion parameters for the error model (σ) are initialized to the identity matrix or 1 as appropriate. They may also be specified. With SAEM, it is reccomended to choose an init larger than the true values to facilitate exploration of the parameter space and avoid getting trapped in local optima early. Currently LaplaceI fits with a PumasEMModel require Ω to be diagonal.

Options for ensemblealg are EnsembleSerial() and EnsembleThreads() (the default). The fit will be parallel if ensemblealg == EnsembleThreads(). SAEM imposes the additional requirement for threaded fits that either the rng used supports Future.randjump or that rng be a vector of rngs, one per thread. The default_rng() supports randjump().

### 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:

StatsAPI.fitMethod

Distributions.fit( model::PumasModel, data::Population, param::NamedTuple, alg::BayesMCMC, )

Samples from the posterior distribution of the model's population and subject-specific parameters given the observations in data using the Hamiltonian Monte Carlo (HMC) based No-U-Turn Sampler (NUTS) algorithm. The MCMC sampler is initialized starting from param for the population parameters. The subject-specific parameters are also apprioriately initialized. The Bayesian inference algorithm's options are passed in the alg struct. Run ?BayesMCMC for more details on the options that can be set.

Info

We use a NUTS sampler with the generalized no-U-turn termination criterion and multinomial sampling on an Hamiltonian system whose kinetic energy is specified with a diagonal metric (diagonal matrix with positive diagonal entries). For numerical integration of the Hamiltonian system, we use the ordinary leapfrog integrator. The adaptation is done using the Stan’s windowed adaptation routine with a default target acceptance ratio of 0.8.

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, allowing utilization of diagnostics and visualization tooling. This is discussed further in the Bayesian Workflow tutorial.

## 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, MAP(FOCE()))

The usual post-processing and diagnostics are available and are calculated according to the approximation wrapped in MAP.