Inference

In Pumas, inference is performed after model fitting to obtain uncertainty estimates on the model's population-level parameters (fixed effects), which can then be used to derive standard errors and confidence intervals. This is accomplished by calling the infer function on the object returned by fit:

infer(fpm::FittedPumasModel; level=0.95, rethrow_error::Bool=false, sandwich_estimator::Bool=true) -> FittedPumasModelInference

infer(fpm::FittedPumasModel, bts::Pumas.Bootstrap; level=0.95)

infer(fpm::FittedPumasModel, alg::MarginalMCMC; level=0.95)

infer(fpm::FittedPumasModel, sir::SIR; level=0.95, ensemblealg = EnsembleThreads()) -> FittedPumasModelInference

The function can be specialized to behave differently by setting the second (optional) argument type to different values. Details of each specialization are given below.

The infer function accepts the following arguments:

• fpm: a fitted Pumas model, typically returned by fit.
• type: the second (optional) argument determines the inferential approach used. Details are given below.
• level: for confidence intervals, the confidence level to use. Typically, 0.95, corresponding to 95% confidence intervals.
• ...: further arguments, the meaning of which is dependent on the type of inference. Details are given below.

The function returns an object of type FittedPumasModelInference. Further functions can be called on this object, such as stderror to obtain the standard errors.

We now describe in more detail the different inferential approaches that can be used by specifying the type argument. The default is to use the asymptotic variance-covariance matrix obtained from the "robust" (also called "sandwich") estimator [8]. Other options include:

• Bootstrap-based inference
• Sampling importance resampling (SIR)

Inference from asymptotic variance-covariance matrices

This inferential approach is the default when no type is specified in infer:

infer(fpm)

The additional optional arguments are:

• rethrow_error: whether to throw an error when it is not possible to compute the variance-covariance matrix (e.g., because the observed information matrix is not numerically positive definite). The default is to return a failed FittedPumasModelInference object but not throw an error.

• sandwich_estimator: whether to use the "robust" variance-covariance estimator [8]. true to default. If false, then the inverse of the observed Fisher information (i.e., negative inverse Hessian of the log-likelihood) is used.

Note that the variance-covariance matrix can also be obtained directly by calling the vcov function directly on a fitted Pumas model (returned by fit).

Here is an example of obtaining confidence intervals by setting sandwich_estimator to false in order to not use the robust estimator:

my_infer = infer(fpm; sandwich_estimator = false)

The FittedPumasModelInference returned by infer has a show method that prints the estimates, standard errors and confidence intervals in the REPL. To extract this information in a DataFrame (e.g. for markdown reporting), the coeftable function can be used:

coeftable(fpm::FittedPumasModel) -> DataFrame

coeftable(vfpm::Vector{<:FittedPumasModel}) -> DataFrame

coeftable(pmi::FittedPumasModelInference; level = pmi.level) -> DataFrame

The returned DataFrame also has relative standard errors (RSEs) included.

Additionally, you can also use the correlation_diagnostic to print a list of parameter pairs with high, medium or low correlations:

correlation_diagnostic(
  pmi::FittedPumasModelInference;
  high_cor_threshold = 0.7,
  medium_cor_threshold = high_cor_threshold / 2,
)

fpmi = infer(fpm)
correlation_diagnostic(fpmi)

Inference using the robust variance-covariance estimator is consistent under heteroscedasticity of residuals, but relies on asymptotic normality of the maximum likelihood estimator. While this is often a reasonable approximation for the parameters associated with the mean of the response variable, it is often a poor approximation for the variance parameters. Furthermore, confidence intervals derived using this approach do not necessarily respects bounds on the parameters, and it is not unlikely for a parameter that by definition is strictly positive (such as a clearance, volume or variance) to have an associated confidence interval that contains negative values. For that reason, inference based on other approaches, such as Bootstrap and SIR, may be preferable in some situations.

Bootstrap-based inference

Bootstrap-based inference is obtained by calling the function Bootstrap and passing the returned object of type Bootstrap as the type argument of infer. The resulting function signature is:

infer(fpm, Bootstrap())

There are no additional keyword arguments to the infer function, but Bootstrap has a number of optional arguments that can be used to control the bootstrap procedure. It has the following signature:

Bootstrap(; rng=Random.default_rng, samples=200, stratify_by=nothing, ensemblealg=EnsembleThreads())

The type of bootstrap procedure used is nonparametric bootstrap, also called paired bootstrap [9]. Bootstrap datasets are obtained by resampling individuals from the original dataset with replacement (all records associated with the sampled individual are included), after stratifying by any covariates specified. The stratification ensures that the number of individuals in each stratum remains the same in each resampled dataset, which can help with the stability of the estimator. For example, if an analysis consists of two studies, a small study with rich sampling, and a large study with sparse sampling, stratifying on the study will ensure that each resampled dataset has the same number of subjects with rich sampling. The fitting algorithm is applied to each resampled dataset to obtain $N$ estimates for each parameter, where $N$ is the number of replicates specified in samples. This collection is treated as a random draw from the sampling distribution of the estimator for the parameters, and a nonparametric 95% confidence interval is obtained from the 2.5th and 97.5th quantiles of this sample.

Inference based on the nonparametric bootstrap approach is often considered more reliable than using the asymptotic variance-covariance matrix, but is computationally intensive. Unlike the latter approach, bounds on the parameters are respected (e.g. the confidence interval for a strictly positive parameter cannot contain negative values).

Here's an example with all options specified:

using Random

my_bootstrap = infer(
    fpm,
    Bootstrap(;
        rn = MersenneTwister(1234),
        sample = 500,
        stratify_by = [:sex, :trt],
        ensemblealg = EnsembleThreads(),
    ),
)

If any of the sample fits requires closer inspection or post-processing, it is possible to get a filtered vector of only the successful fits with the successful_fits function. The included fits are those that did not error. In this case, the appropriate function call would be successful_fits(my_bootstrap). Notice, that for the inference based on the bootstrap procedure to be valid, most fits should have been successful in the sense that an error was not thrown. If several of the fits failed it is important to try to get a better understanding of why that happened before using the results. The coeftable function will work on the returned object, and the confidence interval, relative standard errors, and standard errors will be bootstrap-based.

The bootstrap requires a FittedPumasModel; FittedPumasEMModel support is coming in a future version of Pumas.

Sampling importance resampling (SIR)

As an alternative to the nonparametric bootstrap procedures (which can be computationally intensive or unfeasible with small sample sizes), [10] suggests the sampling importance resampling (SIR) procedure for estimating the uncertainty. This can be obtained by calling the function SIR and passing the returned object of type SIR as the type argument (second optional positional argument) of infer:

SIR(; rng=Random.default_rng, samples, resamples)

Confidence intervals based on SIR respect bounds on the parameters (e.g. the confidence interval for a strictly positive parameter cannot contain negative values).

Here is an example of using how to use SIR in Pumas:

my_sir = infer(my_fit, SIR(; samples = 200, resamples = 70))

The coeftable function will work on the returned object, and the confidence interval, relative standard errors, and standard errors will be based on SIR.

Note that if the robust variance-covariance matrix has already been computed by a call to infer, it is possible to avoid recomputing it by calling infer again with the output of the first call to infer as the first argument, and the result from SIR as the second argument. For example:

my_inference = infer(fpm)
my_sir = infer(my_inference, SIR(; samples = 300, resamples = 100))

SIR requires a FittedPumasModel FittedPumasEMModel support is coming in a future version of Pumas.

Inference based on samples from the marginal likelihood using MarginalMCMC

Instead of using the SIR sampling scheme it is possible to use MarginalMCMC as a way to generate an inference output. It uses a marginal Hamiltonian Monte Carlo (HMC) based No-U-Turn Sampler (NUTS) algorithm where the subject-specific parameters are marginalized and only the population parameters are sampled. The generated chain is the same as would have been generated from

alg = MarginalMCMC()
(; model, data, kwargs) = fpm
param = coef(fpm)
result = fit(model, data, param, alg; kwargs.constantcoef)

but it is set up in a way where it can be used with the usual inference oriented functions such as vcov and stderror.

MarginalMCMC

alg = MarginalMCMC(marginal_alg = LaplaceI(), nsamples = 2000, nadapts = 1000)

Here is an example of using how to use SIR in Pumas:

my_marginalmcmc = infer(my_fit, MarginalMCMC())

The coeftable function will work on the returned object, and the confidence interval, relative standard errors, and standard errors will be based on the MarginalMCMC with a burn-in period.

MarginalMCMC requires a FittedPumasModel, FittedPumasEMModel support is coming in a future version of Pumas.

Calculating mean values and variance-covariance matrices from Bootstrap and SIR

With the output from an infer call using Bootstrap or SIR in hand, it is possible to get mean values of the sampled parameters and a variance-covariance matrix based on the same. If the output of the infer call was named infer_output, these quantities are simply produced using the following code:

mean(infer_output)
vcov(infer_output)

Extracting standard errors (stderror)

The stderror function is used to extract standard errors from either a FittedPumasModel object returned by fit or a FittedPumasModelInference object returned by infer:

stderror(f::AbstractFittedPumasModel) -> NamedTuple

When calling stderror on a infer object, the type of standard errors returned will depend on the options passed to infer. For instance, if nonparametric bootstrap inference was performed, then the standard errors will be bootstrap-based.

Calling stderror directly on a fitted Pumas model object will result in using the default arguments to infer, i.e. standard errors based on the robust variance-covariance matrix. They can be inefficient because it may result in the variance-covariance matrix being re-calculated unnecessarily it is therefore recommended to not call stderror directly on a fitted Pumas model object, but to call infer first and store the object, which is then passed to stderror.

Condition number

Many pharmacometricians use the condition number to gauge if a model is stable or not. While we do not necessarily recommend the condition number as a good measurement of this the number is available by calling the cond function with the inference output as input. This will calculate the condition number of the correlation matrix implied by the variance-covariance matrix. This can be changed to the condition number of the variance-covariance matrix by setting the keyword correlation to false.

cond(pmi::FittedPumasModelInference; correlation = true)