# Inference

In Pumas, inference is performed after model fitting to obtain uncertainty estimates on the model's population-level parameters (fixed effects), which can the be used to derive standard errors and confidence intervals. This is accomplished by calling the infer function on the object returned by fit (an object of type FittedPumasModel or FittedPumasEMModel). The infer function, described below, has several options to control how the inference is performed.

The general signature of the infer function is:

infer(fpm::AbstractFittedPumasModel[, type]; level=0.95, [,type_specific...])

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 FittedPumasModel or FittedPumasEMModel, 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 Halbert White (1982). Other options include:

### Inference from assymptotic variance-covariance matrices

This inferential approach is the default when no type is specified in infer, which then has the following signature:

infer(fpm::FittedPumasModel; level=0.95, rethrow_error=false, sandwich_estimator=true)

• 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 or not to use the "robust" variance-covariance estimator Halbert White (1982). 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 FittedPumasModel object.

Here is an example of obtaining confidence intervals based on robust standard errors:

my_infer = infer(my_fit)

Or, setting sandwich_estimator to false in order to not use the robust estimator:

my_infer = infer(my_fit; 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(my_infer)   # Returns a DataFrame

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 Pumas.Bootstrap and Pumas.SIR, may be preferable in some situations.

### Bootstrap-based inference

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

infer(fpm::FittedPumasModel, type::Bootstrap; level=0.95)

There are no additional keyword arguments to the infer function, but Pumas.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 keyword arguments are as follows:

• rng: used to select a pseudo-random number generator to be used within the bootstrap procedure.

• samples: the number of bootstrap replicates to generate.

• stratify_by: covariate(s) to stratify by when resampling, specified as a single Symbol or an array of Symbols. These must refer to covariates that were declared when the Subjects were created, e.g. by read_pumas. These should be discrete (categorical, not continuous) covariates. In general, the number of strata should not be too large, and each stratum should contain a sufficient number of individuals for resampling to make sense, though it is difficult to give specific guidance.

• ensemblealg: A BasicEnsembleAlgorithm to use for parallel execution. When running Pumas in a distributed setting, it can be beneficial to specify EnsembleThreads when fitting the model and then EnsembleDistributed for the bootstrap. Thereby, the different bootstrap samples will be distributed across Julia processes and each associated model fitting will benefit from multi-threaded execution.

The type of bootstrap procedure used is nonparametric bootstrap, also called paired bootstrap Bradley Efron (1979). 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 that 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 is an example of how to run a nonparametric bootstrap in Pumas using the default options:

my_bootstrap = infer(my_fit, Pumas.Bootstrap())

An here, an example with all options specified:

using Random

my_bootstrap = infer(my_fit, Pumas.Bootstrap(
rng         = MersenneTwister(1234),
samples     = 500,
stratify_by = [:sex, :trt],
ensemblealg = EnsembleThreads()))

The coeftable function will work on the returned object, and the confidence interval 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), Anne-Gaëlle Dosne, Martin Bergstrand, Kajsa Harling, Mats O Karlsson (2016) suggests the sampling importance resampling (SIR) procedure for estimating the uncertainty. This can be obtained by calling the function Pumas.SIR and passing the returned object of type SIR as the type argument of infer. The resulting function signature is:

infer(fpm::FittedPumasModel, type::SIR; level=0.95, ensemblealg=EnsembleThreads())

There is one additional keyword argument:

• ensemblealg: A BasicEnsembleAlgorithm to use to control the mode of parallel execution used for evaluating the weights in the procedure.

The Pumas.SIR accepts some arguments which control the procedure:

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

The keyword arguments are as follows:

• rng: used to select a pseudo-random number generator to be used within the SIR procedure.

• samples: the number of samples to draw from the proposal distribution, a truncated multivariate normal distribution based on the maximum-likelihood estimates and robust variance-covariance matrix.

• resamples: the number of values from the original sample to re-sample (by weighed re-sampling without replacement).

Both samples and resamples must be supplied, as there are no defaults. Additionally, it is required that samples be greater than resamples.

Note that the proposal distribution for the SIR procedure is based on the robust variance-covariance matrix and will therefore fail whenever the computation of the robust variance-covariance matrix fails.

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, Pumas.SIR(samples=200, resamples=70))

The coeftable function will work on the returned object, and the confidence interval 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 Pumas.SIR as the second argument. For example:

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

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

## 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(fpm::FittedPumasModelInference)
stderror(fpm::FittedPumasModel)

When calling stderror on a FittedPumasModelInference object, the type of standard errors returned will depend of 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 AbstractFittedPumasModel object will result in using the default arguments to infer, i.e. standard errors based on the robust variance-covariance matrix. The 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 AbstractFittedPumasModel object, but to call infer first and store the object, which is then passed to stderror.