Sampling Importance Resampling (SIR)

SIR is an optional post-estimation step that provides non-parametric parameter uncertainty estimates. It produces 95% confidence intervals that are more robust than the asymptotic covariance matrix, particularly for models with:

• Non-normal parameter distributions
• Boundary estimates (parameters near constraints)
• Small datasets where asymptotic assumptions may not hold

How It Works

SIR uses the maximum likelihood estimates and their covariance matrix as a proposal distribution, then reweights samples based on the actual likelihood:

1. Sample: Draw M parameter vectors from a multivariate normal distribution centered on the ML estimates, using the estimation covariance matrix
2. Importance weighting: For each sample, compute the objective function value (OFV) and calculate an importance weight based on the ratio of the true likelihood to the proposal density
3. Resample: Draw m vectors (with replacement) proportional to the importance weights

The resampled vectors approximate the true parameter uncertainty distribution. Confidence intervals are derived from their empirical percentiles.

Enabling SIR

Add sir = true to the [fit_options] block. The covariance step must also be enabled (it provides the proposal distribution):

[fit_options]
  method     = focei
  covariance = true
  sir        = true

Options

Output

SIR adds the following to the estimation output:

• 95% CI for each theta, omega, and sigma parameter (2.5th and 97.5th percentiles)
• Effective sample size (ESS): a diagnostic indicating how well the proposal distribution matches the true uncertainty. ESS close to M indicates a good match; ESS much less than m suggests the proposal is a poor fit

Diagnostics

The effective sample size (ESS) is the primary diagnostic:

• ESS > m (resamples): excellent — the proposal distribution is well-matched
• ESS between 100 and m: adequate for most purposes
• ESS < 100: the proposal may be a poor fit; consider a different estimation method or increasing sir_samples

Computational Cost

SIR evaluates the inner loop (EBE optimization) for each of the M proposal samples. With the default M=1000, this is roughly 3-10x the cost of the estimation step itself. The computation is parallelized across samples and warm-started from the ML EBEs to minimize runtime.

The resampling step itself is negligible.

Example

[fit_options]
  method        = focei
  covariance    = true
  sir           = true
  sir_samples   = 1000
  sir_resamples = 250
  sir_seed      = 42

Reference

Dosne A-G, Bergstrand M, Karlsson MO. "Improving the estimation of parameter uncertainty distributions in nonlinear mixed effects models using sampling importance resampling." J Pharmacokinet Pharmacodyn. 2017;44(6):539-562. doi:10.1007/s10928-017-9542-0