Bootstrap uncertainty for the fast AME estimator

Computes bootstrap standard errors and percentile confidence intervals for a fast AME fit produced by ame_als or lame_als. Two strategies are available:

parametric: (default) Simulates fresh outcomes from the fitted model and refits. For normal and IRLS (binary, poisson) fits the simulation is on the calibrated response scale; for a non-normal transform fit – whose estimator is a Gaussian fit to a fixed transformed response – it is on the Gaussian working scale. The fitted residual variance is df-corrected (mean-model degrees of freedom) at fit time, so simulating with it keeps the replicates centred on the point estimate. Available for both cross-sectional and longitudinal fits.
block: Resamples time slices with replacement. With block_length = 1 slices are resampled independently; block_length > 1 draws contiguous blocks (a moving-block bootstrap), which preserves short-run temporal dependence. Longitudinal fits only (requires T > 1); with few slices its intervals are necessarily coarse.

Every replicate is refit by ame_als_refit, warm-started from the original point estimate so that replicates do not drift to different local optima. Replicates that error or return non-finite values are dropped and counted (n_valid / n_total). Standard errors are replicate column standard deviations; confidence intervals use the percentile method.

Usage

ame_als_bootstrap(
  object,
  R = 200,
  type = c("parametric", "block"),
  block_length = 1,
  seed = NULL,
  verbose = TRUE
)

boot_ame(
  object,
  R = 200,
  type = c("parametric", "block"),
  block_length = 1,
  seed = NULL,
  verbose = TRUE
)

Arguments

object: an ame_als object from ame_als or lame_als.
R: integer number of bootstrap replicates (default 200).
type: "parametric" (default) or "block"; see Description.
block_length: block length for the block bootstrap: 1 (default) resamples time slices independently; an integer > 1 draws contiguous blocks of that many slices (a moving-block bootstrap), appropriate when the series has short-run temporal dependence. Capped at T; ignored for the parametric bootstrap.
seed: optional integer random seed.
verbose: logical; print progress (default TRUE).

Value

An object of class "boot_ame" with components including coefs (replicate intercept + regression coefficients), se, ci_lo, ci_hi, point_est, param_names; vc_* for the variance components; a_coefs, b_coefs, se_a, se_b; U_aligned, V_aligned, U_raw, V_raw, se_U, se_V (when R > 0); and n_valid, n_total, type, family.

Details

Choice of inference. The Social Influence Regression paper of Hoff & Minhas (2025) derives its primary standard errors from the observed Hessian (classical \(-H^{-1}\) and the sandwich/robust estimator \(H^{-1} S H^{-1}\)). Two features of the AME model make a Hessian-based variance awkward here. First, without an explicit gauge fix the rank-R multiplicative term is identified only up to a full \(R\times R\) rotation/reflection (\(U \to U R\), \(V \to V R^{-\top}\)), leaving the joint Hessian rank-deficient. Second, even with a gauge fixed, the non-normal families are fit on a Gaussian working response, so a Hessian computed from that working objective is not calibrated to the family likelihood. The bootstrap side-steps both issues and is the recommended uncertainty tool here, mirroring sir::boot_sir().

Alignment-sensitive quantities. The regression coefficients beta, the additive effects a, b and the variance components are rotation-invariant and are aggregated directly. The multiplicative factors U, V are not: each replicate is Procrustes-aligned to the original fit before its standard errors are computed. The object stores both the raw and aligned replicate factors so the effect of alignment can be inspected. The alignment is well-determined only when the multiplicative singular values are well separated; with near-equal singular values the per-column U/V standard errors reflect an unstable rotation, and the subspace — or, for symmetric models, the eigenvalues L — is the summary of record. For a symmetric model the eigenvalues L of the multiplicative term are aggregated and reported as the primary multiplicative-uncertainty summary.

The parametric bootstrap is a full parametric bootstrap of the AME model: each replicate draws fresh additive random effects a, b from their fitted dispersion (va, vb, cab) and fresh residuals, holding mu, beta and the multiplicative term fixed. Regenerating the additive effects (rather than holding the fitted a, b fixed) is what gives the intercept and node-covariate standard errors their actor-level sampling-variability component.

Assumptions. The block bootstrap treats the time slices as exchangeable replicates of the static-effects model — appropriate for that model, but not for strongly trended or serially dependent series (use the dynamic lame there); with only a few time slices it is necessarily coarse, its intervals correspondingly imprecise and somewhat anti-conservative, so parametric is the default. A variance component fit with R > 0 can still carry a small residual parametric-bootstrap bias (the low-rank refit re-absorbs simulated noise); summary() flags any point estimate that falls outside its interval. A binary IRLS fit can carry a finite-sample (incidental-parameters) bias in the point estimator itself, which the bootstrap reproduces rather than removes. The MCMC ame / lame path gives posterior summaries when that is the target.

References

Minhas, S. and Hoff, P. D. (2025). Decomposing Network Dynamics: Social Influence Regression. Political Analysis. The block and parametric bootstrap follow the inference scheme of sir::boot_sir() developed for the SIR estimator.

Author

Cassy Dorff, Shahryar Minhas, Tosin Salau

Examples

# \donttest{
Y <- replicate(6, { m <- matrix(rnorm(225), 15, 15); diag(m) <- NA; m },
               simplify = FALSE)
fit <- lame_als(Y, R = 1, family = "normal", verbose = FALSE)
# post-hoc bootstrap of an existing fit:
bt <- ame_als_bootstrap(fit, R = 50, type = "block",
                            seed = 1, verbose = FALSE)
# equivalent one-shot call:
# fit_b <- lame_als(Y, R = 1, family = "normal", verbose = FALSE,
#                   bootstrap = 50, bootstrap_type = "block",
#                   bootstrap_seed = 1)
print(bt)
#> 
#> Bootstrap results: fast AME estimator
#> Type: block | Family: normal | Replicates: 50/50 valid
#> 
#>           Estimate Boot SE    2.5%  97.5%
#> intercept   0.0092  0.0285 -0.0352 0.0768
#> 
#> beta, a, b and variance components aggregate directly; U/V are
#> Procrustes-aligned.
#> ! Block bootstrap on a static-effects fit: SEs for the intercept,
#> node-covariate coefficients and additive variance components (va, vb, cab) can
#> severely under-cover because slice resampling does not perturb actor-level
#> structure. Use `type = "parametric"` for those parameters.
# }