boot_edges: Bootstrap confidence intervals for between-level edges

In ackwards: Bass-Ackwards Hierarchical Structural Analysis

Bootstrap confidence intervals for between-level edges

Description

Attaches nonparametric bootstrap standard errors and percentile confidence intervals to every between-level correlation (edge) of a fitted bass-ackwards hierarchy. Every edge ackwards() reports is a point estimate; boot_edges() quantifies its sampling uncertainty, which matters most where a hard threshold consumes the estimate – prune()'s ⁠|r| >= redundancy_r⁠ redundancy rule, and the Forbes (2023) practice of interpreting the strongest all-pairs edge.

Usage

boot_edges(x, ...)

## S3 method for class 'ackwards'
boot_edges(x, data, n_boot = 1000L, conf = 0.95, seed = NULL, ...)

Arguments

x	An ackwards object fit with engine = "pca" or "efa" on a "pearson" or "spearman" basis. ESEM objects are not supported (refitting n_boot lavaan hierarchies needs its own performance treatment), nor are polychoric-basis objects (estimating a polychoric matrix in every resample is slow and unstable) or objects fit from a correlation matrix (resampling needs rows).
...	Reserved for future arguments.
data	The raw item data the model was fit on. Required – the ackwards object deliberately does not store raw data (light core), so it must be re-supplied here. Columns are matched by name against the fit; a warning is issued if the data do not look like the fit data.
n_boot	Number of bootstrap replicates. Default 1000L; larger values (2000+) are advisable for published interval endpoints (Efron & Tibshirani, 1993). Each replicate refits the full hierarchy, so cost scales linearly.
conf	Confidence level for the percentile intervals. Default 0.95.
seed	Integer seed for reproducible resampling. NULL (default) uses the current RNG state.

Details

For each of n_boot replicates, n rows are resampled with replacement, the correlation matrix is recomputed with the same basis and missing-data routine used at fit time, and the full hierarchy is refit. Each replicate level is then anchored to the full-sample solution – its factors matched (greedy max-|r| with removal) and sign-oriented against the full-sample factors on the full-sample correlation matrix – before its edges are computed. Without anchoring, factor label switching and sign flipping across replicates would corrupt the pooled edge distributions; this is the same matching machinery comparability() uses.

All resample indices are drawn upfront from seed, so results are reproducible and identical whether replicates run serially or in parallel. Replicate fits are dispatched through future.apply when it is installed and the user has set a future::plan() (serial otherwise, as in ackwards()'s ESEM engine).

Value

x, invisibly modified: the ⁠$boot⁠ element is populated with

edges	Data frame with one row per edge (aligned with tidy(x, what = "edges")): from, to, level_from, level_to, r (the full-sample point estimate), se (bootstrap standard error), lo, hi (percentile interval endpoints), and n_boot_ok (usable replicates for that edge).
n_boot, conf, seed	The request.

After calling boot_edges(), tidy(x, what = "edges") gains se, lo, and hi columns, and print(x)/summary(x) note the interval coverage.

What the intervals do and do not fix

Per-edge intervals make sampling uncertainty visible: an edge whose interval straddles prune()'s redundancy_r threshold should not be treated as decisively above or below it. They do not correct the selection bias of scanning many edges for the strongest one – the maximum of hundreds of correlations capitalizes on chance even when every individual interval is honest. Treat the intervals as per-edge error bars, not a familywise inference.

Failed replicates

A replicate whose hierarchy fails to converge (in full or at some levels) contributes NA to the affected edges and is dropped from their distributions – convergence is data, not an error. The usable replicate count is reported per edge in n_boot_ok, and a message summarises any shortfall.

References

Efron, B., & Tibshirani, R. J. (1993). An introduction to the bootstrap. Chapman & Hall.

Forbes, M. K. (2023). Improving hierarchical models of individual differences: An extension of Goldberg's bass-ackward method. Psychological Methods. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1037/met0000546")}

See Also

prune() for the thresholded rules the intervals contextualise, comparability() for split-half replicability of the factors themselves, tidy.ackwards() for the augmented edge table.

Examples

x <- ackwards(sim16, k_max = 3)
x <- boot_edges(x, sim16, n_boot = 100, seed = 1)
x$boot$edges
head(tidy(x)) # now carries se / lo / hi

ackwards documentation built on July 25, 2026, 1:08 a.m.