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
ackwardsobject fit withengine = "pca"or"efa"on a"pearson"or"spearman"basis. ESEM objects are not supported (refittingn_bootlavaan 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
ackwardsobject 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.
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), andn_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.
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).
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. 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
# \donttest{
x <- ackwards(sim16, k_max = 3)
x <- boot_edges(x, sim16, n_boot = 100, seed = 1)
#> ℹ Fitting 100 bootstrap replicates (pca, k = 1-3)...
#> ✔ Fitting 100 bootstrap replicates (pca, k = 1-3)... [1.4s]
#>
x$boot$edges
#> from to level_from level_to r se lo hi
#> 1 m1f1 m2f1 1 2 0.70729611 0.03334587 0.62894118 0.755307440
#> 2 m1f1 m2f2 1 2 0.70691740 0.03221841 0.65534452 0.777438710
#> 3 m2f1 m3f1 2 3 -0.01134239 0.02483636 -0.08601120 0.007369852
#> 4 m2f1 m3f2 2 3 0.69709298 0.20160298 0.03830045 0.734809599
#> 5 m2f1 m3f3 2 3 0.71689101 0.09283102 0.67793763 0.998542647
#> 6 m2f2 m3f1 2 3 0.99559323 0.09091406 0.69647708 0.999880140
#> 7 m2f2 m3f2 2 3 0.07461577 0.20966388 -0.01269972 0.717035306
#> 8 m2f2 m3f3 2 3 -0.05680322 0.04384198 -0.12671309 0.029286309
#> n_boot_ok
#> 1 100
#> 2 100
#> 3 100
#> 4 100
#> 5 100
#> 6 100
#> 7 100
#> 8 100
head(tidy(x)) # now carries se / lo / hi
#> from to level_from level_to r is_primary above_cut se
#> 1 m1f1 m2f1 1 2 0.70729611 TRUE TRUE 0.03334587
#> 2 m1f1 m2f2 1 2 0.70691740 TRUE TRUE 0.03221841
#> 3 m2f1 m3f1 2 3 -0.01134239 FALSE FALSE 0.02483636
#> 4 m2f1 m3f2 2 3 0.69709298 TRUE TRUE 0.20160298
#> 5 m2f1 m3f3 2 3 0.71689101 TRUE TRUE 0.09283102
#> 6 m2f2 m3f1 2 3 0.99559323 TRUE TRUE 0.09091406
#> lo hi n_boot_ok
#> 1 0.62894118 0.755307440 100
#> 2 0.65534452 0.777438710 100
#> 3 -0.08601120 0.007369852 100
#> 4 0.03830045 0.734809599 100
#> 5 0.67793763 0.998542647 100
#> 6 0.69647708 0.999880140 100
# }