--- title: "Reproducing Forbes (2023): The AMH Applied Example" output: rmarkdown::html_vignette vignette: > %\VignetteIndexEntry{Reproducing Forbes (2023): The AMH Applied Example} %\VignetteEngine{knitr::rmarkdown} %\VignetteEncoding{UTF-8} --- Forbes (2023) introduced the extended bass-ackwards method: all-pairs between-level correlations plus redundancy pruning. A factor is an unobserved dimension that explains why a set of items correlate. A redundant factor restates a factor at another level rather than refining it. She demonstrated the method on a real, deep hierarchy: the **Assessing Mental Health (AMH)** study. That study measured 155 mental-health symptom variables spanning 18 DSM disorders in an Australian general-population sample (N = 3,175, Forbes et al., 2021). Her paper cites this package as the reference implementation of the extension. The Spearman correlation matrix from that applied example ships with the package as the `forbes2023` dataset (redistributed from Forbes's OSF project under CC-BY 4.0, see `?forbes2023`). This vignette reproduces her published analysis end-to-end: the 10-level hierarchy, the skip-level correlations, the redundancy chase, and the pruned-factor diagram. If the *concepts* here are new (what skip-level edges are, how `prune()` decides what to flag), read `vignette("ackwards-forbes")` first. It teaches the method extension on a small didactic dataset. This vignette runs the real thing. ## The data `forbes2023` is a 155 × 155 Spearman correlation matrix, not raw data: ``` r library(ackwards) dim(forbes2023) #> [1] 155 155 forbes2023[1:3, 1:3] #> Impulsivity Reckless_behaviour__i_ Blurting #> Impulsivity 1.0000000 0.5415115 0.6172762 #> Reckless_behaviour__i_ 0.5415115 1.0000000 0.4783731 #> Blurting 0.6172762 0.4783731 1.0000000 ``` A correlation matrix carries no sample size, so we supply the AMH study's N explicitly with `n_obs = 3175`. For the PCA engine used here nothing downstream depends on N. PCA (principal component analysis) summarizes the items with weighted sums called components. Supplying N is still good practice: it is recorded in the object, and it enables N-dependent fit statistics under the factor engines. ## Fitting the 10-level hierarchy Forbes's applied example extracts 1 through 10 components with all between-level pairs: ``` r x <- ackwards(forbes2023, k_max = 10, pairs = "all", n_obs = 3175) x #> #> ── Bass-Ackwards Analysis (ackwards) ─────────────────────────────────────────── #> Engine: pca #> Rotation: varimax #> Basis: (user-supplied matrix) #> n: 3,175 #> k (max): 10 #> #> ── Levels ── #> #> ✔ k = 1: 1 factor, 31.4% variance #> ✔ k = 2: 2 factors, 36.8% variance #> ✔ k = 3: 3 factors, 41.1% variance #> ✔ k = 4: 4 factors, 44.9% variance #> ✔ k = 5: 5 factors, 48.2% variance #> ✔ k = 6: 6 factors, 50.8% variance #> ✔ k = 7: 7 factors, 52.9% variance #> ✔ k = 8: 8 factors, 54.6% variance #> ✔ k = 9: 9 factors, 56.2% variance #> ✔ k = 10: 10 factors, 57.6% variance #> #> ── Edges ── #> #> 312 of 1320 edges have |r| ≥ 0.3 #> ──────────────────────────────────────────────────────────────────────────────── #> Note: This is a series of linked solutions, not a fitted hierarchical model. #> Cross-level edges are descriptive score correlations. Per-level fit indices #> (EFA/ESEM) describe how well a k-factor model fits the items at that level -- #> they do not validate the edges or the hierarchy itself. ``` The hierarchy unfolds a general factor of psychopathology at the top (k = 1, ~31% of variance) down to ten fine-grained components (cumulatively ~58%). With `pairs = "all"`, the edge table holds 1320 correlations across all 45 level pairs. Every one of them is matched against Forbes's own reference implementation by the package regression test (see *Fidelity*, below). **Label mapping.** Forbes labels components by level letter and position within the level. Her `a1` is the single component at k = 1, `b1` and `b2` sit at k = 2, and `j1` through `j10` sit at k = 10. This package uses `m{k}f{j}` for the same positions, so her `d4` is `m4f4` and her `j4` is `m10f4`. Signs may differ component by component, because the package aligns each factor to its primary parent (see `vignette("ackwards-intro")`). That leaves every correlation identical in absolute value. ## Skip-level structure The deep hierarchy is where skip-level correlations earn their keep. Among the non-adjacent pairs alone, 48 correlations exceed |r| = 0.9. These are components that persist almost unchanged across two or more levels:
Strongest skip-level edges in the AMH hierarchy
Top 10 non-adjacent pairs by |r|
From To Level (from) Level (to) r
m6f6 m8f7 6 8 1.000
m7f6 m9f5 7 9 0.999
m6f6 m9f5 6 9 0.998
m5f5 m8f7 5 8 0.998
m5f5 m7f6 5 7 0.998
m8f7 m10f5 8 10 0.998
m7f6 m10f5 7 10 0.998
m6f6 m10f5 6 10 0.997
m5f5 m9f5 5 9 0.996
m7f5 m9f4 7 9 0.996
Correlations this close to 1.0 across multiple levels are the signature of redundancy: the intermediate solutions are relabeling a stable dimension, not refining it. That is exactly what the redundancy chase formalizes. ## The redundancy chase Forbes's `ChaseCorrPaths()` procedure walks each component upward through the hierarchy. At every ancestor level it takes the component with the largest **direct** (skip-level) correlation, and it continues while |r| ≥ 0.9. Chains of near-identical components are then collapsed to a single retained node. `prune(x, "redundant")` implements the same rule as its default (`redundancy_criterion = "direct"`): ``` r xp <- prune(x, "redundant") #> ℹ Redundancy pruning (direct criterion, |r| ≥ 0.9) flagged 37 nodes. #> ℹ Nodes are retained in the object; inspect with `x$prune$nodes` and #> `x$prune$chains`. ``` Of the 55 components in the full hierarchy, **37 are flagged as redundant**, leaving 18 distinct constructs. That is the same decomposition reported in Forbes (2023). As always, `prune()` only *annotates*. Nothing is removed from the object, and every flagged component keeps its loadings (a loading is the correlation between an item and a factor), scores, and edges. ### A chain, worked Forbes's chase traces the chain `d4 → e4 → f5 → g5 → h5 → i4 → j4`: one dimension surfacing at k = 4 and persisting, near-unchanged, to k = 10. In `ackwards` labels, that chain is: ``` r ch <- xp$prune$chains d4 <- ch[ch$chain_id == ch$chain_id[ch$id == "m10f4"], c("id", "level", "retain")] d4 #> id level retain #> 14 m4f4 4 FALSE #> 15 m5f4 5 FALSE #> 16 m6f5 6 FALSE #> 17 m7f5 7 FALSE #> 18 m8f5 8 FALSE #> 19 m9f4 9 FALSE #> 20 m10f4 10 TRUE ``` Because the chain reaches the deepest level, the retention rule keeps its **bottom** node, `m10f4` (her `j4`), the most specific and best-defined manifestation. The rule flags the six components above it as redundant. A chain that stops short of k = 10 instead keeps its *top* node, the broadest manifestation. The full retained set (13 chain-retained components, in level order): m1f1, m4f2, m5f2, m7f7, m10f1, m10f2, m10f3, m10f4, m10f5, m10f6, m10f7, m10f8, m10f9. The other 5 kept components were never part of a |r| ≥ 0.9 chain at all, so no retention rule applies to them. ### Why the *direct* criterion matters here Correlation is not transitive: a chain of adjacent links each ≥ 0.9 does not guarantee the two ends correlate ≥ 0.9 directly. On shallow hierarchies the distinction rarely bites, but on this 10-level hierarchy it does. Forbes's direct rule and an adjacent-hop walk disagree on 7 of the 54 chased components: ``` r xa <- prune(x, "redundant", redundancy_criterion = "adjacent") #> ℹ Redundancy pruning (adjacent criterion, |r| ≥ 0.9) flagged 36 nodes. #> ℹ Nodes are retained in the object; inspect with `x$prune$nodes` and #> `x$prune$chains`. ``` The adjacent opt-in flags 36 components (vs 37 under the default) and retains `m3f3`, which the direct rule chases further up the hierarchy. The default reproduces Forbes's published chase exactly, all 54 components, including those 7 divergences. So stick with it when your goal is comparability with her method. `redundancy_criterion = "adjacent"` remains available for sensitivity analysis. ## The pruned-factor diagram Dropping the flagged components and bridging each retained component directly to its strongest kept ancestor yields the pruned-factor diagram. It uses the same publication style as Forbes (2023): black lines, uniform width, no legend, and correlation labels on each spanning arrow. ``` r autoplot(xp, drop_pruned = TRUE, show_r = TRUE, color_pos = "black", color_neg = "black", edge_linewidth = 0.6, show_arrows = FALSE, legend = FALSE ) ``` ![plot of chunk pruned-diagram](assets/ackwards-forbes2023-pruned-diagram-1.png) Whole levels vanish (every component at those levels was flagged), leaving deliberate gaps in the y-axis. To compact the layout while keeping the original level numbers: ``` r autoplot(xp, drop_pruned = TRUE, compress_levels = TRUE, color_pos = "black", color_neg = "black", edge_linewidth = 0.6, show_arrows = FALSE, legend = FALSE ) ``` ![plot of chunk pruned-diagram-compressed](assets/ackwards-forbes2023-pruned-diagram-compressed-1.png) For the full set of cosmetic controls, see `vignette("ackwards-visualization")`. ## Fidelity This reproduction is not merely visual. It is pinned by the package's regression suite (`tests/testthat/test-forbes-fidelity.R`), which verifies on every check run that: - all 45 between-level correlation matrices match the output of Forbes's own reference implementation entrywise in absolute value (tolerance 1e-12), and - the direct redundancy chase lands on her published `ChaseCorrPaths()` result for **all 54** chased components. The expected values were computed with Forbes's reference implementation (from her OSF project, `https://osf.io/pcwm8/`) on the same md5-pinned matrix that ships as `forbes2023`. So the dataset and the oracle cannot drift apart. Only `ackwards()` runs at test time. ## References Forbes, M. K. (2023). Improving hierarchical models of individual differences: An extension of Goldberg's bass-ackward method. *Psychological Methods*. https://doi.org/10.1037/met0000546 Forbes, M. K., Sunderland, M., Rapee, R. M., Batterham, P. J., Calear, A. L., Carragher, N., Ruggero, C., Zimmerman, M., Baillie, A. J., Lynch, S. J., Mewton, L., Slade, T., & Krueger, R. F. (2021). A detailed hierarchical model of psychopathology: From individual symptoms up to the general factor of psychopathology. *Clinical Psychological Science*, *9*(2), 139--168. https://doi.org/10.1177/2167702620954799 Goldberg, L. R. (2006). Doing it all Bass-Ackwards: The development of hierarchical factor structures from the top down. *Journal of Research in Personality*, *40*(4), 347--358. https://doi.org/10.1016/j.jrp.2006.01.001