Latent-adjustment power comparison — 2026-10-20

Paper-grade power comparison of covariate-aware latent construction against the 0.8.0 covariate-blind basis at the Phase 6 SEA-AD design (openspec/changes/covariate-aware-latent-construction/, design D9, spec latent-adjustment-validation). It is the companion of the Type I check, which admitted stage conditioning. Every number below is read from results/latent-adjustment-power-2026-10-20/ (latent_adjustment_power_decision.json, latent_adjustment_power_rates.csv, latent_adjustment_power_pvalues.csv, merged.jsonl). Rates are rejection rates at p ≤ 0.05 over the available replicates of 500, at 999 permutations. The Monte Carlo SE is at most 0.024.

Run: 14,000 units of three analyses on ing (SLURM array 923444, 200 shards, 158 recorded core-hours, no failed task), from 979657c plus this change's working tree; file hashes are in PROVENANCE.txt. Advisory verdicts: stage candidate_fails, group_stage candidate_fails.

In one paragraph. stage conditioning gains a lot of orientation power. At the top effect, angle rejects 0.922 against the covariate-blind basis's 0.784 with a covariate-aligned nuisance axis, and 0.931 against 0.642 without one. It fails the predeclared rule for three reasons:

it loses shape power at the top shape effect when the nuisance axis is present (0.243 against 0.428);
it loses power for small magnitude effects and on delta under shape effects, without the nuisance axis (for example 0.400 against 0.520 at magnitude e = 0.1);
its translation control rejects angle at 0.085–0.092, above the bound. The covariate-blind basis is above its own bound there too (0.076–0.086).

group_stage conditioning loses most of the orientation and shape power in both columns: angle 0.344 and shape 0.054 at the top effects with the nuisance axis. With the Type I check's angle excess, it is ruled out. No arm in either column ever selected rank 1. Latent adjustment is therefore not a uniform power improvement. It trades shape and small-magnitude sensitivity for orientation sensitivity in this construction, which §6 turns into conditions for the case study.

1. Construction

Cells. The zero-effect anchor, the translation control at e = 1.0, and three modes at the Phase 6 effect levels:
magnitude (joint) at e = 0.02, 0.05, 0.10 and 0.25, below its saturation;
orientation at e = 0.25, 0.5, 0.75 and 1.0;
shape at e = 0.25, 0.5, 0.75 and 1.0.
The design, blocks and estimator are the Type I check's (§1).
Columns.
nuisance: the Type I check's calibrated injection (s_z = 40, s_c = 2).
clean: s_z = s_c = 0. The same eight covariates are declared, but none affects the features.
Matched seeds. The generator, covariate and RRPP seeds depend on the replicate index only. Within a replicate, the two columns share the generated blocks and covariates and differ only in the injection. Every cell starts from the same baseline trajectory, and the arms share the dataset and the RRPP seed. The nuisance column's anchor is therefore the Type I check itself, and its rates reproduce that report's to the third decimal.
Arms. baseline (covariates in the trajectory model only, covariate-blind basis), stage and group_stage.
Rule (fixed in the hashed settings before the run; the study engine's column_decision convention). Each residualized arm is compared against baseline. The pooled SE is √(SE_arm² + SE_baseline²) and the bound is 0.05 + 2·√(0.05·0.95/R).
gain: nuisance column, orientation/angle at e = 1.0: rate_arm − rate_baseline > 2 · pooled SE;
protected: nuisance column, magnitude/delta at e = 0.25 and shape/shape at e = 1.0: rate_baseline − rate_arm ≤ 2 · pooled SE;
no loss: clean column, every statistic at every effect of every mode: the same loss tolerance;
controls: both columns, anchor and translation, every statistic at or below the bound.

D9 also listed magnitude/delta and shape/shape as gain targets. The spec's single gain target is the one applied, because magnitude/delta is at 0.92 or above in every arm and column at its top effect and has little room to gain.

The same 52 (stage) and 38 (group_stage) replicates are unavailable in every cell. Each loses a factor level from a CV training fold, which depends only on the covariates and the folds (Type I report §2). So R = 448 for stage, 462 for group_stage and 500 for baseline throughout.

2. Power on the target statistic

From latent_adjustment_power_rates.csv, rate for the mode's own statistic (SE ≤ 0.024).

mode	e	nuisance: baseline	nuisance: `stage`	nuisance: `group_stage`	clean: baseline	clean: `stage`	clean: `group_stage`
magnitude/`delta`	0.02	0.072	0.042	0.093	0.076	0.042	0.084
	0.05	0.226	0.121	0.229	0.180	0.121	0.240
	0.10	0.560	0.417	0.591	0.520	0.400	0.597
	0.25	0.940	0.969	0.976	0.924	0.960	0.972
orientation/`angle`	0.25	0.208	0.138	0.167	0.128	0.156	0.165
	0.50	0.418	0.263	0.229	0.170	0.254	0.221
	0.75	0.584	0.549	0.255	0.200	0.500	0.251
	1.00	0.784	0.922	0.344	0.642	0.931	0.325
shape/`shape`	0.25	0.104	0.078	0.056	0.066	0.071	0.054
	0.50	0.200	0.080	0.067	0.090	0.074	0.063
	0.75	0.332	0.167	0.056	0.140	0.181	0.058
	1.00	0.428	0.243	0.054	0.128	0.250	0.045

Controls, rate per statistic (delta / angle / shape). The bound is 0.0695 at R = 500, 0.0706 at 448 and 0.0703 at 462.

control	nuisance: baseline	nuisance: `stage`	nuisance: `group_stage`	clean: baseline	clean: `stage`	clean: `group_stage`
anchor	0.048 / 0.062 / 0.078	0.031 / 0.067 / 0.058	0.048 / 0.082 / 0.069	0.044 / 0.076 / 0.062	0.038 / 0.067 / 0.051	0.052 / 0.082 / 0.074
translation	0.052 / 0.076 / 0.070	0.025 / 0.092 / 0.054	0.065 / 0.110 / 0.106	0.050 / 0.086 / 0.050	0.025 / 0.085 / 0.060	0.063 / 0.104 / 0.093

As a cross-check, the clean column's baseline orientation/angle at e = 1.0 (0.642) matches the Phase 6 small-n study's net-displacement rate at the same effect (0.648; power comparison), even though eight covariates are now declared.

3. The verdicts

From latent_adjustment_power_decision.json.

stage — candidate_fails.

gain: passes. 0.922 against 0.784, difference 0.138 > 0.045.
protected: magnitude/delta passes (0.969 against 0.940). shape/shape fails: 0.243 against 0.428, a loss of 0.185 against a tolerance of 0.060.
no loss (clean column) fails in eight reads:
magnitude/delta at e = 0.02, 0.05 and 0.10: 0.042, 0.121 and 0.400 against 0.076, 0.180 and 0.520;
delta under orientation at e = 0.25: 0.029 against 0.066;
delta under shape at all four effects: for example 0.214 against 0.298 at e = 1.0.
Every other clean-column read holds. That includes shape/shape, which gains: 0.250 against 0.128 at e = 1.0.
controls: translation/angle exceeds in both columns (0.092 and 0.085 against 0.0706). Every other control holds.

group_stage — candidate_fails. The gain fails (0.344 against 0.784) and shape/shape fails as a protected pair (0.054 against 0.428). Nine clean-column no-loss reads fail, all under orientation or shape effects. Seven controls exceed the bound: anchor angle in both columns at 0.082, anchor shape in the clean column, and translation angle and shape in both columns.

4. Selected rank

Pooled over the 14 cells of each column, as a share of the available units:

column	arm	rank 2	rank 3	rank ≥ 4
nuisance	baseline	0.381	0.513	0.106
nuisance	`stage`	0.048	0.842	0.110
nuisance	`group_stage`	0.010	0.984	0.006
clean	baseline	0.341	0.621	0.038
clean	`stage`	0.045	0.853	0.102
clean	`group_stage`	0.008	0.987	0.005

Per-cell distributions are in the rank_distribution column of the rates table. Residualization moves the CV almost entirely off rank 2. No unit in any arm or column selected rank 1. This construction therefore does not reproduce the SEA-AD cohorts' rank-1 selection and is not evidence about it. The separate rank-1 angle change needs a construction with a single dominant axis, or real-data evidence (roadmap, Next changes item 4).

5. Reading

Residualization removes signal even when no covariate has an effect. On six clean-column orientation replicates at e = 1.0 (a diagnostic run on the workstation, not part of the records), the subtracted term C γ carries about 19 % of each pooled-standardized feature's variance under stage and about 12 % under group_stage.
Under stage, z differs by group within each stage by construction (a_g + a_gs·s), so γ takes in part of the group difference. About a quarter of the features' between-cell (group × stage) variance is removed from the features the basis is fitted on.
Under group_stage, 12 % is what nine covariates fit by chance on 74 within-cell degrees of freedom. About 40 % of it lands between cells, because z's cell means run from 0.56 to 6.94 against a within-cell SD of about 0.9, so the estimation error of γ is extrapolated into a spurious cell-mean pattern.
The basis is then fitted, at the pooled scale, on features that carry that spurious pattern. This is consistent with, though not proof of, group_stage's loss of orientation and shape power and its control excesses.
stage's profile is a trade. It finds the orientation surgery far more often in both columns, and the shape surgery more often in the clean column. It finds small magnitude effects and the shape surgery less often when the nuisance axis is present. delta rejects less often under shape and orientation effects. Because the orientation gain also appears with no nuisance effect at all, it is not simply the nuisance axis being removed. The basis changes whenever the covariates are confounded with group within stage, as they are here by construction.
The nuisance axis also inflates the covariate-blind basis's power. Its shape rate at e = 1.0 is 0.428 with the axis and 0.128 without, and its orientation angle rate is 0.784 against 0.642. Part of stage's protected-pair shape "loss" is therefore a loss against a nuisance-inflated reference, consistent with the baseline's anchor shape of 0.078 above its bound in the Type I check. The rule does not distinguish the two, and this report does not overrule it.
The translation control. angle under a pure location shift is at or above the bound for every arm here, including the covariate-blind basis (0.076 and 0.086). The net-displacement power comparison already placed translation near the bound (0.058–0.068) without covariates. With eight covariates this construction pushes all arms above it, stage by a further 0.016 in the nuisance column.

6. Scope, limits and consequence

Mediators. Residualizing a variable that mediates the stage effect removes real signal. No simulation can tell a confounder from a mediator; this comparison injects a pure confounder. The case study must choose and justify its covariate set in its own proposal.
Extreme confounding. The covariates here are confounded with group × stage far beyond what technical variables usually are: z's cell means span about 7 within-cell SDs. The power costs above, and the group_stage failure, grow with that confounding. Real covariates that are less tied to the design will remove less signal, but this comparison does not measure how much less.
One design. n = 80, two blocks, 498 features, eight covariates, one injected direction per covariate.
Advisory. The verdicts are advisory; the Type I decision (adopt_stage) is the gate.

Consequence for the case study. --latent-adjustment stage remains size-valid (Type I report §5). It is not a power upgrade to adopt instead of the covariate-adjusted 0.8.0 analysis:

report it as a sensitivity analysis beside the 0.8.0 analysis on the same covariates, not as a replacement;
read an angle rejection that appears only under latent adjustment in light of the translation control above, which is at or above the bound for both bases;
treat a delta or shape non-rejection under latent adjustment as weaker than under the covariate-blind basis;
report the record's latent_adjustment block, in particular feature_r2, which shows how much variance the residualization removed;
do not use group_stage.