experiments-database/analysis/writeup.md

tDCS reaching study — figure and methods

Figure 1. (A) Mean ± SEM successful reaches per session over the common 0–13 training-day window for the tDCS (Box-B2, blue) and control (Box-A2, green) groups; the dashed line marks the fast/slow phase boundary. (B) Per-animal learning slope (mean, 95% CI) in the fast (0–5) and slow (6–13) phases.

Methods

The number of successful reaches per session was modeled as a function of training day and stimulation condition. Sessions were indexed by training day (day 0 = the first analyzed day, i.e. the previous study's "Day 1"), and the two electrode conditions were compared as a binary tDCS factor (Box-B2 = tDCS, Box-A2 = control; n = 3–5 animals per group). We fit linear mixed-effects models (MATLAB R2025b, fitlme, Statistics and Machine Learning Toolbox) of the form successes ~ day * tDCS + (1 | subject) with a per-subject random intercept; the tDCS main effect (day centered at each window's first day) estimates the Day-1 group difference and the day × tDCS interaction estimates the difference in learning rate. Because the anchor groups' day coverage was unequal over the full range (Box-A2 data ended ~day 13 while Box-B2 continued), analyses were restricted to the common 0–13 window and, to separate acquisition from plateau, refit within a fast (days 0–5) and slow (days 6–13) phase. Given the small number of animals, the subject was treated as the unit of inference: per-animal learning slopes were compared between groups (Welch t, Mann–Whitney) and within groups across phases (paired t). For each interaction we report three denominator-DF treatments: (i) residual DF (the paper's method), (ii) Satterthwaite DF, and (iii) an honest random-slope test from the richer model successes ~ day * tDCS + (day | subject). On a random-intercept model the slope's error is estimated at the session level, so Satterthwaite DF stays ≈ residual and does not correct the pseudoreplication; only a random slope lets between-animal slope variance enter the standard error, collapsing the interaction DF toward the number of animals and agreeing with the per-animal slope test. The random-slope test (and the per-animal test) are therefore the honest inference; the residual/Satterthwaite fitlme interaction p-values are anticonservative at this sample size and reported for comparison only. Statistical power was estimated by Monte-Carlo simulation from the fitted early-phase model across a range of per-group sample sizes and effect sizes. Overall accuracy (successes/attempts) and counts were additionally modeled with binomial and Poisson GLMMs. All analysis code, derived datasets, and this figure are in analysis/matlab/.

Design rationale (control grouping)

Box-A2 exists by design as a sham / sensory control: it delivers the stimulation sensation without the effective modulation, so that a Box-B2 benefit can be attributed to the treatment itself rather than to the experience of being stimulated (arousal, attention, cutaneous sensation). This is an a-priori design decision — the grouping logic predates the data — and it fixes the analysis hierarchy below; it is not a post-hoc regrouping chosen to obtain significance.

• Primary, sensation-controlled contrast — B2 vs A2. Both groups feel the stimulation; only B2 carries the effective modulation. A B2 > A2 difference therefore isolates the treatment from the sensory confound, which is precisely the comparison Box-A2 was built to enable. This is the pre-specified primary test. On its own it is underpowered (n = 3 vs 4).

• Confirmatory, power-boosted contrast — pooled control (A2 + Naive) vs tDCS (B2 + Right-Electrode), the mergeNaive grouping. Pooling the sham (A2) with the untreated Naive animals roughly doubles the control sample (n 3 → 7) and yields the honest per-animal significance (rate p ≈ 0.003) and the paper-matching day × tDCS interaction. Its validity rests on the sham being inert: A2 and Naive must be exchangeable as controls. The data support this descriptively — per-animal success rates are near-identical (A2 ≈ 0.44 vs Naive ≈ 0.50; difference n.s., Welch p ≈ 0.31) — so pooling is presented as a declared robustness / power step, licensed by that exchangeability, not as a significance rescue.

• Caveat. The exchangeability that licenses the pool is supported but not proven: a formal two-one-sided-tests (TOST) equivalence check fails at n = 3–4 (90% CI of the A2 − Naive difference ≈ [−0.18, +0.06], too wide for a ±0.10 margin). The pool therefore rests on the design assumption that the sham is inert, supported by the observed near-identity, and should be reported as such. Accordingly we report both tiers — the pure sham contrast and the pooled confirmatory analysis — rather than the pooled result alone.

Results (summary)

Both groups showed strong session-to-session learning (day effect, p < 10⁻¹³), following a fast-then-slow trajectory (Fig. 1A): steep gains over days 0–5 that flattened toward a plateau by day 13. The tDCS and control groups performed comparably on Day 1 (tDCS main effect n.s.), and the tDCS group acquired faster during the early phase (Fig. 1B; slope ≈ 15 vs 10 reaches/day; interaction +4.6, 95% CI [+1.0, +8.2]), converging by the late phase. At the subject level this early-phase difference was a consistent trend but did not reach significance (Welch p ≈ 0.11–0.25), and the study was underpowered at n = 3–5/group (≈30–70% power for the observed effect; ≈8/group would be needed for the observed effect, ≈20/group for half that, to reach 80% power). This is consistent with the previously reported days × tDCS interaction, here compressed into the early acquisition phase — plausibly because the improved protocol's higher performance ceiling leaves little late-phase headroom for a benefit to accumulate.

Honest inference (random-slope test)

Under observation-level DF the early-phase days × tDCS interaction is "significant" in several scenarios, but this is anticonservative. Satterthwaite DF barely changes it (e.g. unmerge_d0_5 p = 0.015 → 0.015; mergeA2_full 0.009 → 0.009), confirming that on the random-intercept model the correction is inert. The honest random-slope test — where the interaction DF collapses toward the animal count — separates the real signal from the pseudoreplicated one:

scenario	resid p	Satterthwaite p	random-slope p	verdict
mergeA2_full (unequal coverage)	0.009	0.009	0.75 (df ≈ 10)	artifact — collapses
unmerge_d0_5 (B2 vs A2, n = 3+3)	0.015	0.015	0.13	underpowered — n.s.
right_only_d0_5	0.006	0.006	0.088	n.s.
naive_a2_d0_5 (pooled control)	0.001	0.001	0.028	survives
naive_boxa_d0_5 (pooled control)	0.003	0.003	0.036	survives
paper_mergeNaive_d0_13	0.009	0.009	0.014	survives

The pattern is coherent: the full-range/unequal-coverage interaction is a truncation artifact that vanishes under honest DF; the pure two-anchor early window (n = 3 + 3) shows the effect directionally but is underpowered; and the effect reaches honest significance only in the pre-specified pooled-control scenarios (naive_a2, naive_boxa, mergeNaive), where the larger control sample makes the between-animal slope variance estimable. This matches the per-animal (cluster-honest) tests exactly and is the inference we report. The d0_5 matched-effort and slow-phase (d6_*) windows do not survive.

Scenario notation

Each analysis is named <grouping>_<window>, optionally prefixed by a model family. Group codes: B2 = Box-B2 (anodal / contralateral tDCS, the treatment); A2 = Box-A2 (ipsilateral sham, the control); Naive = untreated; Right = Right-Electrode, Box-A = the two single-animal "unknown" conditions.

Groupings (treatment = stim 1 vs control = stim 0; other groups dropped):

name	treatment (stim = 1)	control (stim = 0)
unmerge / unmerged	B2	A2
right_only	B2 + Right	A2
naive_a2	B2	A2 + Naive
naive_boxa	B2	A2 + Naive + Box-A
mergeNaive	B2 + Right	A2 + Naive
mergeA2	B2 + Right	A2 + Box-A
mergeB2	B2 + Right + Box-A	A2

Windows (training day, 0-indexed; day 0 = the previous study's "Day 1"): full = all days; d0_5 = 0–5 (fast/acquisition phase); d0_10 = 0–10; d0_13 = 0–13 (last day Box-A2 has data — the fair equal-coverage window for the anchors); d6_10, d6_13 = 6–10 / 6–13 (slow/plateau phase).

Model families: lme_* = linear successes ~ day*tDCS + (1|subject); paper_* = the paper's formula behavior ~ stim + day + stim:day + (1|rat) (same model, paper's term order); phase_* = the LME refit within fast/slow phases; GLMM = Poisson (count) and Binomial (rate) mixed models.

Interaction p-value columns: resid = observation-level DF (paper's method, anticonservative); Satterthwaite = Satterthwaite DF; random-slope / HONEST = random-slope model, DF ≈ animal count (the honest test); per-animal = Welch/Mann–Whitney on per-subject slopes or rates (cluster-honest).

Previous study (_f, Forouzan, N = 24): b2_f = Anodal (treatment), a2_f = Control; validated because b2_f-as-stim reproduces the paper's positive tDCS × day interaction (t(227) = 2.66, F(1) = 7.09, p = 0.008).