# tDCS reaching study — GLM methods

This document explains the statistics in `tdcs_glm.py`: the models, the exact
formulas, how each subject's progress is accounted for, and the caveats.

## The question

Two **known** conditions anchor the performance scale and differ from each other:

- **H2 (anchor check):** `Electrode-Box-B2` performs **better** than `Electrode-Box-A2`.

Two **unknown** conditions are then **classified** against those anchors — for
each of `Electrode-Box-A` and `Right-Electrode`, is it **A2-like** or **B2-like**?
We do *not* assume either belongs to B2; each unknown is compared to *both*
anchors, and the anchor it cannot be distinguished from is its likely class.

There is also a **merged-assumption** mode (`--merge`): assume the two unknowns
resolve as `Right-Electrode == Box-B2` and `Electrode-Box-A == Box-A2`, fold them
into the anchors, and re-estimate everything with 4 subjects per group.

## Data and outcome

- **Outcome:** `success` = successful reaches in a session
  (`analysis_summary.counts.Success`), a non-negative **count**.
- **Attempts:** `total` = reach attempts in the session (`analysis_summary.total`);
  used as the denominator for the rate model. `success_rate == success / total`.
- **Time:** `day` = the "# Days Reach" field (training day, 0..26). Analyses use
  days ≥ 0 (pre-training negative days and zero-attempt sessions are excluded;
  zero-attempt sessions are undefined for the rate model).
- **Groups (subjects):** Naive (4), Box-A2 (3), Box-B2 (3), Box-A (1),
  Right-Electrode (1). Under `--merge`: Box-A2 (4), Box-B2 (4), Naive (4).

**Provenance:** `tdcs_reach_data.csv` was reconstructed from the experiment
database and verified cell-by-cell against the exported matrix (71/71
unambiguous cells on days 0–5 matched exactly).

## The three models

All models use `Electrode-Box-B2` as the **reference** group, so each group term
is that group's contrast *versus Box-B2*. `day_c` is the centered training day and
`day_c2 = day_c²` captures the rise-then-plateau of the learning curve.

### (A) Level — count (primary)

Poisson GEE on the success counts, clustered by subject:

```
success ~ C(group, Treatment('Electrode-Box-B2')) + day_c + day_c2
family        = Poisson (log link)
groups        = subject           # repeated-measures cluster
cov_struct    = Exchangeable      # working within-subject correlation
SE            = robust (sandwich)
```

`exp(coef)` for a group term is an **incidence-rate ratio (IRR)**: expected
successes relative to Box-B2.

### (B) Level — rate

Binomial GLM on successes-out-of-attempts, with cluster-robust SEs by subject:

```
cbind(success, total - success) ~ C(group, Treatment('Electrode-Box-B2')) + day_c + day_c2
family   = Binomial (logit link)
cov_type = cluster (groups = subject)
```

`exp(coef)` is an **odds ratio** for a successful reach relative to Box-B2. This
controls for differing numbers of attempts, so it answers "who is more accurate
per attempt?" rather than "who attempts more?"

### (C) Learning rate

Poisson GEE with a **group × day** interaction, to ask whether groups improve at
different *rates* (not just different levels):

```
success ~ C(group, Treatment('Electrode-Box-B2')) * day_c + day_c2
```

Each `group[T.X]:day_c` term is the difference in log-slope versus Box-B2; a joint
Wald test asks whether *any* group's slope differs. Large p ⇒ parallel learning.

### Anchors-only model

The A2-vs-B2 comparison, refit on **just the two anchor groups** so nothing else
influences the shared day terms or the dispersion/correlation nuisance:

```
# subset to {Box-A2, Box-B2}
success ~ C(group, Treatment('Electrode-Box-B2')) + day_c + day_c2     # count
cbind(success, total-success) ~ C(group, ...) + day_c + day_c2         # rate
```

With B2 as reference the single group term is the A2-vs-B2 effect; H2 is a
one-sided test that this coefficient is below zero.

## How each subject's progress is accounted for

Two distinct pieces:

1. **Progress over training** — the `day_c + day_c2` fixed terms model the average
   learning curve, so groups are compared at comparable points in training rather
   than being confounded by *when* each was measured.

2. **Repeated measures / individual baselines** — each subject contributes many
   correlated sessions and has its own baseline. Handled two ways:

   - **Primary (GEE):** subject is the cluster; an exchangeable working
     correlation plus robust (sandwich) SEs give *population-average* group
     effects whose inference is valid under within-subject correlation and
     Poisson overdispersion.

   - **Sensitivity (mixed model):** a Poisson model with a **per-subject random
     intercept** (`(1 | subject)`) so each animal gets its own baseline level;
     the group effects are estimated after allowing for that individual variation.
     statsmodels has no frequentist Poisson GLMM, so this is the **MAP/Laplace**
     fit (`fit_vb` diverges on these large counts). Its p-values are approximate —
     read it as a direction/magnitude check that should agree with GEE.

## Classification logic (unknowns)

For each unknown group, `diff_contrast` builds a linear contrast of that group
against **each anchor** (both coded vs the B2 reference) and tests it:

- indistinguishable from B2 **and** different from A2 → **B2-like**
- indistinguishable from A2 **and** different from B2 → **A2-like**
- indistinguishable from both → **ambiguous** (report the numerically nearer one)
- different from both → **unlike both**

## Caveats

- **Tiny groups.** Box-A2/B2 have 3 subjects, Naive 4, and each unknown has
  **n = 1 subject**. Classifying a one-subject condition is weak: "matches
  anchor X" means "not statistically distinguishable from X," **not** proof of
  equivalence.
- **Single-cluster fragility.** Cluster-robust/GEE inference with one cluster in a
  group can produce artificially small SEs — most visibly the learning-rate
  interaction for the n=1 groups; do not read those p-values literally.
- **Mixed-model confounding.** With a per-subject random intercept, a group made
  of one subject is partly confounded with that subject's random intercept, so its
  fixed effect is shrunk.
- **Rate vs count.** "Success" alone is a count; the rate model (success/attempts)
  is the fairer accuracy comparison when attempt counts differ.

## Files and usage

- `tdcs_glm.py` — the analysis (run it directly).
- `tdcs_reach_data.csv` — the verified long-format data (`subject, group, day, success, total`).
- `tdcs_learning_curves*.png` — per-group learning curves (count and rate panels).

```
python3 analysis/tdcs_glm.py               # all training days, 5 groups
python3 analysis/tdcs_glm.py --max-day 10  # restrict to days 0–10
python3 analysis/tdcs_glm.py --merge       # assume Right==B2 and Box-A==A2 (3 groups)
```

Requires: pandas, numpy, scipy, statsmodels, matplotlib.