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LINEAR MIXED MODEL (days x tDCS) -- scenario: lme_mergeA2_full
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model: success ~ day * tDCS + (1|subject)   [tDCS: Electrode-Box-B2 = 1 vs Electrode-Box-A2 = 0]
N = 8 subjects, 109 sessions
day coverage: Box-B2 0..22, Box-A2 0..14
(day is raw and 0-indexed: our day 0 = the paper's "Day 1", so the tDCS
 main effect below is the group difference on Day 1 -- comparable to the paper.)
** WARNING: the groups' day coverage is UNEQUAL (differ by 8 days). The
   interaction/slope over this window EXTRAPOLATES the shorter group's line and is
   CONFOUNDED -- prefer the _d0_14 window (both groups have data throughout). **

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FULL MODEL SUMMARY -- fitlme: success ~ day*tDCS + (1|subject)
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Linear mixed-effects model fit by ML

Model information:
    Number of observations             109
    Fixed effects coefficients           4
    Random effects coefficients          8
    Covariance parameters                2

Formula:
    success ~ 1 + day*tDCS + (1 | subject)

Model fit statistics:
    AIC       BIC       LogLikelihood    Deviance
    962.68    978.83    -475.34          950.68  

Fixed effects coefficients (95% CIs):
    Name                   Estimate    SE         tStat      DF     pValue    
    {'(Intercept)'}         20.701      4.9127     4.2138    105    5.3256e-05
    {'day'        }         6.8599     0.69872     9.8178    105    1.5726e-16
    {'tDCS'       }         22.092      6.4372     3.4319    105    0.00085848
    {'day:tDCS'   }        -2.1793     0.81821    -2.6635    105     0.0089525


    Lower      Upper   
      10.96      30.442
     5.4745      8.2453
     9.3279      34.855
    -3.8016    -0.55692

Random effects covariance parameters (95% CIs):
Group: subject (8 Levels)
    Name1                  Name2                  Type           Estimate  
    {'(Intercept)'}        {'(Intercept)'}        {'std'}        2.1043e-15


    Lower    Upper
    NaN      NaN  

Group: Error
    Name               Estimate    Lower     Upper 
    {'Res Std'}        18.954      16.597    21.644


effect                      t(df) / F(df1)       p (resid)    Satterthwaite: p (df)
------------------------------------------------------------------------------
days x tDCS (interaction)   t(105)= -2.66 F(1)=  7.094  p=0.008953    p=0.008906 (df=109)
days (learning)             t(105)=  9.82 F(1)= 96.388  p=1.573e-16    p=1.113e-16 (df=109)
tDCS (main, at Day 1)       t(105)=  3.43 F(1)= 11.778  p=0.0008585    p=0.000848 (df=109)

HONEST LME -- per-animal random slope (day|subject): interaction F(1,9.7)=0.11, p=0.746
  Note: Satterthwaite DF on the random-INTERCEPT model above stays ~= residual
  (the slope's error is at the session level), so it does NOT fix pseudoreplication.
  Letting each animal have its OWN slope collapses the interaction DF toward the animal
  count -- this, and the per-animal slope test, are the honest learning-rate inference.

INTERPRETATION
  - days x tDCS interaction: SIGNIFICANT (p=0.0090, slope diff=-2.18) -> the tDCS (Box-B2) group improves SLOWER -- groups CONVERGE (Box-B2 is ahead early, the gap narrows).
  - days (learning): SIGNIFICANT (p=1.6e-16) -> performance improves with training.
  - tDCS main effect on Day 1 (our day 0): SIGNIFICANT (p=0.0009) -> the groups already DIFFER on Day 1.

Paper reference (N=24): interaction t(227)=2.68, F(1)=7.12, p=0.008;
  days t(227)=9.64, F(1)=267.64, p=1.2e-18; tDCS t(227)=0.23, F(1)=0.053, p=0.81.
  (Our N and exact statistics differ; this replicates the MODEL FORM on our data.)
