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LINEAR MIXED MODEL (days x tDCS) -- scenario: lme_mergeB2_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..13
(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 9 days). The
   interaction/slope over this window EXTRAPOLATES the shorter group's line and is
   CONFOUNDED -- prefer the _d0_13 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
    967.24    983.39    -477.62          955.24  

Fixed effects coefficients (95% CIs):
    Name                   Estimate    SE        tStat      DF     pValue    
    {'(Intercept)'}           21.7     5.9981     3.6178    105    0.00045861
    {'day'        }         6.4863     0.9412     6.8916    105    4.2336e-10
    {'tDCS'       }         17.493     7.1288     2.4538    105      0.015779
    {'day:tDCS'   }        -1.4651      1.025    -1.4294    105       0.15586


    Lower      Upper  
     9.8069     33.593
     4.6201     8.3526
     3.3578     31.628
    -3.4974    0.56726

Random effects covariance parameters (95% CIs):
Group: subject (8 Levels)
    Name1                  Name2                  Type           Estimate  
    {'(Intercept)'}        {'(Intercept)'}        {'std'}        2.1487e-15


    Lower    Upper
    NaN      NaN  

Group: Error
    Name               Estimate    Lower     Upper 
    {'Res Std'}        19.354      16.948    22.101


effect                      t(df) / F(df1)       p (resid)    Satterthwaite: p (df)
------------------------------------------------------------------------------
days x tDCS (interaction)   t(105)= -1.43 F(1)=  2.043  p=0.1559    p=0.1558 (df=109)
days (learning)             t(105)=  6.89 F(1)= 47.494  p=4.234e-10    p=3.734e-10 (df=109)
tDCS (main, at Day 1)       t(105)=  2.45 F(1)=  6.021  p=0.01578    p=0.01572 (df=109)

HONEST LME -- per-animal random slope (day|subject): interaction F(1,11.8)=0.02, p=0.9027
  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: n.s. (p=0.1559, slope diff=-1.47) -> slopes are parallel (no differential change over training).
  - days (learning): SIGNIFICANT (p=4.2e-10) -> performance improves with training.
  - tDCS main effect on Day 1 (our day 0): SIGNIFICANT (p=0.0158) -> 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.)
