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LINEAR MIXED MODEL (days x tDCS) -- scenario: lme_mergeA2_d0_10
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model: success ~ day * tDCS + (1|subject)   [tDCS: Electrode-Box-B2 = 1 vs Electrode-Box-A2 = 0]
N = 8 subjects, 83 sessions
day coverage: Box-B2 0..10, Box-A2 0..10
(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.)

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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              83
    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
    673.84    688.35    -330.92          661.84  

Fixed effects coefficients (95% CIs):
    Name                   Estimate    SE         tStat      DF    pValue    
    {'(Intercept)'}        15.599       3.7298     4.1822    79    7.4136e-05
    {'day'        }        8.3258      0.66962     12.434    79    2.7592e-20
    {'tDCS'       }        4.1737       5.2381    0.79679    79       0.42796
    {'day:tDCS'   }        1.3833       0.9137      1.514    79       0.13402


    Lower       Upper 
       8.175    23.023
      6.9929    9.6586
     -6.2525      14.6
    -0.43535     3.202

Random effects covariance parameters (95% CIs):
Group: subject (8 Levels)
    Name1                  Name2                  Type           Estimate
    {'(Intercept)'}        {'(Intercept)'}        {'std'}        0       


    Lower    Upper
    NaN      NaN  

Group: Error
    Name               Estimate    Lower    Upper 
    {'Res Std'}        13.04       11.2     15.183


effect                      t(df) / F(df1)       p (resid)    Satterthwaite: p (df)
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days x tDCS (interaction)   t(79)=  1.51 F(1)=  2.292  p=0.134    p=0.1338 (df=83)
days (learning)             t(79)= 12.43 F(1)=154.593  p=2.759e-20    p=1.192e-20 (df=83)
tDCS (main, at Day 1)       t(79)=  0.80 F(1)=  0.635  p=0.428    p=0.4278 (df=83)

HONEST LME -- per-animal random slope (day|subject): interaction F(1,20.3)=1.35, p=0.258
  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=0.1340, slope diff=1.38) -> slopes are parallel (no differential change over training).
  - days (learning): SIGNIFICANT (p=2.8e-20) -> performance improves with training.
  - tDCS main effect on Day 1 (our day 0): n.s. (p=0.4280) -> the groups are comparable (as in the paper) 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.)
