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LINEAR MIXED MODEL (days x tDCS) -- scenario: lme_unmerged_d0_10
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model: success ~ day * tDCS + (1|subject)   [tDCS: Electrode-Box-B2 = 1 vs Electrode-Box-A2 = 0]
N = 6 subjects, 61 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              61
    Fixed effects coefficients           4
    Random effects coefficients          6
    Covariance parameters                2

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

Model fit statistics:
    AIC       BIC       LogLikelihood    Deviance
    499.59    512.25    -243.79          487.59  

Fixed effects coefficients (95% CIs):
    Name                   Estimate    SE         tStat      DF    pValue    
    {'(Intercept)'}        17.372       5.1906     3.3469    57     0.0014518
    {'day'        }        7.9666      0.79053     10.077    57    2.8317e-14
    {'tDCS'       }        4.9763       7.2862    0.68298    57       0.49739
    {'day:tDCS'   }        1.5577       1.0483     1.4858    57       0.14283


    Lower      Upper 
     6.9783    27.766
     6.3836    9.5496
     -9.614    19.567
    -0.5416    3.6569

Random effects covariance parameters (95% CIs):
Group: subject (6 Levels)
    Name1                  Name2                  Type           Estimate
    {'(Intercept)'}        {'(Intercept)'}        {'std'}        5.3536  


    Lower     Upper 
    2.1542    13.305

Group: Error
    Name               Estimate    Lower    Upper 
    {'Res Std'}        12.508      10.37    15.086


effect                      t(df) / F(df1)       p (resid)    Satterthwaite: p (df)
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days x tDCS (interaction)   t(57)=  1.49 F(1)=  2.208  p=0.1428    p=0.1428 (df=57)
days (learning)             t(57)= 10.08 F(1)=101.556  p=2.832e-14    p=1.931e-14 (df=59)
tDCS (main, at Day 1)       t(57)=  0.68 F(1)=  0.466  p=0.4974    p=0.5037 (df=17)

HONEST LME -- per-animal random slope (day|subject): interaction F(1,13.0)=1.46, p=0.2479
  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=0.1428, slope diff=1.56) -> slopes are parallel (no differential change over training).
  - days (learning): SIGNIFICANT (p=2.8e-14) -> performance improves with training.
  - tDCS main effect on Day 1 (our day 0): n.s. (p=0.4974) -> 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.)
