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LINEAR MIXED MODEL (days x tDCS) -- scenario: lme_mergeB2_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
    669.06    683.57    -328.53          657.06  

Fixed effects coefficients (95% CIs):
    Name                   Estimate    SE         tStat      DF    pValue   
    {'(Intercept)'}         17.318      5.1725      3.348    79    0.0012488
    {'day'        }         7.9975     0.75795     10.551    79    9.547e-17
    {'tDCS'       }        0.42782      6.5088    0.06573    79      0.94776
    {'day:tDCS'   }         1.7407     0.91342     1.9057    79     0.060324


    Lower       Upper 
      7.0221    27.613
      6.4888    9.5061
     -12.528    13.383
    -0.07739    3.5588

Random effects covariance parameters (95% CIs):
Group: subject (8 Levels)
    Name1                  Name2                  Type           Estimate
    {'(Intercept)'}        {'(Intercept)'}        {'std'}        5.7075  


    Lower     Upper 
    2.7829    11.705

Group: Error
    Name               Estimate    Lower     Upper 
    {'Res Std'}        11.955      10.184    14.034


effect                      t(df) / F(df1)       p (resid)    Satterthwaite: p (df)
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days x tDCS (interaction)   t(79)=  1.91 F(1)=  3.632  p=0.06032    p=0.06034 (df=79)
days (learning)             t(79)= 10.55 F(1)=111.334  p=9.547e-17    p=8.241e-17 (df=80)
tDCS (main, at Day 1)       t(79)=  0.07 F(1)=  0.004  p=0.9478    p=0.9482 (df=21)

HONEST LME -- per-animal random slope (day|subject): interaction F(1,30.6)=3.00, p=0.09319
  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=0.0603, slope diff=1.74) -> slopes are parallel (no differential change over training).
  - days (learning): SIGNIFICANT (p=9.5e-17) -> performance improves with training.
  - tDCS main effect on Day 1 (our day 0): n.s. (p=0.9478) -> 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.)
