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LINEAR MIXED MODEL (days x tDCS) -- scenario: lme_mergeB2_d0_13
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model: success ~ day * tDCS + (1|subject)   [tDCS: Electrode-Box-B2 = 1 vs Electrode-Box-A2 = 0]
N = 8 subjects, 98 sessions
day coverage: Box-B2 0..13, 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.)

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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              98
    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
    803.97    819.48    -395.99          791.97  

Fixed effects coefficients (95% CIs):
    Name                   Estimate    SE         tStat      DF    pValue    
    {'(Intercept)'}        21.528       5.1581     4.1736    94    6.6907e-05
    {'day'        }        6.6603      0.67208       9.91    94     2.855e-16
    {'tDCS'       }        1.8428       6.4496    0.28572    94       0.77572
    {'day:tDCS'   }        1.5375      0.78742     1.9525    94      0.053848


    Lower        Upper 
       11.286    31.769
       5.3259    7.9948
      -10.963    14.649
    -0.025966    3.1009

Random effects covariance parameters (95% CIs):
Group: subject (8 Levels)
    Name1                  Name2                  Type           Estimate
    {'(Intercept)'}        {'(Intercept)'}        {'std'}        5.4251  


    Lower     Upper
    2.6045    11.3 

Group: Error
    Name               Estimate    Lower     Upper 
    {'Res Std'}        13.148      11.362    15.215


effect                      t(df) / F(df1)       p (resid)    Satterthwaite: p (df)
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days x tDCS (interaction)   t(94)=  1.95 F(1)=  3.812  p=0.05385    p=0.05377 (df=97)
days (learning)             t(94)=  9.91 F(1)= 98.208  p=2.855e-16    p=1.981e-16 (df=98)
tDCS (main, at Day 1)       t(94)=  0.29 F(1)=  0.082  p=0.7757    p=0.7776 (df=24)

HONEST LME -- per-animal random slope (day|subject): interaction F(1,29.7)=3.25, p=0.08138
  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=0.0538, slope diff=1.54) -> slopes are parallel (no differential change over training).
  - days (learning): SIGNIFICANT (p=2.9e-16) -> performance improves with training.
  - tDCS main effect on Day 1 (our day 0): n.s. (p=0.7757) -> 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.)
