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LINEAR MIXED MODEL (days x tDCS) -- scenario: lme_unmerged_d0_13
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model: success ~ day * tDCS + (1|subject)   [tDCS: Electrode-Box-B2 = 1 vs Electrode-Box-A2 = 0]
N = 6 subjects, 70 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              70
    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
    586.72    600.22    -287.36          574.72  

Fixed effects coefficients (95% CIs):
    Name                   Estimate    SE         tStat      DF    pValue    
    {'(Intercept)'}          21.7       4.5484     4.7709    66    1.0537e-05
    {'day'        }        6.4863      0.71372     9.0881    66    3.0346e-13
    {'tDCS'       }        6.0226       6.3135    0.95392    66        0.3436
    {'day:tDCS'   }        1.5467      0.93755     1.6497    66       0.10376


    Lower       Upper 
      12.619    30.781
      5.0614    7.9113
     -6.5827    18.628
    -0.32523    3.4185

Random effects covariance parameters (95% CIs):
Group: subject (6 Levels)
    Name1                  Name2                  Type           Estimate
    {'(Intercept)'}        {'(Intercept)'}        {'std'}        0       


    Lower    Upper
    NaN      NaN  

Group: Error
    Name               Estimate    Lower     Upper
    {'Res Std'}        14.676      12.436    17.32


effect                      t(df) / F(df1)       p (resid)    Satterthwaite: p (df)
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days x tDCS (interaction)   t(66)=  1.65 F(1)=  2.721  p=0.1038    p=0.1035 (df=70)
days (learning)             t(66)=  9.09 F(1)= 82.593  p=3.035e-13    p=1.823e-13 (df=70)
tDCS (main, at Day 1)       t(66)=  0.95 F(1)=  0.910  p=0.3436    p=0.3434 (df=70)

HONEST LME -- per-animal random slope (day|subject): interaction F(1,12.8)=1.64, p=0.2227
  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=0.1038, slope diff=1.55) -> slopes are parallel (no differential change over training).
  - days (learning): SIGNIFICANT (p=3e-13) -> performance improves with training.
  - tDCS main effect on Day 1 (our day 0): n.s. (p=0.3436) -> 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.)
