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I have just discovered a better model for participants scores in the experiment. This used to be our prime model: ```prettyprint model <- lmer(score ~ nstates_amp_and_freq_n + (1 | id) + (1 |condition:phase) , data=all, REML=F) ``` I have discovered the following one outperforms this for both the original and the discrete datasets, and is much easier to interpret: ```prettyprint model <- lmer(score ~ nstates_amp_and_freq_n:phase_order:phase + (1|id), data=all, REML=F) ``` Of course, we are using MCMC samples, so the model declaration becomes:

```prettyprint model.glmm <- MCMCglmm(score ~ nstates_amp_and_freq_n:phase_order:phase, random=~id, data=all, nitt=100000, thin=100, pr = TRUE) ``` The result for the original dataset is: ```prettyprint > summary(model.glmm) Iterations = 3001:999501 Thinning interval  = 500 Sample size  = 1994 DIC: -8.19229 G-structure:  ~id       post.mean l-95% CI  u-95% CI eff.samp id 0.0005342        0 3.195e-11    32.48 R-structure:  ~units       post.mean l-95% CI u-95% CI eff.samp units   0.04708   0.0318  0.06755    526.8 Location effects: score ~ nstates_amp_and_freq_n:phase_order:phase                                            post.mean l-95% CI u-95% CI eff.samp   pMCMC     (Intercept)                                  0.63858  0.57880  0.70080     1994 < 5e-04 *** nstates_amp_and_freq_n:phase_order0:phase0   0.06740 -0.02253  0.15539     1994 0.14644     nstates_amp_and_freq_n:phase_order1:phase1   0.17233 -0.01143  0.36640     2316 0.06921 .   nstates_amp_and_freq_n:phase_order2:phase1   0.09099 -0.06046  0.22500     1781 0.19157     nstates_amp_and_freq_n:phase_order1:phase2   0.01965 -0.08214  0.13196     2114 0.73220     nstates_amp_and_freq_n:phase_order2:phase2   0.21595  0.06910  0.36329     3173 0.00301 ** --- Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 ``` The resulting regression for the discrete dataset is: ```prettyprint > summary(model.glmm) Iterations = 3001:999501  Thinning interval  = 500  Sample size  = 1994  DIC: -25.13271  G-structure:  ~id    post.mean   l-95% CI u-95% CI eff.samp id  0.006968 1.794e-204  0.02561    41.72  R-structure:  ~units       post.mean l-95% CI u-95% CI eff.samp units   0.03836  0.02075  0.05622    89.19  Location effects: score ~ nstates_amp_and_freq_n:phase_order:phase                                                 post.mean l-95% CI u-95% CI eff.samp  pMCMC     (Intercept)                                       0.81966  0.76106  0.87812   1994.0 < 5e-04 *** nstates_amp_and_freq_n:phase_order1st:phase1to1  -0.04302 -0.11980  0.03557   1994.0 0.2718     nstates_amp_and_freq_n:phase_order2nd:phase1to2   0.10226 -0.03057  0.25411   1489.9 0.1685     nstates_amp_and_freq_n:phase_order3rd:phase1to2   0.07254 -0.04080  0.18695    313.8 0.2207     nstates_amp_and_freq_n:phase_order2nd:phase2to2   0.14377  0.03020  0.26946   1994.0 0.0191 *   nstates_amp_and_freq_n:phase_order3rd:phase2to2   0.07734 -0.02952  0.18401    887.3 0.1374     --- Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 ``` \\( R^2 \\) values are 0.6153981 and 0.616292, respectively.
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