This supplement reports model checks for the Bayesian models reported in the manuscript. The checks cover:

  1. The reanalysis of existing datasets: M1, M2, M3, and M4, including the individual dataset fits.
  2. Experiment 1: M2, M3, and M4.
  3. Experiment 2: M2, M3, and M4.

The checks include convergence summaries, trace plots, and posterior-predictive checks.

Reanalysis of Existing Datasets

The reanalysis diagnostics include the individual study-level models reported in the model-comparison table. We inspect M1–M4 where the corresponding model was fitted for that dataset.

Parameters with Rhat above 1.01 in the individual reanalysis models.
Study Model Description Parameter Rhat n_eff Mean SD
Roeser et al. (2019, Exp. 3, writing) M3 M3 sigma_w 1.028 170 0.067 0.030
Roeser et al. (2019, Exp. 1, speech) M3 M3 sigma_w 1.021 292 0.015 0.011
Roeser et al. (2019, Exp. 3, writing) M2 M2 sigma_w 1.020 167 0.078 0.032
Roeser et al. (2019, Exp. 3, speech) M2 M2 sigma_w 1.014 349 0.047 0.018
Roeser et al. (2019, Exp. 3, writing) M4 M4 sigma_w 1.013 86 0.063 0.031
Roeser et al. (2019, Exp. 2, writing) M1 M1 sigma_w 1.011 329 0.115 0.034
Roeser et al. (2019, Exp. 1, speech) M4 M4 sigma_w 1.010 193 0.018 0.012

For each study and model, trace plots are shown next to the corresponding posterior-predictive density overlay. The posterior-predictive x-axis is shown on a log scale to make both the bulk of the distribution and the right tail visible. To keep the reanalysis compact, plots are organised in tabs by study.

Reanalysis Checks

Hardy et al. (2020)

Hardy et al. (2022)

Martin et al. (2010, Exp. 3a)

Martin et al. (2010, Exp. 3b)

Martin et al. (2010, Exp. 4a)

Martin et al. (2010, Exp. 4b)

Roeser et al. (2019, Exp. 1, speech)

Roeser et al. (2019, Exp. 1, writing)

Roeser et al. (2019, Exp. 2, speech)

Roeser et al. (2019, Exp. 2, writing)

Roeser et al. (2019, Exp. 3, speech)

Roeser et al. (2019, Exp. 3, writing)

Experiment 1

Convergence summaries for Experiment 1 models.
Model Description n_parameters max_Rhat min_n_eff n_Rhat_above_1.01
M2 Optional-scope mixture 10 1.001 358 0
M3 Optional-scope mixture with off-task noise 13 1.010 150 1
M4 Default-scope mixture with off-task noise 11 1.010 238 1

Experiment 1 Checks

M2

M3

M4

Experiment 2

Convergence summaries for Experiment 2 models.
Model Description n_parameters max_Rhat min_n_eff n_Rhat_above_1.01
M2 Optional-scope mixture 10 1.000 2224 0
M3 Optional-scope mixture with off-task noise 13 1.001 2249 0
M4 Default-scope mixture with off-task noise 11 1.004 1769 0

Experiment 2 Checks

M2

M3

M4

Session Information

sessionInfo()
## R version 4.5.2 (2025-10-31)
## Platform: x86_64-pc-linux-gnu
## Running under: Ubuntu 24.04.4 LTS
## 
## Matrix products: default
## BLAS:   /usr/lib/x86_64-linux-gnu/blas/libblas.so.3.12.0 
## LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.12.0  LAPACK version 3.12.0
## 
## locale:
##  [1] LC_CTYPE=en_GB.UTF-8       LC_NUMERIC=C              
##  [3] LC_TIME=en_US.UTF-8        LC_COLLATE=en_GB.UTF-8    
##  [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_GB.UTF-8   
##  [7] LC_PAPER=en_US.UTF-8       LC_NAME=C                 
##  [9] LC_ADDRESS=C               LC_TELEPHONE=C            
## [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C       
## 
## time zone: Europe/London
## tzcode source: system (glibc)
## 
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
## 
## other attached packages:
##  [1] tidyr_1.3.2         tibble_3.3.1        stringr_1.6.0      
##  [4] rstan_2.32.7        StanHeaders_2.32.10 readr_2.1.6        
##  [7] purrr_1.2.1         patchwork_1.3.2     knitr_1.51         
## [10] ggplot2_4.0.2       dplyr_1.2.0         bayesplot_1.15.0   
## 
## loaded via a namespace (and not attached):
##  [1] sass_0.4.10           generics_0.1.4        stringi_1.8.7        
##  [4] hms_1.1.4             digest_0.6.39         magrittr_2.0.4       
##  [7] evaluate_1.0.5        grid_4.5.2            RColorBrewer_1.1-3   
## [10] fastmap_1.2.0         jsonlite_2.0.0        pkgbuild_1.4.8       
## [13] gridExtra_2.3         QuickJSR_1.8.1        scales_1.4.0         
## [16] codetools_0.2-20      jquerylib_0.1.4       cli_3.6.5            
## [19] rlang_1.1.7           withr_3.0.2           cachem_1.1.0         
## [22] yaml_2.3.12           otel_0.2.0            tools_4.5.2          
## [25] inline_0.3.21         parallel_4.5.2        tzdb_0.5.0           
## [28] curl_7.0.0            vctrs_0.7.1           R6_2.6.1             
## [31] matrixStats_1.5.0     stats4_4.5.2          lifecycle_1.0.5      
## [34] V8_8.0.1              pkgconfig_2.0.3       RcppParallel_5.1.11-1
## [37] pillar_1.11.1         bslib_0.10.0          gtable_0.3.6         
## [40] loo_2.9.0             glue_1.8.0            Rcpp_1.1.1           
## [43] xfun_0.56             tidyselect_1.2.1      rstudioapi_0.18.0    
## [46] farver_2.1.2          htmltools_0.5.9       rmarkdown_2.30       
## [49] compiler_4.5.2        S7_0.2.1