Posterior Model Checks for the Shifted Lognormal Mixture Models

1 Overview

This document reports posterior model checks for the four shifted lognormal mixed-effects mixture models. The checks use the saved Stan fit objects in stanout/smog_exp*.rda. These files are RDS objects despite the .rda extension and are read with readRDS(). They are large, so the document loads and checks one experiment at a time.

2 Diagnostics by Experiment

2.1 Experiment 1

2.1.1 HMC diagnostics

Experiment 1 HMC diagnostics by chain.

chain	divergent	max_treedepth	ebfmi	mean_accept_stat	mean_stepsize
1	0	0	0.735	0.993	0.0201
2	0	0	0.738	0.984	0.0300
3	0	0	0.818	0.992	0.0215

2.1.2 R-hat and effective sample size

Experiment 1 summary of R-hat and effective sample size.

parameters	max_Rhat	parameters_Rhat_above_1_01	parameters_Rhat_above_1_05	min_n_eff	parameters_n_eff_below_400
17	1.007	0	0	658	0

Experiment 1 parameters with the largest R-hat values.

parameter	mean	sd	n_eff	Rhat
beta[2]	5.200	0.040	658	1.007
sigma_u[1]	0.311	0.030	1276	1.006
beta[1]	5.941	0.055	1312	1.003
shift	6.514	0.034	1198	1.003
sigma_diff[2]	0.194	0.007	7989	1.002
sigma	0.552	0.009	5092	1.001
theta[6]	-1.283	0.112	5941	1.001
theta[2]	-1.512	0.126	6112	1.001
theta[4]	-1.399	0.121	5843	1.001
theta[3]	-1.197	0.190	6193	1.000
theta[5]	0.173	0.197	6376	1.000
theta[1]	0.593	0.225	6372	1.000
delta[2]	0.267	0.029	6994	1.000
sigma_diff[1]	0.143	0.013	9058	1.000
sigma_u[2]	0.243	0.024	2296	1.000
sigma_w	0.091	0.009	11623	1.000
delta[1]	0.703	0.039	8052	1.000

2.1.3 Traceplots

2.1.4 Posterior predictive checks

2.2 Experiment 2

2.2.1 HMC diagnostics

Experiment 2 HMC diagnostics by chain.

chain	divergent	max_treedepth	ebfmi	mean_accept_stat	mean_stepsize
1	0	0	0.801	0.985	0.0326
2	0	0	0.817	0.986	0.0310
3	0	0	0.792	0.982	0.0374

2.2.2 R-hat and effective sample size

Experiment 2 summary of R-hat and effective sample size.

parameters	max_Rhat	parameters_Rhat_above_1_01	parameters_Rhat_above_1_05	min_n_eff	parameters_n_eff_below_400
37	1.006	0	0	570	0

Experiment 2 parameters with the largest R-hat values.

parameter	mean	sd	n_eff	Rhat
beta[3]	5.206	0.040	570	1.006
beta[1]	6.109	0.048	817	1.004
beta[2]	6.026	0.053	1029	1.003
sigma_u[2]	0.222	0.019	2978	1.001
sigma_u[1]	0.367	0.028	1185	1.001
shift	6.483	0.029	1495	1.001
theta[15]	-0.146	0.295	13737	1.000
theta[5]	0.102	0.291	13944	1.000
theta[21]	-0.308	0.275	14624	1.000
delta[1]	0.427	0.044	12179	1.000
theta[24]	-1.108	0.131	16329	1.000
sigma_diff[2]	0.200	0.006	9855	1.000
theta[3]	-0.458	0.277	15417	1.000
sigma	0.562	0.007	6400	1.000
theta[17]	0.250	0.245	16046	1.000
theta[11]	-1.089	0.264	16153	1.000
theta[4]	-1.476	0.146	17952	1.000
delta[2]	0.599	0.058	16967	1.000
theta[20]	-1.310	0.136	17189	1.000
theta[7]	-0.238	0.263	15903	1.000

2.2.3 Traceplots

2.2.4 Posterior predictive checks

2.3 Experiment 3

2.3.1 HMC diagnostics

Experiment 3 HMC diagnostics by chain.

chain	divergent	max_treedepth	ebfmi	mean_accept_stat	mean_stepsize
1	0	0	0.792	0.986	0.0332
2	0	0	0.748	0.985	0.0348
3	0	0	0.763	0.993	0.0249

2.3.2 R-hat and effective sample size

Experiment 3 summary of R-hat and effective sample size.

parameters	max_Rhat	parameters_Rhat_above_1_01	parameters_Rhat_above_1_05	min_n_eff	parameters_n_eff_below_400
46	1.010	0	0	341	1

Experiment 3 parameters with the largest R-hat values.

parameter	mean	sd	n_eff	Rhat
sigma_u[1]	0.318	0.023	830	1.010
beta[3]	5.204	0.031	341	1.009
beta[1]	6.228	0.043	580	1.006
beta[2]	6.207	0.041	582	1.006
shift[1]	6.821	0.025	1100	1.003
shift[2]	6.426	0.026	1196	1.003
sigma_u[2]	0.205	0.016	1917	1.001
theta[6]	-1.473	0.131	15976	1.000
theta[9]	1.087	0.394	6057	1.000
sigma	0.536	0.005	6626	1.000
theta[1]	0.620	0.310	10719	1.000
theta[2]	-1.272	0.132	15186	1.000
theta[24]	-1.204	0.124	15511	1.000
theta[21]	-0.936	0.217	13563	1.000
theta[22]	-1.020	0.128	13599	1.000
theta[14]	-1.558	0.146	15374	1.000
theta[11]	0.698	0.296	11463	1.000
theta[8]	-1.305	0.133	14688	1.000
theta[27]	-0.390	0.218	12381	1.000
sigma_diff[2]	0.175	0.004	11745	1.000

2.3.3 Traceplots

2.3.4 Posterior predictive checks

Posterior predictive plots were skipped because the saved posterior predictive draws do not match the reconstructed model data. This usually means that the cleaning or aggregation script has changed since this Stan fit was generated. Refit the model with the current data-processing scripts to regenerate matching y_tilde draws.

Experiment 3 posterior predictive data check.

check	y_tilde observations in saved fit	observations reconstructed from current scripts
n_mismatch	30771	30762

2.4 Experiment 4

2.4.1 HMC diagnostics

Experiment 4 HMC diagnostics by chain.

chain	divergent	max_treedepth	ebfmi	mean_accept_stat	mean_stepsize
1	0	0	0.782	0.986	0.0300
2	0	0	0.781	0.995	0.0170
3	0	0	0.803	0.985	0.0311

2.4.2 R-hat and effective sample size

Experiment 4 summary of R-hat and effective sample size.

parameters	max_Rhat	parameters_Rhat_above_1_01	parameters_Rhat_above_1_05	min_n_eff	parameters_n_eff_below_400
45	1.009	0	0	676	0

Experiment 4 parameters with the largest R-hat values.

parameter	mean	sd	n_eff	Rhat
beta[3]	6.265	0.026	676	1.009
beta[1]	6.660	0.032	912	1.006
beta[2]	6.667	0.034	1041	1.005
sigma_u[1]	0.201	0.016	1273	1.003
delta[1]	0.238	0.020	17238	1.000
sigma	0.392	0.006	6016	1.000
theta[1]	0.017	0.237	14514	1.000
sigma_diff[2]	0.229	0.006	9276	1.000
sigma_u[2]	0.545	0.028	3879	1.000
sigma_diff[1]	0.166	0.006	11671	1.000
theta[31]	-0.487	0.219	14876	1.000
shift	5.442	0.077	1652	1.000
theta[9]	0.459	0.283	12047	1.000
theta[25]	-0.171	0.221	14536	1.000
theta[28]	-0.639	0.173	21959	1.000
theta[14]	-0.390	0.185	20966	1.000
theta[8]	-0.788	0.182	21841	1.000
theta[5]	0.006	0.223	15047	1.000
theta[23]	-0.980	0.240	11538	1.000
theta[27]	-0.031	0.226	14130	1.000

2.4.3 Traceplots

2.4.4 Posterior predictive checks

3 Session Information

## 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         knitr_1.51          ggplot2_4.0.2      
## [10] dplyr_1.2.0        
## 
## 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] timechange_0.3.0      evaluate_1.0.5        grid_4.5.2           
## [10] RColorBrewer_1.1-3    fastmap_1.2.0         jsonlite_2.0.0       
## [13] pkgbuild_1.4.8        gridExtra_2.3         QuickJSR_1.8.1       
## [16] scales_1.4.0          codetools_0.2-20      jquerylib_0.1.4      
## [19] cli_3.6.5             crayon_1.5.3          rlang_1.1.7          
## [22] bit64_4.6.0-1         withr_3.0.2           cachem_1.1.0         
## [25] yaml_2.3.12           otel_0.2.0            tools_4.5.2          
## [28] inline_0.3.21         parallel_4.5.2        tzdb_0.5.0           
## [31] curl_7.0.0            vctrs_0.7.1           R6_2.6.1             
## [34] lubridate_1.9.4       matrixStats_1.5.0     stats4_4.5.2         
## [37] lifecycle_1.0.5       bit_4.6.0             V8_8.0.1             
## [40] vroom_1.6.7           pkgconfig_2.0.3       RcppParallel_5.1.11-1
## [43] pillar_1.11.1         bslib_0.10.0          gtable_0.3.6         
## [46] loo_2.9.0             glue_1.8.0            Rcpp_1.1.1           
## [49] xfun_0.56             tidyselect_1.2.1      rstudioapi_0.18.0    
## [52] farver_2.1.2          htmltools_0.5.9       labeling_0.4.3       
## [55] rmarkdown_2.30        compiler_4.5.2        S7_0.2.1