Bayesian Estimation and Variable Selection for Group Differences in Markov Random Fields

The bgmCompare function estimates group differences in category threshold parameters (main effects) and pairwise interactions (pairwise effects) of a Markov Random Field (MRF) for binary and ordinal variables. Groups can be defined either by supplying two separate datasets (x and y) or by a group membership vector. Optionally, Bayesian variable selection can be applied to identify differences across groups.

Usage

bgmCompare(
  x,
  y,
  group_indicator,
  difference_selection = TRUE,
  main_difference_selection = FALSE,
  variable_type = "ordinal",
  baseline_category,
  difference_scale = 1,
  difference_prior = bernoulli_prior(0.5),
  difference_probability,
  interaction_prior = cauchy_prior(scale = 1),
  threshold_prior = beta_prime_prior(alpha = 0.5, beta = 0.5),
  iter = 2000,
  warmup = 2000,
  na_action = c("listwise", "impute"),
  update_method = c("nuts", "adaptive-metropolis"),
  target_accept,
  nuts_max_depth = 10,
  learn_mass_matrix = TRUE,
  chains = 4,
  cores = parallel::detectCores(),
  display_progress = c("per-chain", "total", "none"),
  seed = NULL,
  standardize = FALSE,
  verbose = getOption("bgms.verbose", TRUE),
  progress_callback = NULL,
  pairwise_scale,
  main_alpha,
  main_beta,
  beta_bernoulli_alpha,
  beta_bernoulli_beta,
  main_difference_model,
  reference_category,
  main_difference_scale,
  pairwise_difference_scale,
  pairwise_difference_prior,
  main_difference_prior,
  pairwise_difference_probability,
  main_difference_probability,
  pairwise_beta_bernoulli_alpha,
  pairwise_beta_bernoulli_beta,
  main_beta_bernoulli_alpha,
  main_beta_bernoulli_beta,
  interaction_scale,
  threshold_alpha,
  threshold_beta,
  burnin,
  save
)

Arguments

x

A data frame or matrix of binary and ordinal responses for Group 1. Variables should be coded as nonnegative integers starting at 0. For ordinal variables, unused categories are collapsed; for Blume–Capel variables, all categories are retained.

y

Optional data frame or matrix for Group 2 (two-group designs). Must have the same variables (columns) as x.

group_indicator

Optional integer vector of group memberships for rows of x (multi-group designs). Ignored if y is supplied.

difference_selection

Logical. If TRUE, spike-and-slab priors are applied to difference parameters. Default: TRUE.

main_difference_selection

Logical. If TRUE, apply spike-and-slab selection to main effect (threshold) differences. If FALSE, main effect differences are always included (no selection). Since main effects are often nuisance parameters and their selection can interfere with pairwise selection under the Beta-Bernoulli prior, the default is FALSE. Only used when difference_selection = TRUE.

variable_type

Character vector specifying type of each variable: "ordinal" (default) or "blume-capel".

baseline_category

Integer or vector giving the baseline category for Blume–Capel variables.

difference_scale

Double. Scale of the Cauchy prior for difference parameters. Default: 1.

difference_prior

An indicator prior specification object for difference selection, created by one of:

bernoulli_prior(): Fixed inclusion probability (default).
beta_bernoulli_prior(): Beta-distributed inclusion.
sbm_prior(): Stochastic Block Model.

Legacy character strings "Bernoulli" and "Beta-Bernoulli" are still accepted but deprecated. Default: bernoulli_prior(0.5).

difference_probability

Numeric. Use difference_prior = bernoulli_prior(probability) instead. Default: 0.5.

interaction_prior

A prior specification object for baseline pairwise interaction parameters, created by one of the prior constructor functions:

cauchy_prior(): Cauchy(0, scale) prior (default).
normal_prior(): Normal(0, scale) prior.

When supplied, overrides pairwise_scale. Default: cauchy_prior(scale = 1).

threshold_prior

A prior specification object for threshold (main effect) parameters, created by one of the prior constructor functions:

beta_prime_prior(): Beta-prime prior (default).
normal_prior(): Normal(0, scale) prior.

When supplied, overrides main_alpha and main_beta. Default: beta_prime_prior(alpha = 0.5, beta = 0.5).

iter

Integer. Number of post–warmup iterations per chain. Default: 2e3.

warmup

Integer. Number of warmup iterations before sampling. Default: 2e3.

na_action

Character. How to handle missing data: "listwise" (drop rows) or "impute" (impute within Gibbs). Default: "listwise".

update_method

Character. Sampling algorithm: "adaptive-metropolis" or "nuts". Default: "nuts".

target_accept

Numeric between 0 and 1. Target acceptance rate. Defaults: 0.44 (Metropolis), 0.80 (NUTS).

nuts_max_depth

Integer. Maximum tree depth for NUTS. Default: 10.

learn_mass_matrix

Logical. If TRUE, adapts a diagonal mass matrix during warmup (NUTS only). Default: TRUE.

chains

Integer. Number of parallel chains. Default: 4.

cores

Integer. Number of CPU cores. Default: parallel::detectCores().

display_progress

Character. Controls progress reporting: "per-chain", "total", or "none". Default: "per-chain".

seed

Optional integer. Random seed for reproducibility.

standardize

Logical. If TRUE, the Cauchy prior scale for each pairwise interaction (both baseline and difference) is adjusted based on the range of response scores. Without standardization, pairs with more response categories experience less shrinkage because their naturally smaller interaction effects make a fixed prior relatively wide. Standardization equalizes relative shrinkage across all pairs, with pairwise_scale itself applying to the unit interval (binary) case. See bgm for details on the adjustment. Default: FALSE.

verbose

Logical. If TRUE, prints informational messages during data processing (e.g., missing data handling, variable recoding). Defaults to getOption("bgms.verbose", TRUE). Set options(bgms.verbose = FALSE) to suppress messages globally.

progress_callback

An optional R function with signature function(completed, total) that is called at regular intervals during sampling, where completed is the number of iterations completed across all chains and total is the total number of iterations. Useful for external front-ends (e.g., JASP) that supply their own progress reporting. When NULL (the default), no callback is invoked.

pairwise_scale

Double. Scale of the Cauchy prior for baseline pairwise interactions. Default: 1.

main_alpha, main_beta

Doubles. Shape parameters of the beta-prime prior for baseline threshold parameters. Defaults: 0.5.

beta_bernoulli_alpha, beta_bernoulli_beta

Doubles. Shape parameters of the Beta prior for inclusion probabilities in the Beta–Bernoulli model. Defaults: 1.

main_difference_model, reference_category, pairwise_difference_scale, main_difference_scale, pairwise_difference_prior, main_difference_prior, pairwise_difference_probability, main_difference_probability, pairwise_beta_bernoulli_alpha, pairwise_beta_bernoulli_beta, main_beta_bernoulli_alpha, main_beta_bernoulli_beta, interaction_scale, threshold_alpha, threshold_beta, burnin, save

Deprecated arguments as of bgms 0.1.6.0. Use difference_scale, difference_prior, difference_probability, beta_bernoulli_alpha, beta_bernoulli_beta, baseline_category, pairwise_scale, and warmup instead.

Value

A list of class "bgmCompare" containing posterior summaries, posterior mean matrices, and raw MCMC samples:

posterior_summary_main_baseline, posterior_summary_pairwise_baseline: summaries of baseline thresholds and pairwise interactions.
posterior_summary_main_differences, posterior_summary_pairwise_differences: summaries of group differences in thresholds and pairwise interactions.
posterior_summary_indicator: summaries of inclusion indicators (if difference_selection = TRUE).
posterior_mean_main_baseline, posterior_mean_pairwise_baseline: posterior mean matrices (legacy style).
raw_samples: list of raw draws per chain for main, pairwise, and indicator parameters.
arguments: list of function call arguments and metadata.

The summary() method prints formatted summaries, and coef() extracts posterior means.

NUTS diagnostics (tree depth, divergences, energy, E-BFMI) are included in fit$nuts_diag if update_method = "nuts".

Details

Group-specific parameters are decomposed into a shared baseline plus group differences that sum to zero. Difference selection uses spike-and-slab priors (Bernoulli or Beta-Bernoulli). Parameters are sampled with NUTS (default) or adaptive Metropolis–Hastings, using the same multi-stage warmup schedule as bgm.

For full details on model specification, prior choices, and output interpretation, see the package website at https://bayesian-graphical-modelling-lab.github.io/bgms-docs/.

References

There are no references for Rd macro \insertAllCites on this help page.

Examples

# \dontrun{
# Run bgmCompare on subset of the Boredom dataset
x = Boredom[Boredom$language == "fr", 2:6]
y = Boredom[Boredom$language != "fr", 2:6]

fit = bgmCompare(x, y, chains = 2)
#> Chain 1 (Warmup): ⦗╺━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 50/4000 (1.2%)
#> Chain 2 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 51/4000 (1.3%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 101/8000 (1.3%)
#> Elapsed: 5s | ETA: 6m 31s
#> Chain 1 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 100/4000 (2.5%)
#> Chain 2 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 100/4000 (2.5%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 200/8000 (2.5%)
#> Elapsed: 10s | ETA: 6m 30s
#> Chain 1 (Warmup): ⦗━╺━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 150/4000 (3.8%)
#> Chain 2 (Warmup): ⦗━╺━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 140/4000 (3.5%)
#> Total   (Warmup): ⦗━╺━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 290/8000 (3.6%)
#> Elapsed: 11s | ETA: 4m 52s
#> Chain 1 (Warmup): ⦗━━╺━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 250/4000 (6.2%)
#> Chain 2 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 155/4000 (3.9%)
#> Total   (Warmup): ⦗━━╺━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 405/8000 (5.1%)
#> Elapsed: 12s | ETA: 3m 45s
#> Chain 1 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 300/4000 (7.5%)
#> Chain 2 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 295/4000 (7.4%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 595/8000 (7.4%)
#> Elapsed: 15s | ETA: 3m 6s
#> Chain 1 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 400/4000 (10.0%)
#> Chain 2 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 400/4000 (10.0%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 800/8000 (10.0%)
#> Elapsed: 16s | ETA: 2m 24s
#> Chain 1 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 500/4000 (12.5%)
#> Chain 2 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 498/4000 (12.4%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 998/8000 (12.5%)
#> Elapsed: 17s | ETA: 1m 59s
#> Chain 1 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 600/4000 (15.0%)
#> Chain 2 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 596/4000 (14.9%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1196/8000 (14.9%)
#> Elapsed: 18s | ETA: 1m 42s
#> Chain 1 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 700/4000 (17.5%)
#> Chain 2 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 684/4000 (17.1%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1384/8000 (17.3%)
#> Elapsed: 18s | ETA: 1m 26s
#> Chain 1 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 800/4000 (20.0%)
#> Chain 2 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 775/4000 (19.4%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1575/8000 (19.7%)
#> Elapsed: 19s | ETA: 1m 17s
#> Chain 1 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 900/4000 (22.5%)
#> Chain 2 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 893/4000 (22.3%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1793/8000 (22.4%)
#> Elapsed: 20s | ETA: 1m 9s
#> Chain 1 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1000/4000 (25.0%)
#> Chain 2 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 989/4000 (24.7%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1989/8000 (24.9%)
#> Elapsed: 21s | ETA: 1m 3s
#> Chain 1 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1100/4000 (27.5%)
#> Chain 2 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1091/4000 (27.3%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 2191/8000 (27.4%)
#> Elapsed: 21s | ETA: 56s
#> Chain 1 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1200/4000 (30.0%)
#> Chain 2 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1193/4000 (29.8%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 2393/8000 (29.9%)
#> Elapsed: 22s | ETA: 52s
#> Chain 1 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1300/4000 (32.5%)
#> Chain 2 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1294/4000 (32.4%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 2594/8000 (32.4%)
#> Elapsed: 23s | ETA: 48s
#> Chain 1 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1400/4000 (35.0%)
#> Chain 2 (Warmup): ⦗━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1404/4000 (35.1%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 2804/8000 (35.0%)
#> Elapsed: 23s | ETA: 43s
#> Chain 1 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1500/4000 (37.5%)
#> Chain 2 (Warmup): ⦗━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1521/4000 (38.0%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━━━━⦘ 3021/8000 (37.8%)
#> Elapsed: 24s | ETA: 40s
#> Chain 1 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1600/4000 (40.0%)
#> Chain 2 (Warmup): ⦗━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━━━⦘ 1625/4000 (40.6%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━━━⦘ 3225/8000 (40.3%)
#> Elapsed: 25s | ETA: 37s
#> Chain 1 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1700/4000 (42.5%)
#> Chain 2 (Warmup): ⦗━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━━⦘ 1712/4000 (42.8%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━━⦘ 3412/8000 (42.6%)
#> Elapsed: 28s | ETA: 38s
#> Chain 1 (Warmup): ⦗━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━━⦘ 1750/4000 (43.8%)
#> Chain 2 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1766/4000 (44.1%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 3516/8000 (44.0%)
#> Elapsed: 29s | ETA: 37s
#> Chain 1 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1800/4000 (45.0%)
#> Chain 2 (Warmup): ⦗━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━⦘ 1815/4000 (45.4%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━⦘ 3615/8000 (45.2%)
#> Elapsed: 30s | ETA: 36s
#> Chain 1 (Warmup): ⦗━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━⦘ 1850/4000 (46.2%)
#> Chain 2 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1866/4000 (46.7%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 3716/8000 (46.5%)
#> Elapsed: 31s | ETA: 36s
#> Chain 1 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1900/4000 (47.5%)
#> Chain 2 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━⦘ 1931/4000 (48.3%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━⦘ 3831/8000 (47.9%)
#> Elapsed: 31s | ETA: 34s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 2000/4000 (50.0%)
#> Chain 2 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━⦘ 2028/4000 (50.7%)
#> Total   (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━⦘ 4028/8000 (50.3%)
#> Elapsed: 32s | ETA: 32s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 2100/4000 (52.5%)
#> Chain 2 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━⦘ 2125/4000 (53.1%)
#> Total   (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━⦘ 4225/8000 (52.8%)
#> Elapsed: 33s | ETA: 29s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 2200/4000 (55.0%)
#> Chain 2 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━⦘ 2218/4000 (55.5%)
#> Total   (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━⦘ 4418/8000 (55.2%)
#> Elapsed: 33s | ETA: 27s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 2300/4000 (57.5%)
#> Chain 2 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━⦘ 2320/4000 (58.0%)
#> Total   (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━⦘ 4620/8000 (57.8%)
#> Elapsed: 34s | ETA: 25s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 2400/4000 (60.0%)
#> Chain 2 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━⦘ 2422/4000 (60.6%)
#> Total   (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━⦘ 4822/8000 (60.3%)
#> Elapsed: 34s | ETA: 22s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 2500/4000 (62.5%)
#> Chain 2 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━⦘ 2516/4000 (62.9%)
#> Total   (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━⦘ 5016/8000 (62.7%)
#> Elapsed: 35s | ETA: 21s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 2600/4000 (65.0%)
#> Chain 2 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━⦘ 2614/4000 (65.3%)
#> Total   (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━⦘ 5214/8000 (65.2%)
#> Elapsed: 36s | ETA: 19s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 2700/4000 (67.5%)
#> Chain 2 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━⦘ 2707/4000 (67.7%)
#> Total   (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━⦘ 5407/8000 (67.6%)
#> Elapsed: 36s | ETA: 17s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 2800/4000 (70.0%)
#> Chain 2 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 2796/4000 (69.9%)
#> Total   (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 5596/8000 (70.0%)
#> Elapsed: 37s | ETA: 16s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 2900/4000 (72.5%)
#> Chain 2 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 2897/4000 (72.4%)
#> Total   (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 5797/8000 (72.5%)
#> Elapsed: 37s | ETA: 14s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 3000/4000 (75.0%)
#> Chain 2 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 2986/4000 (74.7%)
#> Total   (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 5986/8000 (74.8%)
#> Elapsed: 38s | ETA: 13s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 3100/4000 (77.5%)
#> Chain 2 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 3082/4000 (77.0%)
#> Total   (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 6182/8000 (77.3%)
#> Elapsed: 38s | ETA: 11s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 3200/4000 (80.0%)
#> Chain 2 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 3177/4000 (79.4%)
#> Total   (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 6377/8000 (79.7%)
#> Elapsed: 39s | ETA: 10s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 3300/4000 (82.5%)
#> Chain 2 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 3272/4000 (81.8%)
#> Total   (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 6572/8000 (82.2%)
#> Elapsed: 40s | ETA: 9s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 3400/4000 (85.0%)
#> Chain 2 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 3374/4000 (84.4%)
#> Total   (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 6774/8000 (84.7%)
#> Elapsed: 40s | ETA: 7s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 3500/4000 (87.5%)
#> Chain 2 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 3477/4000 (86.9%)
#> Total   (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 6977/8000 (87.2%)
#> Elapsed: 41s | ETA: 6s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 3600/4000 (90.0%)
#> Chain 2 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 3566/4000 (89.1%)
#> Total   (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 7166/8000 (89.6%)
#> Elapsed: 41s | ETA: 5s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 3700/4000 (92.5%)
#> Chain 2 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 3672/4000 (91.8%)
#> Total   (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 7372/8000 (92.2%)
#> Elapsed: 42s | ETA: 4s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 3800/4000 (95.0%)
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#> Total   (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 7564/8000 (94.5%)
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#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 3900/4000 (97.5%)
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# Posterior inclusion probabilities
summary(fit)$indicator
#>                            parameter    mean        mcse        sd
#> 1                  loose_ends (main) 1.00000          NA 0.0000000
#> 2    loose_ends-entertain (pairwise) 0.02075 0.002465560 0.1425463
#> 3   loose_ends-repetitive (pairwise) 0.03400 0.004292323 0.1812291
#> 4  loose_ends-stimulation (pairwise) 0.18525 0.015385589 0.3885002
#> 5    loose_ends-motivated (pairwise) 0.02700 0.003277419 0.1620833
#> 6                   entertain (main) 1.00000          NA 0.0000000
#> 7    entertain-repetitive (pairwise) 0.02550 0.003236806 0.1576380
#> 8   entertain-stimulation (pairwise) 0.19600 0.014945503 0.3969685
#> 9     entertain-motivated (pairwise) 0.05275 0.005091077 0.2235340
#> 10                 repetitive (main) 1.00000          NA 0.0000000
#> 11 repetitive-stimulation (pairwise) 0.03250 0.003959743 0.1773239
#> 12   repetitive-motivated (pairwise) 0.02575 0.003076864 0.1583886
#> 13                stimulation (main) 1.00000          NA 0.0000000
#> 14  stimulation-motivated (pairwise) 0.02150 0.002717689 0.1450440
#> 15                  motivated (main) 1.00000          NA 0.0000000
#>    n0->0 n0->1 n1->0 n1->1 n_eff_mixt      Rhat
#> 1      0     0     0  3999         NA        NA
#> 2   3842    74    74     9  3342.5691 0.9998241
#> 3   3782    81    81    55  1782.6716 1.0015384
#> 4   3092   166   166   575   637.6088 1.0056350
#> 5   3812    80    79    28  2445.7532 1.0025602
#> 6      0     0     0  3999         NA        NA
#> 7   3823    74    74    28  2371.8576 1.0153935
#> 8   3026   189   189   595   705.4903 1.0011247
#> 9   3658   130   130    81  1927.8257 1.0125021
#> 10     0     0     0  3999         NA        NA
#> 11  3785    84    84    46  2005.3968 1.0335602
#> 12  3817    79    80    23  2649.9102 1.0021137
#> 13     0     0     0  3999         NA        NA
#> 14  3843    70    70    16  2848.3920 1.0008539
#> 15     0     0     0  3999         NA        NA

# Bayesian model averaged main effects for the groups
coef(fit)$main_effects_groups
#>                      group1     group2
#> loose_ends(c1)  -0.95016614 -0.9117283
#> loose_ends(c2)  -2.74702132 -2.2364094
#> loose_ends(c3)  -4.01588140 -3.5409158
#> loose_ends(c4)  -5.32208706 -4.8121972
#> loose_ends(c5)  -7.62241777 -7.3964166
#> loose_ends(c6)  -9.86066047 -9.9215537
#> entertain(c1)   -0.74624512 -1.0395753
#> entertain(c2)   -2.19242809 -2.2775450
#> entertain(c3)   -3.98731883 -3.6870683
#> entertain(c4)   -5.05364349 -5.1653601
#> entertain(c5)   -7.02064761 -6.9615941
#> entertain(c6)   -9.68859337 -9.4460012
#> repetitive(c1)  -0.04906668 -0.2836083
#> repetitive(c2)  -0.50124957 -0.9170959
#> repetitive(c3)  -1.03147910 -1.1356898
#> repetitive(c4)  -1.96462050 -1.7341976
#> repetitive(c5)  -3.56189321 -2.9714787
#> repetitive(c6)  -5.29598764 -4.6905399
#> stimulation(c1) -0.34748453 -0.8538983
#> stimulation(c2) -1.75528308 -1.8505253
#> stimulation(c3) -2.44253926 -2.6726572
#> stimulation(c4) -3.42234052 -3.8450436
#> stimulation(c5) -5.05128231 -5.2768116
#> stimulation(c6) -6.72244716 -7.3737635
#> motivated(c1)   -0.46094605 -0.7073385
#> motivated(c2)   -1.74051862 -1.8692412
#> motivated(c3)   -3.41909357 -3.1581994
#> motivated(c4)   -5.04804954 -4.5797147
#> motivated(c5)   -6.63256734 -6.6827406
#> motivated(c6)   -9.29664271 -8.8925731

# Bayesian model averaged pairwise effects for the groups
coef(fit)$pairwise_effects_groups
#>                            group1     group2
#> loose_ends-entertain   0.16906685 0.16925516
#> loose_ends-repetitive  0.05688985 0.05778803
#> loose_ends-stimulation 0.12297675 0.13176343
#> loose_ends-motivated   0.14052529 0.13992543
#> entertain-repetitive   0.06415804 0.06472679
#> entertain-stimulation  0.10422423 0.11309280
#> entertain-motivated    0.08383659 0.08549094
#> repetitive-stimulation 0.05581212 0.05657889
#> repetitive-motivated   0.13477296 0.13528928
#> stimulation-motivated  0.10738071 0.10765510
# }