# Getting started with bayesqm

This vignette walks through one Q study end to end in bayesqm: import, fit, diagnose, interpret, select K, and export. The running example is a synthetic 33-participant, 42-statement Q-sort.

    library(bayesqm)
    library(ggplot2)

## Setup

bayesqm performs inference with Stan, so you need a working Stan backend before the modelling functions will run. The package supports cmdstanr (recommended) and rstan.

cmdstanr is distributed through the Stan R-universe rather than CRAN, and installing the R package does not install CmdStan itself; that is a separate step. Run the following once (not evaluated here):

    install.packages("cmdstanr", repos = c("https://stan-dev.r-universe.dev", getOption("repos")))
    cmdstanr::check_cmdstan_toolchain(fix = TRUE)
    cmdstanr::install_cmdstan()
    cmdstanr::cmdstan_version()   # should print a version

On Windows you additionally need Rtools matching your R version; on macOS, run `xcode-select --install`. If you prefer rstan, install it from CRAN with a C++ toolchain; bayesqm uses whichever backend is available.

The remainder of this vignette uses small precomputed demonstration objects (`demo_fit()`, `demo_run()`) so it renders without a Stan backend. To reproduce the examples on real data you will need the backend installed as above.

## The Q-sort data

A Q study asks each participant to rank-order a set of statements into a forced distribution. `bayesqm` represents that as a `J × N` numeric matrix (statements as rows, participants as columns) plus a vector of forced-distribution counts. `read_qsort()` auto-detects the file format:

    qdata <- read_qsort("mystudy.csv")   # CSV / Excel / HTMLQ / FlashQ
    qdata <- read_qsort("mystudy.DAT")   # PQMethod
    qdata <- read_qsort("mystudy.zip")   # KADE project

The example dataset:

    sim   <- generate_data(N = 33, J = 42, K = 3, seed = 1)
    qdata <- qsort_data(sim$Y, distribution = sim$distribution)
    qdata
    #> Q-sort data
    #>   statements  : 42   participants : 33
    #>   distribution: 2 3 4 5 7 7 5 4 3 2   (sum = 42 )
    #>   value range : [-5, 5]
    #>   source      : manual

## Fitting the model

`fit_bayesian()` samples the posterior of a low-rank factor model with a Student-t likelihood (by default) and a hierarchical normal prior on the loadings, then resolves rotational ambiguity with the MatchAlign post-processing of Poworoznek et al. (2025):

    fit <- fit_bayesian(qdata, K = 3, chains = 4, iter = 2000,
                        warmup = 1000, seed = 1)

    fit <- demo_fit(N = 33, J = 42, K = 3, seed = 1)
    fit
    #> Bayesian Q-methodology factor model
    #>   Call:      fit_bayesian(Y, K = K)
    #>   Family:    Student-t (nu = estimated)
    #>   Factors:   K = 3
    #>   Data:      N = 33 persons, J = 42 statements
    #>   Draws:     4 chains x 1000 post-warmup = 4000 total
    #>   Backend:   demo
    #>   Fitted:    2026-06-09 18:15:53
    #>   Max Rhat:  1.010
    #>   Min ESS:   bulk 820 / tail 950
    #>   Divergent: 0
    #>
    #> Factor loadings (posterior median [95% CI], first 10 of 33 persons):
    #>     f1                  f2                  f3
    #> P1  0.82 [0.66, 0.97]   -0.04 [-0.21, 0.12] 0.02 [-0.15, 0.19]
    #> P2  -0.09 [-0.24, 0.07] 0.74 [0.58, 0.91]   0.13 [-0.03, 0.29]
    #> P3  0.04 [-0.13, 0.19]  -0.12 [-0.30, 0.04] 0.60 [0.44, 0.77]
    #> P4  0.69 [0.53, 0.86]   -0.03 [-0.20, 0.12] 0.08 [-0.06, 0.24]
    #> P5  0.06 [-0.08, 0.21]  0.78 [0.63, 0.94]   -0.04 [-0.19, 0.11]
    #> P6  0.13 [-0.02, 0.32]  -0.08 [-0.26, 0.08] 0.59 [0.42, 0.71]
    #> P7  0.67 [0.50, 0.82]   -0.04 [-0.19, 0.13] -0.15 [-0.32, 0.01]
    #> P8  0.11 [-0.03, 0.25]  0.74 [0.59, 0.89]   -0.00 [-0.16, 0.15]
    #> P9  -0.01 [-0.15, 0.12] -0.10 [-0.25, 0.04] 0.75 [0.60, 0.90]
    #> P10 0.68 [0.51, 0.83]   -0.12 [-0.27, 0.04] 0.06 [-0.08, 0.22]
    #>   ... (23 more; see fit$loa_median / fit$ci_lower / fit$ci_upper)
    #>
    #> Hyperparameters:
    #>  parameter mean median    sd lower upper
    #>         nu 20.0   20.1 3.818 12.41 26.89
    #>      sigma  0.5    0.5 0.083  0.34  0.67
    #>        tau  0.5    0.5 0.082  0.34  0.66
    #>
    #> Use summary() for factor characteristics, the divergence summary, and LOO.

`summary(fit)` adds factor characteristics, the PSIS-LOO ELPD (when available), a divergence summary, and the MatchAlign diagnostic.

## Diagnostics

Convergence statistics are stored on `fit$diagnostics`. The recommended thresholds are R-hat below 1.01, bulk and tail effective sample sizes above 400, and zero divergent transitions (Vehtari et al. 2021):

    fit$diagnostics
    #> $rhat_max
    #> [1] 1.01
    #>
    #> $ess_bulk
    #> [1] 820
    #>
    #> $ess_tail
    #> [1] 950
    #>
    #> $divergences
    #> [1] 0

`plot_tucker()` visualises the per-draw Tucker phi between each aligned factor column and the MatchAlign pivot. Values near 1 indicate a stable alignment; bimodality signals residual label-switching.

    plot_tucker(fit)

MatchAlign alignment quality by factor.

## Factor loadings

Each participant has a posterior distribution over their loading on every factor. `plot_loading_posterior()` draws the loading forest, with nested 50% and 95% credible-interval whiskers and the classical Brown (1980) cut-off as a faint reference. Filled points are participants with `P(factor k dominant) > 0.5`.

    plot_loading_posterior(fit)

Loadings with nested 50% and 95% credible intervals.

The same information as a tidy data frame:

    head(compute_loadings(fit$Lambda_draws, prob = 0.95))
    #>   participant      f1_loa    f1_lower   f1_upper      f2_loa   f2_lower
    #> 1          P1  0.82218583  0.65967029 0.97347665 -0.04267266 -0.2145922
    #> 2          P2 -0.08886156 -0.24309784 0.07209832  0.74522036  0.5791526
    #> 3          P3  0.03940137 -0.12972350 0.19405389 -0.12287342 -0.2986770
    #> 4          P4  0.69365900  0.53025182 0.85839822 -0.03571152 -0.1965369
    #> 5          P5  0.05989579 -0.07856949 0.21054518  0.78202658  0.6258714
    #> 6          P6  0.12982142 -0.02363006 0.31509987 -0.08730420 -0.2556340
    #>     f2_upper      f3_loa    f3_lower  f3_upper
    #> 1 0.12335192  0.01982447 -0.14521623 0.1878235
    #> 2 0.91064671  0.13579980 -0.02821464 0.2948461
    #> 3 0.03507329  0.60443673  0.44105818 0.7671924
    #> 4 0.12328690  0.08182326 -0.06409235 0.2354870
    #> 5 0.93757307 -0.03495072 -0.19214191 0.1134495
    #> 6 0.07939546  0.58198954  0.41551318 0.7077884

## Factor z-scores

`plot()` produces a cross-panel z-score dotchart; rows share an order (by factor 1’s posterior mean, configurable via `sort_by`) so factors can be compared horizontally.

    plot(fit)

Posterior z-scores across factors with 50% and 95% credible intervals.

For a single statement across factors, `plot_zscore_posterior()` shows the per-factor view:

    plot_zscore_posterior(fit, statement = 1)

Posterior z-score for a single statement across factors.

## Choosing K

`run_bayes()` fits the model for each K in a range and applies the peak-plus-Sivula protocol: the ELPD peak is the automated choice and the Sivula parsimony rule (Sivula et al. 2025) is reported alongside as a diagnostic.

On real data:

    run <- run_bayes(qdata, K_max = 4, seed = 1,
                     chains = 2, iter = 1500, warmup = 800)

    run <- demo_run(K_max = 5, k_peak = 3, k_sivula = 1, case = "gap")
    run
    #> Bayesian Q-methodology: multi-K comparison
    #>   K range:      1..5
    #>   ELPD peak:    K = 3  (automated adoption)
    #>   Sivula rule:  K = 1  (parsimony diagnostic)
    #>   Case:         gap  (ELPD peak > Sivula (weak discrimination between adjacent models))
    #>
    #> LOO comparison:
    #>  K    elpd   se delta_elpd se_delta ratio
    #>  1 -180.09 8.00
    #>  2 -170.91 7.25      -9.18     3.00  3.06
    #>  3 -165.42 6.50      -5.49     3.00  1.83
    #>  4 -170.20 5.75       4.78     3.00  1.59
    #>  5 -179.61 5.00       9.41     3.00  3.14
    #>
    #> Case 'gap': k_peak is adopted; Sivula is reported as a parsimony diagnostic only.

    make_elpd_diff(run)

ΔELPD across K with the Sivula non-promotion band at |ΔELPD| < 4. Triangle marks the Sivula K, square marks the ELPD peak (the adopted K).

The ELPD peak is always the adopted K. The Sivula diagnostic is reported alongside as a parsimony check. The `run$case` field labels the relationship between the two: when Sivula lands on the same K as the peak, the case is `agree`; when Sivula points to a smaller K than the peak, the case is `gap`; when Sivula exceeds the peak, the case is `reversed` (rare in practice).

## Distinguishing, consensus, and membership

`bayesqm` replaces the classical z-score test with the posterior of an explicit viewpoint-divergence measure. For each statement, D_j is the mean absolute pairwise difference of the K viewpoint scores; the package reports P(D_j > delta | Y) and P(D_j < delta | Y), the probabilities that the viewpoints diverge meaningfully or practically agree. By default the separation delta is the reliability-adjusted critical difference from classical Q analysis (Brown 1980), computed from the posterior dominant-factor counts (`critical_delta()`); `suggest_delta()` (one forced-distribution category on the standardised scale) is an alternative, and results are reported with sensitivity across the choice of delta. No fixed probability cutoff defines a statement.

The full distinguishing/consensus table is stored on `fit$qdc` (per viewpoint: grid position and z-score with 95% CrI, then `D_j` with 95% CrI, `pi_D`, `pi_C`). `critical_delta()` returns the separation the fit used:

    critical_delta(fit$Lambda_draws)
    #> [1] 0.4131967
    head(fit$qdc)
    #>   statement f1_grid     f1_zsc    f1_lower   f1_upper f2_grid     f2_zsc
    #> 1        S1      -1 -0.3436199 -0.75598677  0.1066338      -1 -0.5924826
    #> 2        S2       1  0.3356155 -0.07696179  0.7385355      -1 -0.2080451
    #> 3        S3       4  1.4979823  1.11477391  1.9446729      -3 -1.2921554
    #> 4        S4      -2 -1.2847601 -1.75639810 -0.8298201       4  1.4848346
    #> 5        S5       5  1.5174761  1.01119678  1.9237407      -5 -1.3155732
    #> 6        S6       1 -0.1660726 -0.64869137  0.2149788      -1 -0.1410127
    #>     f2_lower   f2_upper f3_grid      f3_zsc    f3_lower   f3_upper  D_median
    #> 1 -1.0039154 -0.1486896      -1 -0.15075033 -0.55964521  0.2793994 0.3381550
    #> 2 -0.6461072  0.2744403       2  0.34784928 -0.06702773  0.7702793 0.4427912
    #> 3 -1.7067359 -0.8610203      -3 -1.29522175 -1.79044451 -0.8143189 1.9525495
    #> 4  1.0077216  1.9142497      -5 -1.30747085 -1.76660521 -0.8738692 1.9386399
    #> 5 -1.7515059 -0.8978920      -4 -1.30529856 -1.77439164 -0.8363799 1.9742330
    #> 6 -0.5929771  0.3163001       1  0.01668352 -0.42748367  0.4427847 0.2651122
    #>      D_lower   D_upper   pi_D   pi_C
    #> 1 0.06190555 0.7146339 0.3725 0.6275
    #> 2 0.13551050 0.8479464 0.5825 0.4175
    #> 3 1.58449898 2.2832984 1.0000 0.0000
    #> 4 1.55679839 2.3175306 1.0000 0.0000
    #> 5 1.54433252 2.3631369 1.0000 0.0000
    #> 6 0.04975712 0.6005598 0.1900 0.8100

    plot_dist_cons(fit)

Posterior viewpoint divergence D_j with 95% credible intervals; statements ordered by P(D_j > delta | Y).

Participant-level membership is also probabilistic. `classify_membership()` turns posterior dominance into a `Strong / Moderate / Weak` tier:

    head(classify_membership(fit$Lambda_draws))
    #>   participant dominant_factor dominant_label max_prob   tier
    #> 1          P1               1             f1        1 Strong
    #> 2          P2               2             f2        1 Strong
    #> 3          P3               3             f3        1 Strong
    #> 4          P4               1             f1        1 Strong
    #> 5          P5               2             f2        1 Strong
    #> 6          P6               3             f3        1 Strong

    make_dominant_panel(fit)

Blue tiles: posterior probability that each factor is dominant for a given participant (values printed in each cell). The right column shows the overall assignment verdict on an orange-red scale.

## Posterior predictive check

`plot_ppc()` shows the posterior distribution of the RMSE between `cor(Y_rep)` and `cor(Y_obs)`. A well-specified model puts most posterior mass at small RMSE.

    plot_ppc(fit)

PPC on the by-person correlation matrix.

## Hyperparameters

    plot_hyper(fit)

Posterior densities of nu, sigma, and tau.

## Reporting and exporting

`save_bayesqm_plot()` writes any plot to PDF, SVG, PNG, TIFF, or JPEG at the chosen dimensions:

    save_bayesqm_plot("fig_loadings.pdf", plot_loading_posterior(fit))
    save_bayesqm_plot("fig_elpd.pdf",      plot_elpd(run),
                      width = 3.5, height = 3)

`caption_bayesqm()` returns a figure caption with model configuration, sample sizes, interval probability, and convergence diagnostics:

    caption_bayesqm(fit)
    #> [1] "Bayesian Q-methodology factor model (K = 3, N = 33, J = 42); fitted with a Student-t likelihood via demo (4 chains, 4,000 post-warmup draws); intervals shown at 95% posterior coverage; max Rhat = 1.010, min bulk ESS = 820, 0 divergent transitions. Fitted with the bayesqm R package."

Standard R accessors work on the fit (`coef`, `fitted`, `residuals`, `sigma`, `family`, `as.matrix`), and `as.matrix(fit)` returns a Stan-style draws matrix that `posterior` and `bayesplot` consume natively.

## Theming

Every plot reads its palette through `bayesqm_colors()`. Switch the scheme once and every subsequent base-R plot follows:

    bayesqm_set_colors("teal")   # "blue" (default), "red", "purple", "grey"
    plot(fit)
    bayesqm_set_colors("blue")

## Where next

- `?fit_bayesian`: every prior and sampler option
- `?run_bayes`: peak-plus-Sivula thresholds
- `?bayesqm-membership`: the full set of membership summaries
- `?rename_factors`: relabel `f1..fK` with substantive factor names
- `ggplot2::autoplot()`: ggplot2 / ggdist versions of every view above, when `ggplot2` and `ggdist` are installed

## References

Brown, Steven R. 1980. Political Subjectivity: Applications of q Methodology in Political Science. Yale University Press.
Poworoznek, Evan, Niccolo Anceschi, Federico Ferrari, and David Dunson. 2025. “Efficiently Resolving Rotational Ambiguity in Bayesian Matrix Sampling with Matching.” Bayesian Analysis, 1–22. https://doi.org/10.1214/25-BA1544.
Sivula, Tuomas, Måns Magnusson, Asael Alonzo Matamoros, and Aki Vehtari. 2025. “Uncertainty in Bayesian Leave-One-Out Cross-Validation Based Model Comparison.” Bayesian Analysis, 1–31. https://doi.org/10.1214/25-BA1569.
Vehtari, Aki, Andrew Gelman, Daniel Simpson, Bob Carpenter, and Paul-Christian Bürkner. 2021. “Rank-Normalization, Folding, and Localization: An Improved \(\widehat{R}\) for Assessing Convergence of MCMC (with Discussion).” Bayesian Analysis 16 (2): 667–718. https://doi.org/10.1214/20-BA1221.