fit_mfrm: Fit many-facet ordered-response models with a flexible number...

In mfrmr: Estimation and Diagnostics for Many-Facet Measurement Models

Fit many-facet ordered-response models with a flexible number of facets

Description

This is the package entry point. It wraps mfrm_estimate() and defaults to method = "MML". Any number of facet columns can be supplied via facets. The RSM / PCM branches are the package's many-facet Rasch-family reference route; the bounded GPCM branch is available where explicitly documented.

Usage

fit_mfrm(
  data,
  person,
  facets,
  score,
  rating_min = NULL,
  rating_max = NULL,
  weight = NULL,
  keep_original = FALSE,
  missing_codes = NULL,
  model = c("RSM", "PCM", "GPCM"),
  method = c("MML", "JML", "JMLE"),
  step_facet = NULL,
  slope_facet = NULL,
  facet_interactions = NULL,
  min_obs_per_interaction = 10,
  interaction_policy = c("warn", "error", "silent"),
  anchors = NULL,
  group_anchors = NULL,
  noncenter_facet = "Person",
  dummy_facets = NULL,
  positive_facets = NULL,
  anchor_policy = c("warn", "error", "silent"),
  min_common_anchors = 5L,
  min_obs_per_element = 30,
  min_obs_per_category = 10,
  quad_points = 31,
  maxit = 400,
  reltol = 1e-06,
  mml_engine = c("direct", "em", "hybrid"),
  population_formula = NULL,
  person_data = NULL,
  person_id = NULL,
  population_policy = c("error", "omit"),
  facet_shrinkage = c("none", "empirical_bayes", "laplace"),
  facet_prior_sd = NULL,
  shrink_person = FALSE,
  attach_diagnostics = FALSE,
  checkpoint = NULL
)

Arguments

data	A data.frame in long format with one row per observed rating event.
person	Column name for the person (character scalar).
facets	Character vector of facet column names.
score	Column name for the observed ordered category score. Values must be coercible to numeric integer category codes. Fractional values are rejected. Binary 0/1 or 1/2 responses are supported as the ordered two-category special case. When keep_original = FALSE, unused intermediate categories are collapsed to a contiguous internal scale and the mapping is recorded in fit$prep$score_map. If rating_min / rating_max are supplied and the observed scores are a contiguous subset of that range (for example a 1-5 scale with only 2-5 observed), the supplied full range is retained so zero-count boundary categories remain part of the fitted score support.
rating_min	Optional minimum category value. Supply this with rating_max when the intended score scale includes unobserved boundary categories.
rating_max	Optional maximum category value. Supply this with rating_min when the intended score scale includes unobserved boundary categories.
weight	Optional weight column name.
keep_original	Logical. FALSE (the current default) collapses non-consecutive observed categories to a contiguous internal scale and records the mapping in fit$prep$score_map (the downstream Count = 0 rows are consequently absent). TRUE preserves the declared scale so unused intermediate categories remain visible in rating_scale_table() and APA outputs, which is recommended for publication reporting.
missing_codes	Optional pre-processing step that converts sentinel missing-code values to NA across the person, facets, and score columns before any downstream logic. One of: NULL (default): no recoding; strictly backward-compatible. TRUE or "default": FACETS / SPSS / SAS convention set ("99", "999", "-1", "N", "NA", "n/a", ".", ""). Character vector: an explicit code set, e.g. c("99", "999", ".a"). Replacement counts are recorded in fit$prep$missing_recoding and surfaced by build_mfrm_manifest(). Equivalent to calling recode_missing_codes() manually before the fit.
model	"RSM", "PCM", or bounded "GPCM".
method	"MML" (default) or "JML". "JMLE" is accepted as a backward-compatible alias for the same joint-maximum-likelihood path.
step_facet	Step facet for PCM and the bounded GPCM branch. For GPCM, this should be supplied explicitly rather than relying on an implicit default.
slope_facet	Slope facet for the bounded GPCM branch. The current release requires slope_facet == step_facet and uses a positive-slope identification convention on the log scale with geometric mean discrimination fixed to 1.
facet_interactions	Optional confirmatory two-way interaction terms between non-person facets, supplied as explicit character terms such as "Rater:Criterion" or as a list of length-two character vectors. These interactions are estimated simultaneously as fixed effects in RSM and PCM fits. Person-involving interactions, higher-order interactions, and random-effect interaction terms are outside the current scope.
min_obs_per_interaction	Minimum weighted observations recommended for each interaction cell. Cells below this value are flagged in interaction_effect_table() and handled according to interaction_policy.
interaction_policy	How to handle sparse interaction cells: "warn" (default), "error", or "silent".
anchors	Optional anchor table.
group_anchors	Optional group-anchor table.
noncenter_facet	One facet to leave non-centered.
dummy_facets	Facets to fix at zero.
positive_facets	Facets with positive orientation.
anchor_policy	How to handle anchor-review issues: "warn" (default), "error", or "silent".
min_common_anchors	Minimum anchored levels per linking facet used in anchor-review recommendations.
min_obs_per_element	Minimum weighted observations per facet level used in anchor-review recommendations.
min_obs_per_category	Minimum weighted observations per score category used in anchor-review recommendations.
quad_points	Integer number of Gauss-Hermite quadrature points used for MML integration over the person distribution. Default is 31, chosen so that marginal log-likelihood values are stable enough for direct manuscript reporting. Recommended tiers: 7 fast exploratory scan; in-package helpers such as predict_mfrm_population() and reference_case_benchmark() use this value. 15 intermediate analysis when runtime matters. 31 default / publication tier. ⁠61+⁠ ultra-precise runs when benchmarking or working on very narrow score supports. Internal benchmarks show the marginal log-likelihood still drifts by ~0.5-1 logit between quad_points = 15 and quad_points = 61 on moderately sized designs, which is why the default now sits at the publication tier; set a lower value explicitly for exploratory runs.
maxit	Maximum optimizer iterations.
reltol	Optimization tolerance.
mml_engine	MML optimization engine for method = "MML": "direct" (default) uses direct BFGS on the marginal log-likelihood, "em" uses an EM loop for RSM / PCM with population = NULL, and "hybrid" uses EM as a warm start before the direct optimizer. Unsupported combinations currently fall back to "direct" and record that fallback in fit$summary.
population_formula	Optional one-sided formula for a person-level latent-regression population model, for example ~ grade + ses. In the current release, latent regression is implemented only for method = "MML" with a unidimensional conditional-normal population model.
person_data	Optional one-row-per-person data.frame holding background variables for population_formula. Numeric, logical, factor, ordered factor, and character predictors are expanded through stats::model.matrix(); categorical xlevels and contrasts are stored for replay and scoring. Required when population_formula is supplied.
person_id	Optional person-ID column in person_data. Defaults to person when that column exists in person_data.
population_policy	How missing background data are handled for a latent-regression fit. "error" (default) requires complete person-level covariates; "omit" fits the model on the complete-case subset and records omitted persons / omitted response rows in the returned population metadata while retaining the observed-person-aligned pre-omit table for replay/export provenance.
facet_shrinkage	Character. "none" (default) keeps the 0.1.5 fixed-effects behaviour. "empirical_bayes" applies a post-hoc James-Stein / empirical-Bayes shrinkage to each non-person facet (Efron & Morris, 1973); fit$facets$others gains ShrunkEstimate, ShrunkSE, and ShrinkageFactor columns, and fit$shrinkage_report records the per-facet prior variance and effective degrees of freedom. "laplace" currently aliases to "empirical_bayes" and is reserved for a future penalised-MML implementation.
facet_prior_sd	Optional numeric scalar. When supplied, the shrinkage prior variance is fixed at facet_prior_sd^2 instead of being estimated by method of moments. Useful for eliciting a prior from domain knowledge or a previous fit.
shrink_person	Logical. When TRUE and facet_shrinkage is active, the same empirical-Bayes shrinkage is applied to fit$facets$person. Default FALSE, since MML already integrates over an N(0, 1) prior on theta; the option mainly benefits JML.
attach_diagnostics	Logical. When TRUE, diagnose_mfrm() is run once after the fit with residual_pca = "none", and the per-level SE, Infit, Outfit, InfitZSTD, OutfitZSTD, and PtMeaCorr columns from diagnostics$measures are merged onto fit$facets$others (non-person facets) and fit$facets$person (Person rows). This is convenient when downstream code expects a FACETS Table 7 style facet table with fit statistics in one place, and lets summary(fit) show per-person fit columns alongside the measure. For person rows, an existing posterior SE (typical for method = "MML") is preserved and the diagnostic SE is only attached when the existing column is empty. Adds diagnostic runtime (typically +1-2 s on moderate designs) and sets fit$config$attached_diagnostics = TRUE. Default FALSE preserves the minimal Facet / Level / Estimate layout.
checkpoint	Optional list(file = ..., every_iter = ...). When supplied, the MML EM engine writes its state to file every every_iter outer EM iterations using saveRDS(). If the file already exists when the fit starts, the engine resumes from the recorded iteration. Only the EM engine (mml_engine = "em" or the EM warm-start step of mml_engine = "hybrid") honours the checkpoint; the direct optim() engine ignores it. Use this to make long MML EM fits crash-resilient on shared compute environments.

Details

Data must be in long format (one row per observed rating event).

Value

An object of class mfrm_fit (named list) with:

• summary: one-row model summary (LogLik, AIC, BIC, convergence) including public Method, internal MethodUsed, and MMLEngineRequested, MMLEngineUsed, and EMIterations for MML fits

• facets$person: person estimates (Estimate; plus SD for MML)

• facets$others: facet-level estimates for each facet

• steps: estimated threshold/step parameters as a one-row-per-step tibble with Estimate. Bare fits keep this table as point estimates. diagnose_mfrm() exposes MML observed-information step uncertainty in diagnostics$parameter_uncertainty$steps; when attach_diagnostics = TRUE, those SE, confidence-limit, and status columns are attached to fit$steps when the Hessian is available. For step-structure quality, also use the step-collapse and disordering warnings from diagnose_mfrm() and category_structure_report().

• slopes: estimated discrimination parameters for GPCM fits as a one-row-per-slope-element tibble with LogEstimate and Estimate. Bare fits keep this table as point estimates. For MML bounded-GPCM fits, diagnose_mfrm() exposes log-slope SEs plus positive-scale delta-method SEs and confidence limits in diagnostics$parameter_uncertainty$slopes; when attach_diagnostics = TRUE, those columns are attached to fit$slopes when the Hessian is available. The identification convention pins the geometric mean of slopes at 1.

• interactions: model-estimated facet interaction effects and metadata when facet_interactions is supplied

• population: population-model metadata. Ordinary fits keep an inactive scaffold (active = FALSE, posterior_basis = "legacy_mml"). Active latent-regression fits store the fitted design matrix, regression coefficients, residual variance, omission review, the complete-case estimation table (person_table), and the observed-person-aligned replay/export provenance table retained before complete-case omission (person_table_replay), plus stored categorical xlevels / contrasts for model-matrix replay and scoring, together with posterior_basis = "population_model".

• config: resolved model configuration used for estimation, including config$anchor_review

• prep: preprocessed data/level metadata

• opt: raw optimizer result from stats::optim()

Model

fit_mfrm() estimates many-facet ordered-response models. The RSM and PCM branches follow the many-facet Rasch-family tradition (Linacre, 1989); the bounded GPCM branch extends the partial-credit kernel with estimated positive slopes under the package's documented identification constraints. For the equal-slope RSM/PCM branch, a two-facet design (rater j, criterion i) is:

\ln\frac{P(X_{nij} = k)}{P(X_{nij} = k-1)} = \theta_n - \delta_j - \beta_i - \tau_k

where \theta_n is person ability, \delta_j rater severity, \beta_i criterion difficulty, and \tau_k the k-th Rasch-Andrich threshold. Any number of facets may be specified via the facets argument; each enters as an additive term in the linear predictor \eta.

With model = "RSM", thresholds \tau_k are shared across all levels of all facets. With model = "PCM", each level of step_facet receives its own threshold vector \tau_{i,k} on the package's shared observed score scale.

With bounded model = "GPCM", the adjacent-category kernel is multiplied by a positive slope for the designated slope-facet level:

\ln\frac{P(X_{nij} = k)}{P(X_{nij} = k-1)} = \alpha_g(\eta - \tau_{g,k}),\quad \alpha_g > 0.

The current implementation requires slope_facet == step_facet and identifies slopes by a sum-to-zero constraint on log slopes, so their geometric mean is 1.

With only two ordered categories (K = 1), the RSM/PCM branch reduces to the usual binary Rasch logit for the single category boundary:

\ln\frac{P(X_{n\cdot} = 1)}{P(X_{n\cdot} = 0)} = \eta - \tau_1

Bounded GPCM uses the slope-scaled counterpart \alpha_g(\eta - \tau_{g,1}).

With method = "MML", person parameters are integrated out using Gauss-Hermite quadrature and EAP estimates are computed post-hoc. With method = "JML", all parameters are estimated jointly as fixed effects. "JMLE" remains an accepted compatibility alias, but package output now uses "JML" as the public label. See the "Estimation methods" section of mfrmr-package for details.

Weighting policy

mfrmr treats RSM / PCM as the equal-weighting reference route for operational many-facet measurement. In that Rasch-family branch, discrimination is fixed, so the scoring model does not differentially reweight item-facet combinations through estimated slopes.

Bounded GPCM is supported as an alternative when users explicitly accept discrimination-based reweighting. This often improves model fit, but the package does not treat better fit alone as a sufficient reason to replace an equal-weighting Rasch-family model.

The weight argument is separate from that modeling choice. It supplies an observation-weight column; it does not create a free-form facet-weighting scheme and does not change the fixed-discrimination contract of RSM / PCM.

Input requirements

Minimum required columns are:

• person identifier (person)

• one or more facet identifiers (facets)

• observed score (score)

Scores are treated as ordered categories. Non-numeric score labels are dropped with a warning after coercion, whereas fractional numeric scores are rejected with an error instead of being silently truncated.

The fitted many-facet ordered-response model assumes conditional independence of observations given the person and facet parameters (Linacre, 1989). Repeated ratings of the same person-criterion combination by the same rater violate this assumption. When such structures may be present, follow fitting with diagnose_mfrm(fit, diagnostic_mode = "both"); its strict_pairwise_local_dependence screen is an exploratory check for residual dependence beyond what the additive linear predictor absorbs.

Binary responses are therefore supported as ordered two-category scores (for example 0/1 or 1/2) under the same ordered-response interface. If your observed categories do not start at 0, set rating_min/rating_max explicitly to avoid unintended recoding assumptions. For example, if the intended instrument is a 1-5 scale but the current sample only uses 2-5, set ⁠rating_min = 1, rating_max = 5⁠ to retain the zero-count category 1 in the score support. If these bounds are omitted, the observed score range is used and the provenance is stored in fit$prep and summary(fit)$settings_overview. Set options(mfrmr.show_inferred_rating_range = TRUE) when you want an interactive reminder whenever a bound is inferred. Data-preparation events such as row drops, ID trimming, duplicate person-by-facet cells, and single-level facets are stored in fit$prep$row_retention and fit$prep$preparation_notes. Routine row-drop/trim/single-level messages are quiet by default; set options(mfrmr.show_preparation_messages = TRUE) to show them during interactive checks.

When keep_original = FALSE, observed gaps such as ⁠1, 3, 5⁠ are recoded internally to a contiguous scale (⁠1, 2, 3⁠) and the mapping is stored in fit$prep$score_map. To retain zero-count intermediate categories as part of the original scale, set keep_original = TRUE in addition to supplying the full rating_min / rating_max range.

Fixed effects assumption (facets have no prior)

fit_mfrm() follows the Linacre (1989) many-facet Rasch specification: person ability is integrated out under a N(0, 1) prior (or under the ⁠N(X\beta, \sigma^2)⁠ latent-regression population model when population_formula is supplied), but every facet parameter (Rater, Criterion, Task, ...) is estimated as a fixed effect identified by a sum-to-zero constraint. There is no hierarchical prior, no shrinkage, and no variance component for the facets.

Practical implication: when a facet has very few observed levels (for example 3 raters) or some of its levels have very few ratings (for example 5 ratings per rater), the fixed-effect estimates retain wide SEs, and extreme estimates are not pulled toward the facet mean. Jones and Wind (2018) note that rater estimates in particular are "more sensitive to link reductions" than examinee or task estimates. For a publication-workflow review of this, use:

• facet_small_sample_review() for per-level N and SE bands against Linacre (1994) sample-size guidelines.

• detect_facet_nesting() and analyze_hierarchical_structure() when raters are nested in regions, schools, or other strata that the additive fixed-effects MFRM cannot partition out.

• compute_facet_icc() and compute_facet_design_effect() for descriptive variance- component summaries based on lme4 (optional).

fit$summary$FacetSampleSizeFlag summarizes the worst Linacre band across non-person facet levels ("sparse" < 10, "marginal" < 30, "standard" < 50, "strong" >= 50).

JML estimator caveat (use MML for final reporting)

Joint maximum likelihood (method = "JML" / "JMLE") estimates both the structural parameters (facets, thresholds, slopes) and every person measure as fixed parameters in one optimization. This is the incidental-parameter problem of Neyman & Scott (1948): the structural parameter estimates are inconsistent as the number of persons grows with the number of items per person held fixed, carrying a bias of order 1/L (where L is the number of items per person) that does not vanish with sample size. Wright & Stone (1979) and Wright & Masters (1982, ch. 5) document an empirical (L-1)/L correction that approximately removes the bias for the dichotomous Rasch model; mfrmr does not apply that correction (no bias_correction argument exists). The JML branch also does not produce a profile-likelihood Hessian for the structural parameters: SEs reported under JML are observation-table approximations (1/\sqrt{\sum \mathrm{Var}(X_{pi})}) and are marked as exploratory in the diagnostics output.

Practical recommendation:

• Use method = "MML" for any value reported in a manuscript or operational decision. MML integrates the person measures out under a population prior and produces consistent structural estimates with marginal observed-information SEs.

• Use method = "JML" only for fast exploratory iteration, the classical FACETS-style workflow, or contexts where the bias is tolerable (large L per person, descriptive screening, or teaching).

• When a third-party CML estimator is needed (the only consistent Rasch-family estimator under the incidental-parameter setting), fit with eRm and import via import_erm_fit().

Model-estimated facet interactions

facet_interactions adds confirmatory fixed-effect interaction terms to the linear predictor. For example, facet_interactions = "Rater:Criterion" estimates a rater-by-criterion deviation matrix in the same likelihood as the main MFRM fit. The additive reference is

\eta_{nij} = \theta_n - \delta_j - \beta_i

and the interaction extension is

\eta_{nij} = \theta_n - \delta_j - \beta_i + \gamma_{ji}

where the interaction block is identified by zero marginal sums:

\sum_j \gamma_{ji} = 0,\quad \sum_i \gamma_{ji} = 0.

With J levels of the first facet and I levels of the second facet, this contributes (J - 1)(I - 1) free parameters. Positive interaction estimates indicate scores higher than expected under the additive main-effects model for that facet-level combination; negative estimates indicate lower-than-expected scores.

This is a model-estimated interaction term, not the residual screening reported by estimate_bias() or estimate_all_bias(). In line with the MFRM bias-interaction literature, the facet pair should be named explicitly before fitting. Exploratory use is possible, but should be reported as screening, with sparse-cell and multiplicity caveats. The current implementation is intentionally narrow: two-way non-person facet interactions for RSM and PCM only, estimated as fixed effects. GPCM interactions, person interactions, higher-order interactions, and random-effect facet interactions are deferred.

This is ordered binary support, not a separate nominal-response model. In PCM, a binary fit still uses one threshold per step_facet level on the shared observed-score scale.

Supported model/estimation combinations in the current release:

• model = "RSM" with method = "MML" or "JML"/"JMLE"

• model = "PCM" with a designated step_facet (defaults to first facet)

• facet_interactions with model = "RSM" or "PCM" for explicit two-way non-person facet interactions

• model = "GPCM" is currently implemented only for the narrow bounded branch with slope_facet == step_facet; MML and JML fitting, core summaries, fixed-calibration posterior scoring, compute_information(), Wright/pathway/CCC fit plots, diagnose_mfrm(), residual-PCA follow-up, interrater_agreement_table(), unexpected_response_table(), displacement_table(), measurable_summary_table(), rating_scale_table(), facet_quality_dashboard(), reporting_checklist(), category_structure_report(), category_curves_report(), and graph/scorefile facets_output_file_bundle() routes are available with score-side caveats. Direct simulation specifications and data generation are also supported through build_mfrm_sim_spec(), extract_mfrm_sim_spec(), and simulate_mfrm_data() when the slope-aware generator contract is stored explicitly; direct recovery checks are available through evaluate_mfrm_recovery() and assess_mfrm_recovery(). Slope-aware fair_average_table() and estimate_bias() are available with their documented caveats. Role-based design evaluation, population forecasting, diagnostic-screening, and signal-detection helpers are available as caveated sensitivity evidence. Full FACETS-style score-side contract review, posterior predictive checks, and heavy backend routes should be treated as unsupported unless documented otherwise. Use gpcm_capability_matrix() as the formal boundary statement for the current GPCM scope.

Latent-regression status:

• population_formula = NULL keeps the legacy unconditional MML / JML behavior.

• Supplying population_formula activates a first-version latent-regression branch for method = "MML" only.

• The current branch assumes a one-dimensional conditional-normal population model with person-specific quadrature nodes \theta_{nq} = x_n^\top \beta + \sigma z_q.

• Background variables must be supplied in person_data; numeric/logical columns and categorical factor/character columns are expanded through stats::model.matrix().

• Current overlap with the ConQuest latent-regression documentation is limited to direct estimation from response data under a unidimensional MML population model with package-built model-matrix covariates. It should not be described as numerical equivalence for arbitrary imported design matrices, multidimensional models, or the full ConQuest plausible-values workflow.

• predict_mfrm_units() and sample_mfrm_plausible_values() can score latent-regression fits under the fitted population model, but they require one-row-per-person background data for scored units when the fitted population model includes covariates. Intercept-only latent-regression fits (population_formula = ~ 1) can reconstruct that minimal person table internally during scoring.

Latent-regression workflow

For an initial latent-regression run, keep the setup explicit:

1. Put response data in data, with one row per rating event.

2. Put background variables in person_data, with exactly one row per person. The ID column must match person, or be supplied through person_id.

3. Use method = "MML" and a one-sided formula such as population_formula = ~ Grade + Group.

4. Numeric/logical and factor/character predictors are expanded with stats::model.matrix(). After fitting, inspect summary(fit)$population_coding to see the fitted levels, contrasts, and encoded design columns that will be reused for scoring/replay.

5. Start with population_policy = "error" while preparing data. Use "omit" only when complete-case removal is intended, and then inspect summary(fit)$population_overview and summary(fit)$caveats before reporting results.

6. Report summary(fit)$population_coefficients as coefficients of the conditional-normal latent population model, not as a post hoc regression on EAP or MLE scores.

Latent-regression standard-error caveat

summary(fit)$population_coefficients reports point estimates of \hat{\boldsymbol{\beta}} and \hat{\sigma}^2 only. mfrmr does not currently compute standard errors, confidence intervals, or asymptotic z / Wald statistics for the population-model parameters: no Hessian on (\boldsymbol{\beta}, \log\sigma^2) is extracted from the marginal log-likelihood, and no vcov() method is exposed for these coefficients. Treat the coefficient table as point estimates suitable for descriptive reporting; do not quote \hat{\beta}_j \pm 1.96 \cdot \mathrm{SE} bounds because the SE column is not provided. A marginal-Hessian-based SE for (\boldsymbol{\beta}, \sigma^2) is planned for a future release.

Identification: the latent-regression intercept is identifiable only under the default noncenter_facet = "Person" (which sum-to-zero- centers all non-Person facets). If you re-anchor identification on a non-Person facet, the intercept becomes confounded with the freed Person-facet mean and the coefficient table becomes unidentified; mfrmr does not currently warn about this failure mode in the design-matrix check.

Anchor inputs are optional:

• anchors should contain facet/level/fixed-value information.

• group_anchors should contain facet/level/group/group-value information. Both are normalized internally, so column names can be flexible (facet, level, anchor, group, groupvalue, etc.).

Anchor review behavior:

• fit_mfrm() runs an internal anchor review.

• invalid rows are removed before estimation.

• duplicate rows keep the last occurrence for each key.

• anchor_policy controls whether detected issues are warned, treated as errors, or kept silent.

Facet sign orientation:

• facets listed in positive_facets are treated as +1

• all other facets are treated as -1 This affects interpretation of reported facet measures.

Performance tips

For exploratory work, method = "JML" is usually faster than method = "MML", but it may require a larger maxit to converge on larger datasets.

For MML runs, quad_points is the main accuracy/speed trade-off. The ⁠@param quad_points⁠ tier table is the authoritative reference; in short:

• quad_points = 7 is a lightweight setting for quick iteration.

• quad_points = 15 is an intermediate option when runtime matters.

• quad_points = 31 is the package default and the publication tier: the marginal log-likelihood is stable enough for direct manuscript reporting.

• quad_points = 61 (or higher) is reserved for ultra-precise benchmarking on very narrow score supports.

• mml_engine = "direct" remains the most stable general-purpose path.

• mml_engine = "em" or "hybrid" currently target RSM / PCM fits without a latent-regression population model.

• Benchmark your own workload before using mml_engine = "em" or "hybrid" for final reporting; direct remains the safer default when you have not compared engines for your data.

• For RSM and PCM fits only, an opt-in C++ MML backend can be enabled with options(mfrmr.use_cpp11_backend = TRUE). The backend implements the same physicist Gauss-Hermite quadrature and sum-to-zero identification as the pure-R engine, validated against the pure-R reference at tolerance = 1e-12 on a fixed regression fixture. It is opt-in for this release; the default flip to ON is planned for a follow-up release after a cycle of community testing. GPCM fits stay on the pure-R engine regardless of the option.

Downstream diagnostics can also be staged:

• use diagnose_mfrm(fit, residual_pca = "none") for a quick first pass

• add residual PCA only when you need exploratory residual-structure evidence

Downstream diagnostics report ModelSE / RealSE columns and related reliability indices. For MML, non-person facet ModelSE values are based on the observed information of the marginal log-likelihood and person rows use posterior SDs from EAP scoring. For JML, these quantities remain exploratory approximations and should not be treated as equally formal.

For bounded GPCM, residual-based mean-square fit screens are also best treated as exploratory diagnostics rather than strict Rasch-style invariance tests, because the discrimination parameter is free.

Interpreting output

A typical first-pass read is:

1. fit$summary for convergence and global fit indicators.

2. summary(fit) for human-readable overviews.

3. for RSM / PCM, diagnose_mfrm(fit) for element-level fit, approximate separation/reliability, and warning tables.

4. for bounded GPCM, use diagnose_mfrm() and the residual-based table helpers as exploratory screens, together with posterior scoring / compute_information() where documented.

Typical workflow

1. Fit the model with fit_mfrm(...).

2. Validate convergence and scale structure with summary(fit).

3. For RSM / PCM, run diagnose_mfrm() and proceed to reporting with build_apa_outputs().

4. For bounded GPCM, use the fitted object, slope summary, diagnose_mfrm(), residual-based table helpers, posterior scoring helpers, compute_information(), direct simulation/recovery helpers, fair_average_table(), and estimate_bias() with their documented caveats. Use gpcm_capability_matrix() to confirm which helper families are currently supported, caveated, blocked, or deferred.

References

The ordered-category many-facet formulation follows Linacre (1989), with the RSM and PCM branches grounded in Andrich (1978) and Masters (1982). The bounded GPCM branch follows the generalized partial credit formulation of Muraki (1992) under a package-specific positive log-slope identification convention. The MML route follows the quadrature-based marginal-likelihood framework of Bock and Aitkin (1981).

• Andrich, D. (1978). A rating formulation for ordered response categories. Psychometrika, 43(4), 561-573.

• Bock, R. D., & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm. Psychometrika, 46(4), 443-459.

• Linacre, J. M. (1989). Many-facet Rasch measurement. MESA Press.

• Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47(2), 149-174.

• Myford, C. M., & Wolfe, E. W. (2003). Detecting and measuring rater effects using many-facet Rasch measurement: Part I. Journal of Applied Measurement, 4(4), 386-422.

• Myford, C. M., & Wolfe, E. W. (2004). Detecting and measuring rater effects using many-facet Rasch measurement: Part II. Journal of Applied Measurement, 5(2), 189-227.

• Muraki, E. (1992). A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16(2), 159-176.

• Robitzsch, A., & Steinfeld, J. (2018). Item response models for human ratings: Overview, estimation methods, and implementation in R. Psychological Test and Assessment Modeling, 60(1), 101-139.

See Also

diagnose_mfrm(), estimate_bias(), build_apa_outputs(), gpcm_capability_matrix, mfrmr_workflow_methods, mfrmr_reporting_and_apa

Examples

# Fast smoke run: a JML fit on the bundled `example_core` toy
# dataset finishes in well under a second and returns a populated
# `summary` overview ready for inspection.
toy <- load_mfrmr_data("example_core")
fit_quick <- fit_mfrm(toy, "Person", c("Rater", "Criterion"), "Score",
                      method = "JML", maxit = 30)
fit_quick$summary[, c("Model", "Method", "N", "Converged")]


# Full run with the package default MML estimator (recommended for
# final reporting because person parameters are integrated out under
# an N(0, 1) prior). The default `quad_points = 31` is the
# publication tier; `quad_points = 7` below is an exploratory speed
# setting and should not be used as the final manuscript fit.
fit <- fit_mfrm(
  data = toy,
  person = "Person",
  facets = c("Rater", "Criterion"),
  score = "Score",
  model = "RSM",
  quad_points = 7,
  maxit = 30
)
fit$summary
s_fit <- summary(fit)
s_fit$overview[, c("Model", "Method", "Converged")]
# Look for: Converged = TRUE. If FALSE, raise `maxit`, relax `reltol`,
#   or inspect `summary(fit)$key_warnings` for sparse-cell or
#   identification flags.
s_fit$person_overview
# Look for: Mean ~ 0 logits and SD ~ 1 logit are typical when the
#   sample is centred on the test difficulty. SD < 0.5 suggests the
#   test is too easy / hard for this group; SD > 1.5 suggests strong
#   targeting mismatch or extreme-score persons (see `Extreme` flag).
s_fit$targeting
# Look for: |Targeting| < ~0.5 logits is comfortable; larger absolute
#   values mean persons sit systematically above or below the facet
#   means under the package's sum-to-zero identification.
p_fit <- plot(fit, draw = FALSE)
p_fit$wright_map$data$plot

# JML is available for exploratory / fast iteration passes:
fit_jml <- fit_mfrm(
  data = toy,
  person = "Person",
  facets = c("Rater", "Criterion"),
  score = "Score",
  method = "JML",
  model = "RSM",
  maxit = 30
)
summary(fit_jml)$overview[, c("Model", "Method", "Converged")]

# Latent regression (MML only) uses person-level background variables:
person_tbl <- unique(toy[c("Person")])
person_tbl$Grade <- seq_len(nrow(person_tbl))
person_tbl$Group <- rep(c("A", "B"), length.out = nrow(person_tbl))
fit_pop <- fit_mfrm(
  data = toy,
  person = "Person",
  facets = c("Rater", "Criterion"),
  score = "Score",
  method = "MML",
  population_formula = ~ Grade + Group,
  person_data = person_tbl
)
summary(fit_pop)$population_overview
summary(fit_pop)$population_coding

# Binary responses are supported as ordered two-category scores:
set.seed(1)
binary_toy <- expand.grid(
  Person = paste0("P", 1:30),
  Item = paste0("I", 1:4),
  stringsAsFactors = FALSE
)
theta <- stats::rnorm(length(unique(binary_toy$Person)))
beta <- seq(-0.8, 0.8, length.out = length(unique(binary_toy$Item)))
eta <- theta[match(binary_toy$Person, unique(binary_toy$Person))] -
  beta[match(binary_toy$Item, unique(binary_toy$Item))]
binary_toy$Score <- stats::rbinom(nrow(binary_toy), 1, stats::plogis(eta))
fit_binary <- fit_mfrm(
  data = binary_toy,
  person = "Person",
  facets = "Item",
  score = "Score",
  model = "RSM",
  method = "JML",
  maxit = 30
)
fit_binary$summary[, c("Model", "Categories", "Converged")]

# Next steps after fitting:
diag <- diagnose_mfrm(fit, residual_pca = "none")
chk <- reporting_checklist(fit, diagnostics = diag)
head(chk$checklist[, c("Section", "Item", "DraftReady")])

mfrmr documentation built on June 13, 2026, 1:07 a.m.