What mfrmr Calculates and Where to Read More

Find an analysis by its question, understand what its numbers mean, and follow links to the calculation details and original references. For a first runnable analysis, use mfrmr_workflow_methods or vignette("mfrmr-workflow", package = "mfrmr") first. This is a help topic, opened with help("mfrmr_methods_reference"), not a function to run. For an integrated guide to when to use each model and estimator, read vignette("mfrmr-model-choice", package = "mfrmr") or its online version. It compares practical uses and limitations, including why a more complex model is not automatically preferable. The sections below provide the calculation references after that choice.

Start with the question

Suppose several judges score performances on several criteria. A low score might reflect the performance, a stricter judge, a harder criterion, or a combination the chosen model does not explain. MFRM estimates these effects under a stated model. Raw agreement asks a different question: did judges give the same ratings to the same performances?

A facet is a modeled role, such as Rater or Task; a level is one rater or task. Person is specified separately in fit_mfrm(), so facets = c("Rater", "Criterion") gives three total facets including Person. More facets do not mean more ability dimensions. The rating assignments must contain information that distinguishes the requested effects. Arbitrary names and flexible facet counts do not make every crossed, nested or sparse design identifiable. Before interpreting a diagram, read the score-direction and diagnostic-term section of mfrmr_visual_diagnostics. It explains reverse-coded comprehensibility, calibrated judges and fit, conditional interaction bias, and adjacent-category Steps. A higher coded score is not inherently better.

Check the rating design before fitting

Start with describe_mfrm_data() and the intended comparison. A successful data review is not a certificate of identification or adequate precision.

Count the information, not only Persons. Inspect ratings per Person, observations per facet level and shared ratings. Adding raters while each Person still receives two ratings spreads the same workload across more rater effects. More Persons do not repair an assignment that cannot separate the effects. There is no package-wide minimum sample size or facet-count rule that guarantees reliable estimates or intervals.
Separate the effects you want to compare. For example, if Task A always uses Criterion A and Task B always uses Criterion B, their separate difficulties are confounded without additional information or constraints. Each Person-facet graph can be connected while this joint ambiguity remains. Review design_connectivity, then the estimator-specific identification and readiness results from fit_mfrm(). A local nonlinear rank check is not a proof of global identification.
Declare the assignments and rubric. Supply expected_design to the data review when the planned roster is known. Compare planned cells, observed cells and retained rating rows; repeated events need their own interpretation. An unassigned cell is not a zero score. Set the intended category bounds and inspect category use within the selected step owner. A category observed elsewhere may still be unsupported for that owner. See rating_scale_table() and fit_mfrm() for the model-specific checks.
State what missingness and anchors mean. Using available ratings does not correct ability-dependent assignment or score-dependent missingness. Missing population predictors have a separate population_policy. Shared Persons provide linking observations; they are not fixed anchors. review_mfrm_anchors() checks declared anchors and constraints. Treating an external estimate as an exact anchor does not propagate its uncertainty. Person anchors require JML; MML rejects them. By default, centering_scope = "facet" centers the whole declared facet, including fixed values. Use "remainder" only to reproduce legacy centering of remaining ungrouped levels. Read mfrmr_linking_and_dff for a complete example, the distinction from FACETS, and the references supporting each linking claim.
Check the requested output separately. A fitted point, covariance, interval, residual analysis and G/D projection have different requirements. Flexible facet names/counts in one route do not extend another route. For example, two-family GPCM currently has exactly two non-Person facets; incomplete G-study observations do not authorize arbitrary unbalanced future D-study plans. Follow gpcm_capability_matrix(), mfrmr_interval_guide() and mfrm_d_study() for those limits.

Find the appropriate calculation

Follow the help link for arguments, equations, returned fields, examples and references. These are complementary analyses; running every row is not a requirement for every assessment.

Can my ratings support the intended comparison? Retained ratings, category use, exposure and design/identification checks. Connectedness alone does not establish precision. describe_mfrm_data(), subset_connectivity_report() and fit_mfrm(); Linacre (1989).
How do ability, severity and category use explain scores? Ordered-category response probabilities with specified facet effects. RSM shares steps; PCM assigns steps to one selected facet; GPCM additionally allows positive slopes. fit_mfrm() and gpcm_capability_matrix(); Andrich (1978), Masters (1982), Muraki (1992).
How are the parameters estimated? MML integrates over a declared ability distribution; JML estimates Person parameters jointly with structural parameters. Native RSM/PCM CML conditions on each Person's total, eliminating ability without estimating a population distribution. It supplies structural conditional-information intervals and no ability population SD. Explicit comparison-only CML also supports identified interaction contrasts when other structural directions are unidentified; unavailable targets stay missing. These routes have different assumptions and uncertainty targets. The estimator and population sections of fit_mfrm(); Bock and Aitkin (1981), and the JML references on that page.
How uncertain is a measure or difference? A model-specific SE or interval for a named target, conditional on specified assumptions. A difference may need the covariance between estimates. mfrmr_interval_guide(), mfrm_facet_intervals() and mfrm_population_intervals(); each page describes its calculation and limitations.
Which ratings are unexpected? Infit/Outfit and residual summaries compare ratings with model predictions; ZSTD transforms fit using the selected reference convention. diagnose_mfrm() and fit_measures_table(); Wright and Masters (1982), Linacre (2002). Extended models use mfrm_response_diagnostics().
Are categories functioning as intended? Usage counts, owner-specific step estimates and probability curves describe category use and transitions. rating_scale_table(), category_structure_report() and category_curves_report(); Andrich (1978), Masters (1982). RSM has a common step set; PCM/GPCM has one declared step-owning facet, which can differ from the single slope-owning facet in GPCM. Review owner_category_table before interpreting pooled counts. Stored step SEs describe centered offsets, not full crossing, gap or curve uncertainty. Disorder does not automatically invalidate the model or justify collapsing categories. Two-family curves use their separate fitted-curve route with unavailable curve/step intervals; these ordinary helpers do not extend to it.
Is a rater unusually strict in a particular context? Conditional bias screens describe departures from the additive model for a combination of facet levels. estimate_bias() and estimate_all_bias(); Linacre (1989) and the FACETS sources in their help. These screens hold calibration fixed. Use bias_interaction_report() to keep the chosen review rules with saved tables and plots. Its scatter compares observed-minus-expected score averages with conditional logit shifts; neither its normal/profile bounds nor repeated views provide a calibrated fairness test.
Are group differences specific to a facet? DFF screens and separately matched MH/ordinal-logistic analyses use different adjustment models and sampling assumptions. analyze_dff(), mantel_haenszel_table() and ordinal_logistic_table(); see their References and matching rules.
How well are measures separated relative to error? Separation, reliability and strata summarize estimated spread and measurement error for a facet. The interpretation and References sections of diagnose_mfrm() and plot(x, type = "reliability"); Wright and Masters (1982).
Do raters or scoring procedures agree? Matched observed scores supply agreement coefficients, mean differences or ICCs. These do not estimate MFRM severity. pairwise_kappa_table(), gwet_ac2_table(), krippendorff_alpha_table(), pairwise_icc_table() and score_comparison_table(); each has method-specific References.
Do unexplained scores vary together? Residual association concerns departures from the fitted model, not agreement of the original scores. q3_statistic() uses standardized residuals averaged within Person-by-level cells; it differs from Yen's raw-residual Q3. Its help identifies the original definition and the limits of the package's screening cutoffs. For organized residual patterns, analyze_residual_pca() distinguishes overall combined-facet columns from separate facet-level matrices. Eigenvalues across these different matrices do not rank facet importance. Its reference comparisons are exploratory; neither component counts nor fixed cutoffs establish a number of substantive abilities. Save the PCA result before repeated plots so the reference is not generated again.
Do raters match an expert standard or show centrality? Agreement with supplied references and concentration of category use are separate rater properties. fit_rater_accuracy() and rater_centrality_table(); their References identify the accuracy and centrality methods and current restrictions. Centrality's RSM residual correlation, supporting fit and rater-owned PCM threshold SD are indirect indices with separate availability and source status, not an estimated centrality parameter or a severity ranking. plot(fit, type = "rater_severity", diagnostics = diagnostics) instead orders facet locations and returns SE sources and interval-use labels. Its whiskers are normal approximations, not likelihood-profile intervals or pairwise tests. See plot_rater_severity_profile() for interpretation and mfrmr_plot_migration for the public plot() entry point. rater_halo_network_analysis() instead compares observed-score profiles across shared contexts. Its graph centrality is a network property; neither its edge filters nor high same-rater correlations establish causal halo or residual independence. See the rater-effects section below for the distinction from centrality and the horn effect.
What does a cut score imply? Posterior ability probabilities concern a Person now; classification accuracy/consistency concerns a specified future rating plan and population. score_mfrm_persons(), predict_mfrm_units() and classification_table(); the latter gives Rudner (2001, 2005) and Lee (2010), with DOI links.
Would another assessment plan be more dependable? G-studies estimate observed-score variance/covariance components; D-studies project specified future plans and score weights. mfrm_g_study() and mfrm_d_study(); Brennan (2001) and the cited design-specific methods. These are separate from MFRM separation reliability.
How do I retain the analysis and its caveats? Saved results, figures, notes and exports preserve the calculation and its interpretation. mfrm_results(), mfrm_report(), plot_data() and export_mfrm_results(). See mfrm_calibration_workflow for eligible calibration reuse. Saving an analysis supports later review; a frozen calibration supports scoring new Persons at known facet levels; saving a score result supports later tables and plots without rescoring. These are different objects. Freezing makes a checked calibration available for scoring. It does not establish that the model or its prior suits another cohort. The calibration help links Bock and Mislevy (1982) to conditional EAP scoring, separately from calibration-estimation and interval-performance claims.

Use results as evidence about a construct

A construct is the attribute a score is intended to represent, such as speaking proficiency. Begin with a claim about that score and its use: "Content and Organization describe related aspects of performance across the specified tasks and raters." State the population, rubric, score units, intended composite or separate scores, and an expected pattern before inspecting results. A different claim, such as diagnosing an individual's weakness in Organization, needs evidence about that specific score or difference, not just the dependability of a total.

The Standards (AERA, APA, & NCME, 2014, Standards 1.13–1.14) require evidence for the proposed internal structure, subscore interpretation and composite rationale. Kane (2013) connects score-use claims to their assumptions and supporting evidence. The following is a guide to applying those principles with mfrmr, not a validity test implemented by the package.

Proposed claim	Inspect with mfrmr	Alternative explanation and warranted conclusion
Related criterion scores reflect shared differences between people.	G-study Person covariance, its uncertainty, and the other source matrices; `plot.mfrm_multivariate_gstudy()`.	Positive Person covariance is compatible with the claim under the model. It does not identify one latent dimension. A near-zero or negative estimate with a broad interval leaves the proposed relation unresolved, rather than establishing distinct constructs.
The association is not solely a common scoring condition.	Rater, Task, Person-by-Rater and Person-by-Task covariances alongside Person covariance.	Rater covariance describes judges shifting both scores together; task covariance describes tasks shifting both. Interactions describe person-specific shifts. Shared rubric interpretation, genuine performance demands and impressions can compete as explanations. No covariance sign identifies halo or horn.
The rubric and model represent the intended ordering.	Owner-specific category counts/curves, Wright-map targeting, and matching Infit/Outfit or residual diagnostics; `rating_scale_table()`, `plot.mfrm_fit()`.	Sparse categories, poor targeting or unexplained residuals can challenge the proposed measurement. Good fit alone does not establish the meaning of the trait; an unexpected pattern calls for inspection of the rubric and performances before recoding or excluding ratings.
A proposed score is dependable enough for its stated decision.	G/Phi and SEM for the prespecified score, population and future plan; `summary.mfrm_multivariate_d_study()`.	A heterogeneous Person population can raise a coefficient without reducing error in score units. High G/Phi concerns the declared replication, not construct validity. High MFRM rater separation means distinguishable severities, not agreement or evidence against shared scoring bias.

Work through conflicting results. If G is high but Person covariance between the criteria is uncertain, the composite may be dependable while the proposed shared relation remains unresolved. If Person covariance is positive but Person-by-Task variation is substantial, consider whether the intended generalization spans those tasks. A task-specific strength may be substantive; it is error only relative to the chosen across-task target. If fit looks satisfactory but the claimed proficiency is poorly represented by the tasks, fit does not repair content underrepresentation.

Inspect all components rather than choosing favorable ones. Covariances depend on units, and a raw component is not its contribution to a future average: D-studies apply the relevant count divisors. Review negative or indefinite component estimates and interval qualifications. Separate pointwise intervals do not test equality between components. An interval containing zero neither proves independence nor establishes equivalence.

Choose reporting and follow-up from the claim. Prespecify composite weights for substantive reasons; optimizing G/Phi after inspecting the data can change the score being interpreted. Dependable subscores do not by themselves establish dependable or useful differences between them. MFRM and multivariate G-theory provide complementary evidence, not a joint fitted model or a latent-dimensionality test. Agreement between outputs based on the same ratings is not independent replication.

Report the claim, design, observed pattern with uncertainty, competing explanation, justified conclusion and unresolved evidence. Where the claim requires it, examine task content, how raters reach judgments, or relations with an appropriate external measure. These require substantive evidence beyond the returned tables; the package does not certify validity or prescribe another analysis for every assessment. See the G-theory worked example and the figure reading guide.

Choose an agreement target and keep its denominator

Start with the same rated response, identified by all relevant keys such as Person, Task and Criterion. Matching several responses from one Person does not make them independent sampling units. The runnable vignette("mfrmr-rating-agreement", package = "mfrmr") shows matching, exclusions, interpretation and saving without fitting a measurement model.

Identical scores and model expectations: interrater_agreement_table() reports exact agreement on matched contexts. Its ExpectedExact averages products of fitted category probabilities, with calibration fixed. This is a conditional model expectation, not the chance-agreement term in kappa or AC2. MML uses a plug-in EAP, not posterior integration. Unavailable primary estimates withhold expectations with an ExpectedReason; observed agreement is retained. Overall exact agreement counts every matched pair-context opportunity; a context rated by more judges supplies more pairs.
Category agreement for two raters: pairwise_kappa_table() uses the same common complete units for both rater margins. QWK penalizes category differences quadratically. Numeric categories use their actual spacing; character categories use their declared positions. Each pair may use a different sample, so these coefficients are not a panel average.
Pooled agreement: krippendorff_alpha_table() excludes singleton units from both disagreements and weights pairable units by rating count. Its ordinal distances depend on pairable-category frequencies. gwet_ac2_table() weights pairable units equally for observed agreement; singletons enter its expected-agreement margins. Its declared unused categories also matter. Choose the definition before seeing which value is larger; neither numerical missing-data convention corrects selective assignment or missingness.
Reliability on an interval score scale: pairwise_icc_table() requires choices about selected versus randomly sampled raters, absolute agreement versus consistency, and one rating versus the mean of two. Consistency excludes systematic rater offsets; agreement includes them. The default concerns a single rating from selected raters. An average-rating pairwise ICC concerns two ratings, not the whole panel; numeric category codes alone do not justify an interval scale.
A score source against a supplied reference: score_comparison_table() separates QWK, correlation and standardized mean difference. Adding a second reference uses common complete records for all statistics and can change the sample. A chosen reference is not automatically correct, and the optional Williamson et al. (2012) screen is an application-specific choice rather than a universal quality standard.

The raw-score helpers above return descriptive point estimates with their counts, cases, settings and notes; they do not provide sampling intervals. Save their complete lists with base::saveRDS() and export chosen tables with utils::write.csv(). They are not fitted-model result bundles for export_mfrm_results(), and their custom tutorial figures are not new plot() methods. The separate mfrm_interrater bundle supports summary() and plot(); see interrater_agreement_table() for its different repeated- cell handling. Low residual association does not establish raw agreement, and high agreement does not establish accuracy against an expert standard.

Distinguish rater effects before choosing a diagnostic

Consider a judge scoring content, organization and language on a four-category rubric. These questions concern different features of the ratings:

Severity: does the judge generally give lower scores after accounting for the modeled Person and criterion effects? Review fitted Rater locations.
Centrality: does the judge compress ratings toward middle categories? rater_centrality_table() combines the supplied RSM residual/fit and rater-specific PCM threshold descriptions. A severity near zero does not answer this question; neither does graph-theoretic centrality.
Halo or horn: does a favorable or unfavorable impression spill over into ratings on other criteria? For example, a poor impression of language could lower content and organization ratings too. The latter pattern is called a horn effect; it is not defined by a negative criterion correlation. rater_halo_network_analysis() describes observed-score associations and cannot determine their psychological cause or classify halo versus horn. Genuine relationships among performance criteria also produce correlations.
Local dependence: do ratings share variation beyond the chosen model? A supported fit_mfrm_testlet() model can account for a declared shared effect, but its variance does not identify halo or horn.

Infit/Outfit summarize squared residuals. Low values can have several causes and do not identify which effect occurred or its positive/negative direction. Inspect shared-rating counts, category use and the actual performances before drawing conclusions about a rater. Agreement among several indices computed from the same ratings is not independent confirmation.

Direct models answer a further question: how would estimated measurements change when a specified rater effect is included in the response model? The FM-SC severity/centrality model and the MRFM-H halo mixture models are not implemented. See the original references and model distinctions in rater_centrality_table() and rater_halo_network_analysis(). These models require their own identifying assumptions and uncertainty evaluation; adding GPCM slopes or a testlet variance does not implement them.

Choose the G-theory target and uncertainty

Start with the rating design and the intended score, not the word "reliability" alone. The multivariate route also accepts one score; multiple score columns are fixed score components, not additional random facets or latent dimensions. All methods below require explicit assumption = "normal", prespecified scores/weights and the documented sampling/assignment assumptions. An incomplete source is not permission to use a sparse future plan: current D-studies project complete balanced plans.

First choose the quantity whose uncertainty you need:

Question	Interface	Meaning of the uncertainty
How uncertain is a source variance or covariance?	`confint.mfrm_multivariate_gstudy()`	Component interval; variance units are squared score units, covariance units are products of score units.
How uncertain are the original component estimates jointly?	`vcov.mfrm_multivariate_gstudy()`	Sampling covariance and SEs of ANOVA/MINQUE components, including identifiable incomplete sources. Not covariance between individual ratings.
How uncertain is a future plan's G, Phi or SEM?	`confint.mfrm_multivariate_d_study()`	One selected population coefficient or measurement-error target at prespecified counts. Coefficient SE and measurement SEM are different quantities.
How uncertain is the change between two plans?	`confint(d, reference = 1, ...)` or `mfrm_multivariate_d_compare()`	Paired comparison using the shared source; never subtraction of separate marginal endpoints.

Then select a method allowed by the observed source. Here g is a G-study and d its D-study. The table describes available calculations, not a ranking by interval width or a guarantee of nominal coverage.

Target and source	Default or explicit choices	Qualification
Component variance, balanced ANOVA	`confint(g)`: `auto`/`mls`	Exact residual chi-square; other component MLS intervals are approximate.
Signed covariance between two scores, balanced ANOVA	Explicit `gci` or `regression_region`	Generalized approximate coverage or conservative enclosure; no automatic covariance default.
One-facet G/Phi, balanced ANOVA	`confint(d)`: `exact_f` for G, `gci` for Phi	Normal-model exact F versus generalized approximate coverage.
Two-crossed-facet G/Phi, balanced ANOVA	Default `mls`; explicit `delta`, `approximate_f`, `mean_square_region`, `reml_profile`; G also permits `score`	Distinct approximations/conservative enclosures. Read formula-completion flags for MLS; these are not interchangeable calibrations.
Nested G/Phi, balanced ANOVA	With unchanged source child count per parent, the one-facet defaults apply. Otherwise explicitly select `delta`, `approximate_f`, `mean_square_region` or `reml_profile`.	Both facets remain random. The default reduction is unavailable when the child count changes.
Relative/absolute SEM, balanced crossed/nested ANOVA	Default `mls`; two-facet sources also permit `mean_square_region`	Error-variance intervals transformed by square root; exactness depends on the contributing mean squares.
G/Phi, identifiable MINQUE source	Explicit `delta` or `reml_profile`; one-facet G also permits `spectral_f`	Delta concerns the original coefficient; REML has a separate center. Spectral F has an exact normal-model pivot with its own residual-information requirements.
Component variance/covariance or SEM, identifiable MINQUE source	Explicit `reml_profile`; `residual_contrast` additionally supplies residual components and one-facet RelativeSEM	REML is experimental; residual contrasts have an exact variance law or conservative signed covariance enclosure. Residual-only inference does not supply other components or AbsoluteSEM.

delta is also an explicit G/Phi option for balanced one-facet sources; reml_profile accepts every supported balanced source as well as identifiable incomplete/unequal sources. residual_contrast and one-facet spectral_f also accept retained complete ratings. These methods do not become an automatic fallback when another calculation fails. MINQUE requests require an explicit interval method even when the source happens to be balanced. For derivations, required inputs, optional dependencies and references, use confint.mfrm_multivariate_gstudy() and confint.mfrm_multivariate_d_study(). Almehrizi's (2021) coefficient-SE formula and the limits of its extension are in the latter's Delta coefficient SEs and intervals section; SE accuracy does not establish Wald coverage.

For paired plans, choose the relevant source law separately:

Balanced one-facet G/Phi map the whole common F/generalized set. The nested G/Phi reduction requires every future child count to equal the source count. One-facet SEM and nested SEM with a common future child count map one error-SD interval; that future child count may differ from the source.
General balanced two-facet comparisons have explicit mean_square_region enclosures from one shared source region. These retain dependence but can be wider than the tight difference image. Two crossed random facets also permit mfrm_multivariate_d_compare() for a separately qualified paired delta approximation, including identifiable MINQUE sources.
Explicit reml_profile supplies G/Phi/SEM comparisons for supported complete or incomplete sources. Proportional one-facet/common-child nested plans map a common profile; other plans profile the difference directly. Calibration remains approximate, especially at zero-contrast singularities.
For retained one-facet ratings, spectral_f G and residual_contrast RelativeSEM differences map their shared confidence sets. Marginal method = "delta" in confint(d) does not supply paired differences.

Read and save the result. summary.mfrm_multivariate_d_study() with intervals = TRUE collects supported methods, original points, any separate IntervalCenter, source/future counts and weights. Select parm explicitly when the chosen method supports only some metrics. Ordinary summary(d) returns the coefficient table. confint(d, method = c(...)) compares compatible methods for one target; confint(g) selects one component method at a time.

Wide computable intervals remain results. Delta limits outside [0,1], negative original components, empty sets, failed numerical checks and absent methods are different states. Read Status, availability and saved notes. A REML or residual-projection center must not be relabeled as MINQUE. General bootstrap intervals are not implemented. Nonnormal robustness, selection-adjusted inference and sparse/unequal future plans are not supplied. These limits do not make an implemented interval calculation nonexistent.

There is no universal acceptable width or G/Phi cutoff. Define the intended score use, a meaningful improvement or threshold, and task/rater costs before choosing a plan. An interval spanning a decision threshold describes uncertainty about that decision; selecting the narrowest method after seeing its result does not resolve it. A method comparison on one dataset is sensitivity analysis, not a coverage study. Neither high G/Phi nor a narrow interval alone establishes construct validity or fairness.

Use plot(d) for point curves, intervals = TRUE for an explicit interval request, or supply saved intervals for reuse. plot(ci) compares D-study intervals on one axis; plot(g, intervals = TRUE, ...) shows component intervals and retains them in the returned plot object. Save complete results with saveRDS(); as_ggplot() renders saved plot data without fitting. Ribbons are pointwise, not simultaneous bands; overlap is not a paired test. The G-theory tutorial supplies a bundled-data reading path, and the function help holds mathematical details and references. The simpler mfrm_generalizability() / mfrm_d_study() route is a separate main-effects approximation; its residual sensitivity is not a confidence interval.

Read the method before interpreting the number

Model: RSM/PCM use unit discrimination. One-family GPCM assigns slopes to one selected facet; MML allows a different step owner, whereas JML requires the same owner. The experimental two-family MML route has exactly two non-Person facets and additional restrictions. Check gpcm_capability_matrix() before transferring an ordinary-model analysis. The original response-model papers do not establish the performance of every mfrmr estimator, extension or interval.

Estimation and scoring: MML uses numerical integration of the marginal likelihood. JML includes Person parameters in the joint likelihood. EAP scoring subsequently averages over a conditional ability distribution; its posterior SD is not a sampling SE for a structural parameter. Ordinary RSM/PCM MML fixes N(0,1) when population_formula = NULL; estimating a population model is an explicit choice. A histogram of EAP estimates is not itself an estimate of the latent population density. Numerical integration, population assumptions and the source of uncertainty are documented separately in fit_mfrm() and the scoring help pages.

Fit: Infit weights squared standardized residuals by their model variance; Outfit gives each retained residual equal weight when observation weights are all one. Values above one indicate more residual variation than the reference model expects. A flag is a reason to inspect ratings and assumptions, not a verdict on a rater. Read the selected residual basis, weighting and ZSTD convention in diagnose_mfrm(). Posterior diagnostics for extended models do not inherit ordinary fit cutoffs.

Reliability and strata: the ordinary summaries adjust the spread of finite measures using their matching squared SEs. Invalid or missing SEs can make the summary unavailable. High rater separation indicates distinguishable severities, not agreement; low separation can also reflect poor precision. Strata is a hypothetical separation index, not a discovered number of groups or a set of pass/fail boundaries. Model and fit-adjusted summaries are not confidence bounds. See the Facets explanation alongside mfrmr's estimator-specific conventions in diagnose_mfrm(). These marginal-SE summaries omit covariance among calibrated measures. Applying variance subtraction to EAP estimates is not posterior-variance EAP reliability, and neither quantity is a G-theory coefficient.

Post-fit shrinkage: apply_empirical_bayes_shrinkage() adds a descriptive zero-centered normal-model adjustment while retaining the fitted estimates. Its plug-in SD conditions on the chosen prior variance; it is not a calibrated sampling SE or a general improvement guarantee. Additional Person shrinkage requires ordinary JML sampling SEs. MML EAP estimates and posterior SDs must not be fed through a second such update.

Categories and missingness: use the intended rubric, not only the observed score range. category_policy = "preserve" retains that ladder; "collapse" changes a gapped ladder and records the mapping. Missing assigned scores, unassigned ratings and unavailable rubric categories are different situations. The current ordinary models use a common category range; owner-specific ranges are not implied by PCM step ownership. See describe_mfrm_data() and fit_mfrm() for support checks and retained rows.

Uncertainty: distinguish the estimate, its numerical status and the interpretation allowed by its assumptions. A finite experimental interval is not evidence of nominal coverage. A missing endpoint is not infinity. Conditional bias screens and fixed-calibration Person intervals omit calibration uncertainty. Use mfrmr_interval_guide() to select the target before choosing a method.

Find and cite the supporting literature

Open the function's Details, named method sections and References; report its selected settings as well as the paper. For example, help("diagnose_mfrm") connects the Infit/Outfit definitions to the ZSTD conventions and reliability formulas; help("classification_table") explains how the Lee recursion differs from the Rudner approximation. The foundational sources below identify the model/estimation framework. Specialist help pages provide the references for their own formulas and adaptations. A citation to a general framework is not validation of a new implementation or of your data's assumptions.

Use citation("mfrmr") for the software citation and packageVersion("mfrmr") for the installed version. Cite the relevant methodological sources as well. A DOI identifies a paper even if access to its full text requires a library subscription.

References

American Educational Research Association, American Psychological Association, & National Council on Measurement in Education. (2014). Standards for educational and psychological testing. American Educational Research Association. Internal structure and score-combination rationale: Standards 1.13–1.14, pp. 26–27; reliability/precision, Chapter 2. https://www.testingstandards.net/uploads/7/6/6/4/76643089/standards_2014edition.pdf.

Kane, M. T. (2013). Validating the interpretations and uses of test scores. Journal of Educational Measurement, 50(1), 1–73. doi:10.1111/jedm.12000 . A framework for claims, assumptions and evidence; not validation of a particular mfrmr calculation.

Andrich, D. (1978). A rating formulation for ordered response categories. Psychometrika, 43, 561-573. doi:10.1007/BF02293814 . RSM foundation.

Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 149-174. doi:10.1007/BF02296272 . PCM foundation.

Muraki, E. (1992). A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16(2), 159-176. doi:10.1177/014662169201600206 . GPCM foundation.

Bock, R. D., & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm. Psychometrika, 46, 443-459. doi:10.1007/BF02293801 . Marginal-likelihood estimation framework; this citation does not mean every mfrmr engine uses EM.

Linacre, J. M. (1989). Many-facet Rasch measurement. MESA Press. Many-facet formulation and interpretation.

Wright, B. D., & Masters, G. N. (1982). Rating scale analysis. MESA Press. Fit, separation and reliability conventions.

Linacre, J. M. (2002). What do Infit and Outfit, Mean-square and Standardized mean? Rasch Measurement Transactions, 16(2), 878. https://www.rasch.org/rmt/rmt162f.htm. An introductory fit explanation.

Brennan, R. L. (2003). Coefficients and indices in generalizability theory (CASMA Research Report No. 1). University of Iowa. Pages 3–4 introduce random-facet G/D error and coefficient targets. https://education.uiowa.edu/sites/education.uiowa.edu/files/2026-04/casma-research-report-1-archived.pdf.