MML estimation and marginal-fit diagnostics

Why the MML calculations are shared

Earlier versions computed closely related quantities multiple times in separate paths:

• marginal log-likelihood evaluation
• optimization-time gradient calculation
• posterior weights for EAP summaries
• strict marginal expected counts used in diagnostics

The current implementation reuses the same latent-integrated quantities across estimation and diagnostics. This keeps EAP summaries, gradients, and strict marginal expected counts aligned.

Mathematical Core

For a response vector \(\mathbf{x}_n\) and parameter vector \(\beta\), the current MML path targets the marginal likelihood

\[ L(\beta) = \prod_{n=1}^{N} \int p(\mathbf{x}_n \mid \theta, \beta) g(\theta) \, d\theta \approx \prod_{n=1}^{N} \sum_{q=1}^{Q} w_q \, p(\mathbf{x}_n \mid \theta_q, \beta), \]

where \((\theta_q, w_q)\) are Gauss-Hermite nodes and weights. In mfrmr, the integral is approximated with Gauss-Hermite quadrature, the marginal log-likelihood is optimized from the same shared kernel, and person summaries are computed post hoc from the posterior bundle. When a latent-regression population model is active, the package uses person-specific transformed nodes derived from the same quadrature basis rather than one unconditional fixed grid.

The posterior weight for person \(n\) at node \(q\) is

\[ \omega_{nq} = \frac{w_q \, p(\mathbf{x}_n \mid \theta_q, \hat{\beta})} {\sum_{r=1}^{Q} w_r \, p(\mathbf{x}_n \mid \theta_r, \hat{\beta})}. \]

Expected a posteriori (EAP) scoring then uses

\[ \hat{\theta}_n^{\mathrm{EAP}} = \sum_{q=1}^{Q} \theta_q \, \omega_{nq}. \]

This is the kernel that now feeds logLik, the gradient, EAP summaries, and strict marginal expected values.

Current MML scope

For the current public RSM / PCM release:

• the person distribution is integrated with Gauss-Hermite quadrature
• mml_engine = "direct" uses gradient-based direct optimization of the marginal log-likelihood
• mml_engine = "em" and mml_engine = "hybrid" are also available for RSM / PCM, while unsupported branches fall back to direct
• person summaries are reported post-hoc from the integrated posterior

This is the implemented scope for the current release.

Strict Marginal Diagnostic Target

The strict marginal branch is not based on plugging \(\hat{\theta}_n^{EAP}\) back into the response model. Instead, it works with posterior-integrated expectations. For a grouped summary \(g\) and category \(c\),

\[ \mathbb{E}_{\hat{\beta}}(N_{gc}) = \sum_{n=1}^{N} \sum_{q=1}^{Q} \omega_{nq} \, I(n \in g) \, P(X_n = c \mid \theta_q, \hat{\beta}). \]

The corresponding residual compares the observed count to that latent-integrated expectation rather than to an EAP plug-in prediction.

For pairwise local-dependence follow-up, the package keeps the same posterior weights but replaces the one-category event with agreement or adjacency events for the relevant pair of facet levels. That is why top_marginal_cells and top_marginal_pairs are conceptually related but not numerically comparable.

Diagnostic Basis In The Package

diagnose_mfrm() now keeps two evidence paths explicit:

• legacy: residual/EAP-oriented diagnostics inherited from the earlier stack
• marginal_fit: strict latent-integrated first-order and pairwise screens
• both: returns both without collapsing them into one decision rule

The object returned by summary(diag) exposes diagnostic_basis so the two paths can be interpreted separately.

Literature Positioning

The current design is deliberately aligned with five strands of the IRT fit literature.

1. Limited-information item-fit logic. Orlando and Thissen (2000, 2003) show why grouped or score-conditioned comparisons can be more stable than full-information contingency-table statistics in realistic IRT settings. The current package borrows that limited-information logic, but it does not implement S-X2 or S-G2 literally. Instead, it applies posterior-integrated grouped residual screens to many-facet cells and levels.

2. Generalized residual logic. Haberman and Sinharay (2013) define a generalized residual for a summary statistic \(T\) as

\[ r = \frac{T - \hat{\mathbb{E}}(T)}{\hat{s}_D}, \]

where \(\hat{\mathbb{E}}(T)\) and \(\hat{s}_D\) are computed under the fitted model. This is the clearest template for thinking about the current marginal_fit outputs. The current pairwise local-dependence summaries are informed by the same observed-versus-expected logic, but they should still be read as exploratory agreement screens rather than as formal Haberman- Sinharay generalized residual tests.

3. Multi-method fit assessment and practical significance. Sinharay and Monroe (2025) review limited-information statistics, generalized residuals, posterior predictive checking, and practical significance, and recommend prioritizing fit procedures by intended use rather than treating one index as universally decisive.

4. Posterior predictive follow-up. Sinharay et al. (2006) treat posterior predictive checking as a separate model-checking family built around replicated datasets and discrepancy measures. That is the intended follow-up role of the package’s currently scaffolded posterior_predictive_follow_up path.

5. Many-facet reporting context. Linacre’s FACETS framework and applied MFRM studies such as Eckes (2005) remain the primary references for severity/leniency, mean-square fit, separation, and inter-rater agreement. The current strict marginal branch is designed to sit alongside that many-facet toolkit, not to replace it.

Interpretation Boundaries

The strict marginal branch is currently a screening layer, not a fully calibrated inferential test battery.

• well-specified simulation rows are interpreted as Type I proxies
• misspecified rows are interpreted as sensitivity proxies
• posterior-predictive checks remain a follow-up path rather than a completed default computation

This package therefore treats strict marginal diagnostics as structured evidence about possible misfit, not as a single definitive accept/reject rule. That design choice follows the broader review logic in Sinharay and Monroe (2025): use several complementary diagnostics, match them to the intended use of the scores, and examine practical significance before making strong claims.

For many-facet reporting, one additional boundary matters. Facet-level separation/reliability and inter-rater agreement answer different questions. High rater separation reliability can coexist with weak observed agreement, and strong observed agreement does not imply that raters are interchangeable on the latent severity scale. That is why mfrmr reports diagnostics$reliability and diagnostics$interrater as separate objects.

Validation Scope In The Current Release

The current simulation-based validation covers:

• well-specified baselines
• local dependence misspecification
• latent distribution misspecification
• step-structure misspecification

These checks target RSM and PCM. GPCM is now supported only within a bounded core route: fitting, slope summaries, posterior scoring, information curves, direct curve/category reports, and exploratory residual-based follow-up. Broader APA/report bundles, fair-average semantics, and planning/forecasting helpers remain out of scope for GPCM in this release.

Why GPCM Is The Current Upper Scope

GPCM is the current upper supported scope for three reasons.

1. The shared MML kernel and the response-probability core already generalize to the bounded GPCM branch without changing the main package architecture.
2. The package has direct checks for that bounded route.
3. The helpers that still depend on Rasch-family score semantics or on the role-based planning layer are already blocked explicitly, so formal support does not require pretending that every downstream helper has full coverage.

This is a narrower but more defensible claim than saying the whole package is uniformly generalized to free-discrimination many-facet work.

Equal weighting as a model-choice principle

Robitzsch and Steinfeld (2018) are helpful because they separate two arguments that are often conflated in applied many-facet work.

1. A generalized many-facet model with discrimination parameters will often fit empirical data better than a Rasch-MFRM.
2. That fit advantage does not, by itself, settle the operational scoring question.

If the intended score interpretation requires equal contributions of items and raters, then the Rasch-family route remains substantively attractive even when a slope-aware model fits better. mfrmr therefore treats RSM / PCM as the equal-weighting reference models and bounded GPCM as a supported alternative for users who explicitly want to inspect or allow discrimination-based reweighting.

This is also why some score-side helpers remain out of scope for bounded GPCM. FACETS-style fair averages are Rasch-family score transformations, and a slope-aware analogue should not silently reuse the Rasch-family calculation.

One additional distinction matters for implementation. The weight argument in fit_mfrm() is an observation-weight column. It changes how rating events enter estimation and summaries, but it is not the same thing as the equal-weighting versus discrimination-weighting question discussed above.

Future extensions

Posterior-predictive checking, MCMC engines, and heavier runtime infrastructure remain future extensions. They are not required for the current quadrature-based MML route or for the bounded GPCM support described here.

Recommended Expert Reading Of Package Output

For the current release, the most defensible interpretation sequence is:

1. Read summary(fit) for estimation status and precision basis.
2. Read summary(diag) with diagnostic_mode = "both" to keep legacy and strict evidence separate.
3. Treat marginal_fit and marginal_pairwise as screening layers for first-order and local-dependence follow-up.
4. Use plots and tables to judge magnitude and practical importance, not only presence/absence of a flag.
5. If a use case demands stronger confirmation, treat posterior predictive checking as the next methodological step rather than over-reading the current screening statistics.

Key References

• Andrich, D. (1978). A rating formulation for ordered response categories. Psychometrika, 43, 561-573.
• Bock, R. D., & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm. Psychometrika, 46, 443-459.
• Haberman, S. J., & Sinharay, S. (2013). Generalized residuals for general models for contingency tables with application to item response theory. Journal of the American Statistical Association, 108, 1435-1444.
• Eckes, T. (2005). Examining rater effects in TestDaF writing and speaking performance assessments: A many-facet Rasch analysis. Language Assessment Quarterly, 2, 197-221.
• Linacre, J. M. (1989). Many-facet Rasch measurement. MESA Press.
• Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 149-174.
• Orlando, M., & Thissen, D. (2000). Likelihood-based item-fit indices for dichotomous item response theory models. Applied Psychological Measurement, 24, 50-64.
• Orlando, M., & Thissen, D. (2003). Further investigation of the performance of S-X2: An item fit index for use with dichotomous item response theory models. Applied Psychological Measurement, 27, 289-298.
• Robitzsch, A., & Steinfeld, J. (2018). Modeling rater effects in achievement tests by item response models: Facets, generalized linear mixed models, or signal detection models? Journal of Educational and Behavioral Statistics, 43, 218-244.
• Sinharay, S., & Monroe, S. (2025). Assessment of fit of item response theory models: A critical review of the status quo and some future directions. British Journal of Mathematical and Statistical Psychology, 78, 711-733.
• Sinharay, S., Johnson, M. S., & Stern, H. S. (2006). Posterior predictive assessment of item response theory models. Applied Psychological Measurement, 30, 298-321.