Pointwise intervals for fixed facet estimates and contrasts

Describe uncertainty about a fixed facet estimate, such as rater severity, or a specified difference between two raters. Start from a supported fitted model; the point estimates stay the same when you change the interval method.

Usage

mfrm_facet_intervals(
  fit,
  facet = NULL,
  contrasts = NULL,
  method = c("model", "sandwich"),
  clusters = NULL,
  adjust = FALSE,
  level = 0.95,
  scale = c("native", "standardized"),
  comparison_review = NULL,
  anchor_covariance = NULL,
  independent_anchors = FALSE,
  interaction = NULL
)

# S3 method for class 'mfrm_facet_intervals'
print(x, ...)

# S3 method for class 'mfrm_facet_intervals'
summary(object, ...)

# S3 method for class 'mfrm_facet_intervals'
confint(object, parm = NULL, level = object$settings$level, ...)

# S3 method for class 'mfrm_facet_intervals'
vcov(object, ...)

# S3 method for class 'mfrm_facet_intervals'
plot(
  x,
  comparison = TRUE,
  draw = TRUE,
  preset = "standard",
  title = paste(x$settings$facet, "estimates and contrasts"),
  subtitle = paste0(format(100 * x$settings$level, trim = TRUE),
    "% pointwise normal intervals | ", if (x$settings$method == "sandwich")
    paste0("Sandwich | ", x$settings$clusters, " independent clusters assumed") else if
(identical(x$settings$procedure, "cml_conditional_information"))
    "Conditional information" else if (identical(x$settings$uncertainty,
    "independent_anchor_delta")) "Information + anchor source" else
    "Observed information"),
  caption = if (any(!is.finite(x$table$Estimate)))
    paste0("Not estimated: no point or interval | ",
    "Crosses: estimated points with unavailable intervals") else if (any(x$table$Status
    != "available"))
    "Open diamonds: fixed targets | Crosses: unavailable selected intervals" else NULL,
  reference = 0,
  show_legend = TRUE,
  ...
)

Arguments

fit

An inference-ready RSM/PCM MML fit from fit_mfrm(), using a fixed standard-normal person distribution, fixed quadrature and unit observation weights. A fit whose category review concerns singleton counts can additionally use method = "model" after the separate checks described under "Small samples and singleton categories". Experimental two-family GPCM MML also supports method = "model" with fixed-grid EM or direct adaptive integration; see "Two-family location targets" below. Direct MML native locations additionally support estimated normal RSM/PCM populations, including conditional-normal latent regression, and one-family GPCM with fixed N(0,1) or an estimated intercept-only normal population, using fixed or adaptive integration; see "Native locations" below for their scope and separate numerical checks. Native RSM/PCM CML results additionally support method = "model", scale = "native", including the saved targets from fit_scope = "comparisons". Matching full RSM/PCM MML interaction fits use the comparison_review extension below; their explicit model and source checks remain separate.

facet

A non-person facet, for example "Rater". May be omitted when comparison_review supplies a facet-by-origin interaction target, or for a CML comparison-only fit, which already retains its targets.

contrasts

Optional numeric matrix with distinct target row names and columns named by every level of facet. Columns are aligned by name. A row with coefficients c(1, -1, 0) estimates the first level minus the second. By default, each level is reported.

method

"model" (default) uses ordinary observed information; "sandwich" uses independent-cluster marginal-likelihood scores. The native-location and experimental two-family extensions support only "model"; sandwich support remains limited to the ordinary fixed-population RSM/PCM route.

clusters

For sandwich inference, an optional data frame with exactly one row per fitted person and complete Person and Cluster identifiers. Without it, persons are the independent units. A larger cluster could be a school or clinic if different schools or clinics are independent. A person cannot be split across clusters. Supply original person labels, even when the input person column has another name.

adjust

Logical; with method = "sandwich", optionally multiply the covariance by G/(G-1), where G is the cluster count. Default FALSE. This scaling does not guarantee small-sample coverage.

level

Pointwise confidence level, strictly between zero and one.

scale

"native" (default) retains the fitted ability units. "standardized" divides an explicit within-facet zero-sum contrast, or the interaction contrasts declared by comparison_review, by the population SD, using full joint delta-method uncertainty. Requires method = "model" and a fixed N(0,1) or estimated intercept-only normal population; individual locations and latent regression are not supported.

comparison_review

Optional describe_mfrm_data() review or its $comparison_review for a matching full native RSM/PCM MML interaction fit. Uses its declared facet-by-origin contrasts instead of facet locations; omit contrasts and optionally facet. Requires method = "model". See "MML interaction contrasts" for scope and assumptions. Omit this argument for a CML comparison-only fit; its saved review is reused.

anchor_covariance

Optional full sampling covariance of the supplied anchor estimates, including their correlations, in native ability units squared. Row and column names must match every directly anchored level on the one noncentered facet. Requires the fixed-grid direct RSM/PCM anchor-origin route with an intercept-only population and fixed SD. NULL retains the existing conditional-on-anchors calculation.

independent_anchors

Logical, default FALSE. Set TRUE with anchor_covariance only when source calibration and receiving responses are independent. Shared respondents or dependent clusters require cross-sample covariance and are unsupported. This is an explicit assumption, not a check inferred from different Person identifiers.

interaction

Optional exact name of one fitted two-facet interaction, for example "Rater:Criterion". Omit facet and comparison_review. This selects jointly fitted cell effects rather than fixed-baseline bias screens. Requires method = "model", scale = "native", full fixed-grid RSM/PCM MML (direct, EM or hybrid) with fixed N(0,1), one interaction, no anchors, groups, dummy facets or shrinkage. Every engine must pass the common numerical and full-joint information checks; EM stopping alone does not qualify intervals. See the joint-cell section below.

x

Output from mfrm_facet_intervals().

Also for print(): An object returned by mfrm_facet_intervals().

...

For print.mfrm_facet_intervals(): Unused by print and summary methods.

For plot(): Unused.

object

An object returned by mfrm_facet_intervals().

parm

For confint(), optional distinct target names from object$table$Target. Extracts the stored limits without recalculation; level must match the saved level. Use mfrm_facet_intervals() to calculate a different level. Unavailable limits remain NA; inspect summary(object) for their eligibility and reasons.

comparison

Logical. With a sandwich result, also show its ordinary model-based interval. With source-propagated anchors, show the separate conditional-on-anchors interval. Both use the same point estimate.

draw

Logical; FALSE returns exact plotted data without opening a device.

preset

A package plot preset, for example "standard" or "monochrome".

title, subtitle, caption

Optional returned text; NULL omits it. Defaults describe the facet, exact confidence level, method and fixed/unavailable symbols. This text is not drawn; use plot_data(p, "notes") and plot_data(p, "title") to retrieve it.

reference

Optional finite vertical reference; default zero. Use NULL to omit it. This is not a rater-quality threshold.

show_legend

Logical; show the method legend (default TRUE).

Value

An mfrm_facet_intervals object with table, selected target covariance, model_covariance, free-parameter parameter_covariance, model_parameter_covariance, person_scores, cluster_scores, the clusters mapping, exact contrasts, settings, and source fit. The model-based SE and interval remain alongside the selected method. If ordinary MML joint information is unavailable, free-parameter covariances are NULL and target covariances contain NA (zero for constraint-fixed rows/columns); unavailable_reason and checks explain the failure. Sandwich scores and their rank are then uncomputed, not zero. Score rows are derivatives of a person's marginal log likelihood, not observed category scores or ability estimates. Scores are absent for method = "model". When weak information passes numerical refinement, cautions and information_review retain the warning and checks. InferenceCaution also accompanies the interval table when applicable. Each row retains Status, CIEligible and a plain-language InferenceReview. CIEligible concerns the selected interval, not the validity of the point estimate or a guarantee of coverage. fixed means the target is fixed by constraints, with no inferential interval. insufficient_cluster_rank withholds sandwich bounds because independent cluster scores do not span all free parameters. The point and separate ModelSE/ModelLower/ModelUpper are retained; those model bounds are not a replacement robust interval. Review the independent sampling units and model dimension; do not split dependent clusters to obtain an interval. nonpositive_variance or nonfinite_bounds denotes an unusable target-specific calculation. Missing bounds are not zero uncertainty. Two-family results additionally retain checks, numerical_checks, score_rank, source identity, scale references and interval refusal reasons. Their table includes CIEligible; failed numerical checks leave bounds and SEs missing while preserving the fitted point estimates. Native-location results retain checks, numerical comparisons, full and target covariance, target Jacobian and source identity, without an empirical Person-score rank requirement. Standardized results also retain NativeEstimate, the population SD, whether it was estimated, and the full transformation Jacobian. CIEligible describes numerical availability; CoverageQualified = FALSE does not claim finite-sample coverage for the transformed interval.

For plot(): Invisibly, a ggplot object; plot_data() retains the table, contrast coefficients, settings and labels, and as_ggplot() remains idempotent on the returned object.

Details

Compare ordinary observed-information intervals with a sandwich covariance that treats a person's complete response vector, or an explicitly declared larger cluster, as the independent sampling unit. Estimates are not refitted.

For cluster score s_g and observed negative-log-likelihood Hessian H, the unadjusted sandwich is solve(H) %*% sum_g(s_g %*% t(s_g)) %*% solve(H). Scores aggregate all responses of a person before forming the outer products; grouping individual rating rows would be a different and incorrect calculation for this marginal likelihood. The full covariance is transformed through the fitted constraints and requested contrasts. Both methods use normal critical values. No automatic method selection, multiplicity adjustment or significance flag is supplied.

The sandwich requires many independent sampling units and appropriate regularity. It permits dependence within the declared unit but does not establish independence between units. A small number of units can give poor intervals even with nonsingular covariance. If their scores do not span the free-parameter space, sandwich intervals are unavailable; point estimates and the reason remain. For ordinary fixed-population RSM/PCM, an ineligible source fit still causes an error. If the source is eligible but joint information is unavailable, nonpositive or regularized, finite points remain with Status = "information_unavailable", CIEligible = FALSE, missing model and sandwich uncertainty, and the explanation in InferenceReview. This includes information-workspace limits. A sandwich also needs the inverse information; switching methods cannot repair its absence. Failed information can limit confidence in point accuracy along weak directions. No inverse is regularized for this output. Invalid arguments and cluster mappings remain errors. Targets fixed by constraints have no inferential interval. Known anchors exclude anchor uncertainty. An ill-conditioned but numerically verified unregularized inverse can be used with a warning. Successful inversion does not establish reliable normal intervals. Review interval width, boundary proximity and quadrature sensitivity; changing to sandwich covariance does not remove this concern.

plot() method: Plot fixed-facet interval methods

For plot(): Fixed targets use open diamonds without inferential intervals. A conditionally fixed anchor can have a source-propagated interval; the comparison displays its conditional fixed value separately. Method labels distinguish source propagation from conditional observed information. Unavailable intervals in either displayed method use crosses; an ordinary comparison interval does not substitute for a missing sandwich interval. The zero line is a reference, not a threshold of practical importance or a rater-quality rule. No fit or covariance is recalculated. Target order is preserved. Weak-information cautions remain in the returned data and appear in the default subtitle. Custom or omitted subtitles change display only. For experimental two-family GPCM locations, the axis uses fixed-N(0,1) ability units and the default subtitle states the location reference and unqualified coverage, with a deficient Person-score rank caution when applicable. These are not uniform expected-rating differences. Standardized contrast axes use population SD units; returned notes identify whether SD uncertainty is estimated or the known SD is one. Reviewed interaction contrasts retain group direction, panel weights and scale in their source result; returned labels distinguish them from facet locations. Native CML intervals are labeled conditional information. Their returned notes describe asymptotic validity without implying weak information or an estimated ability population SD. When a comparison has no estimate, its row is labeled "not estimated" and has no point or interval. An empty display range is only an axis placeholder; it does not assign a value to unavailable targets. Joint interaction-cell results retain cell identities, counts and sparse flags in plot_data(p, "interaction_cells"). Their axis uses additive predictor shifts; row labels show the two level names in fitted facet order. Units follow the fixed N(0,1) ability scale. Positive effects raise expected coded scores relative to the jointly fitted main effects. These pointwise intervals are not fixed-baseline bias screens or fairness labels.

Jointly fitted interaction cells and contrasts

Fit a prespecified two-facet interaction with fit_mfrm(facet_interactions = "Rater:Criterion"), then use mfrm_facet_intervals(fit, interaction = "Rater:Criterion"). The default returns all fitted cells, including the dependent last row and column. Cell labels, counts, sparse flags and identification constraints are retained in interaction_cells. This bounded route uses a fixed standard-normal ability population; latent regression, estimated population variance, adaptive integration, JML, CML, GPCM and multiple fitted interaction terms are outside this extension.

An interaction effect is an additive predictor shift after jointly fitting the main effects. Positive values increase expected coded ratings; they do not automatically imply favorable performance or unfairness. Effects have zero row sums and zero column sums across the declared levels, without weighting those constraints by cell counts. Their values depend on that definition and the included main effects. This differs from estimate_bias(), which holds the fitted calibration and person measures fixed while estimating an additional residual shift. Neither result proves substantive rater bias.

Write free interaction coordinates as gamma and their expansion to all cells as G gamma. Cell covariance is G V_gamma G', where V_gamma is the interaction block of the inverse full marginal information, including main effects and thresholds. It is not the inverse of only the interaction information block. Dependent cells consequently have correlated estimates and a singular joint cell covariance; this alone does not invalidate individual pointwise intervals. For a prespecified cell contrast C, the covariance is C G V_gamma G' C'. Supply every cell Target as a named column of contrasts, including zeros. A contrast fixed at zero by the row/column constraints has Status = "fixed" and no inferential interval; it is not evidence for a null interaction.

Source consistency, identification/boundary records, full information, stationarity and finer-quadrature checks must pass. Unavailable intervals retain finite point estimates and reasons. Sparse-cell flags are descriptive, not a sample-size guarantee. Full-design identification and numerical success do not establish nominal coverage. These are pointwise Wald intervals, not simultaneous bounds or multiplicity-adjusted cell decisions. Saved tables, plots, apa_table() and mfrm_results() retain the same values and cautions.

A bundled-data example uses the existing estimation and interval APIs:

ratings <- load_mfrmr_data("example_core")
fit <- fit_mfrm(ratings, "Person", c("Rater", "Criterion"), "Score",
  model = "PCM", step_facet = "Criterion", noncenter_facet = "Criterion",
  facet_interactions = "Rater:Criterion", quad_points = 61)
cells <- mfrm_facet_intervals(fit, interaction = "Rater:Criterion")
summary(cells)[, c("Target", "Estimate", "SE", "Lower", "Upper", "Status")]
plot(cells)
# Compare the first two displayed cells; choose the scientific contrast first.
C <- matrix(0, 1, nrow(cells$table),
  dimnames = list("First minus second", cells$table$Target))
C[1, 1:2] <- c(1, -1)
difference <- mfrm_facet_intervals(fit, interaction = "Rater:Criterion",
  contrasts = C)
vcov(difference) # Uses covariance between both cells

Uncertainty from independently estimated anchors

The default intervals condition on the exact anchor values. Supplying anchor_covariance and independent_anchors = TRUE adds source-calibration uncertainty without changing any point estimates. Use the full covariance, not only a vector of SEs or a diagonal replacement for correlated anchors. Its numerical validity is checked; its statistical validity, units and correspondence to the supplied anchor values remain the analyst's responsibility.

Write the free receiving parameters as u, supplied anchors as a, and the negative log likelihood Hessian blocks as H_uu and H_ua. Implicit differentiation gives S = -inverse(H_uu) H_ua. For a reported target t(u,a), its source derivative is D = t_u S + t_a. The selected target covariance is t_u V_conditional t_u' + D V_anchor D'. The direct derivative t_a matters for targets containing anchored elements; adding variance only to the free parameters would omit that contribution and their cross-covariance. This is a first-order approximation for independent samples, not joint recalibration, a bootstrap or a correction for anchor drift or biased anchors.

Source derivatives use central differences of the analytic marginal score, checked by halving the difference step and refining the quadrature. Failed source-derivative checks preserve point estimates and the conditional results, with unavailable selected uncertainty. ConditionalSE, ConditionalLower, ConditionalUpper and ConditionalStatus retain that separate calculation; facet Model* columns also retain it. AnchorSourceSE is the square root of the added variance contribution, not a separate confidence interval. The selected SE, bounds and vcov() include both components. A conditionally fixed anchor may therefore have a selected interval for its source-estimated value. A zero covariance contribution does not create an interval for a target fixed under both specifications.

anchor_uncertainty retains the supplied and numerically symmetrized/PSD covariance, any floating-point adjustment, anchor values, sensitivity, independent-sample assumption and derivative checks. Singular PSD matrices are allowed. The free-parameter covariance includes source propagation; it does not include anchored values as extra optimizer coordinates. Saved population intervals can reuse this sensitivity and the conditional information: mfrm_population_intervals(rater_ci) propagates it once, without refitting or differentiating again. Reports and exported tables retain the supplied covariance and source-derivative checks; RDS retains the full result.

Only fixed-grid direct RSM/PCM MML with unit weights, no interactions, shrinkage or groups, an intercept-only population, fixed SD and all anchors on its noncentered facet is supported. Adaptive integration, free population variance, GPCM, dependent calibration samples and uncertainty in the supplied SD are outside this extension. No nominal coverage guarantee follows from numerical agreement or positive covariance. Other facets retain their centering; changing an item pool still requires a defensible scale link.

A complete interface example starts from bundled ratings. This split is illustrative: independent Persons and stable anchors must be justified for the actual sampling design, and different IDs alone do not establish them.

ratings <- load_mfrmr_data("example_core")
ids <- unique(ratings$Person)
source <- ratings[ratings$Person %in% ids[1:20], ]
receiving <- ratings[!ratings$Person %in% ids[1:20], ]
baseline <- fit_mfrm(source, "Person", c("Rater", "Criterion"), "Score",
  model = "RSM", noncenter_facet = "Rater", quad_points = 61)
baseline_ci <- mfrm_facet_intervals(baseline, "Rater")
common <- c("R01", "R02")
anchors <- data.frame(Facet = "Rater", Level = common,
  Anchor = baseline_ci$table$Estimate[match(common, baseline_ci$table$Target)])
cohort <- fit_mfrm(receiving, "Person", c("Rater", "Criterion"), "Score",
  model = "RSM", anchors = anchors, noncenter_facet = "Rater",
  population_formula = ~1, person_data = unique(receiving["Person"]),
  population_sd = 1, quad_points = 61)
rater_ci <- mfrm_facet_intervals(cohort, "Rater",
  anchor_covariance = vcov(baseline_ci)[common, common], independent_anchors = TRUE)
summary(rater_ci) # Selected and conditional intervals side by side
population_ci <- mfrm_population_intervals(rater_ci)

Small samples and singleton categories

A small sample or a wide interval is not, by itself, a reason to suppress an estimate or an otherwise available asymptotic interval. RSM/PCM model intervals can retain singleton categories (one observed response in a fitted or local category cell) when all declared categories are represented in the fitted and local scopes. They require fresh category review, source and convergence checks, positive unregularized joint information and a finer-quadrature score/covariance comparison. Empty categories, unsupported coordinates, known identification failures and failed numerical checks are not excused by this route. Failed interval checks retain the point with unavailable uncertainty and a reason.

The fit keeps its category-review state and can still have InferenceReady = FALSE. Inspect this interval object's CIEligible, checks, numerical_checks and InferenceCaution for the specific calculation. This does not qualify other analyses or guarantee small-sample coverage. The same model covariance is used by vcov(fit) and default confint(fit). Singleton counts do not gain a sandwich or GPCM Wald extension here. For supported one-Person RSM/PCM calibration, model-information intervals may be available after the same numerical checks. Returned cautions and InferenceCaution state that calibration relies on one response vector and its assumed conditional independence; CoverageQualified is FALSE. A finite point or information matrix does not establish small-sample coverage or between-Person variation. One Person cannot provide sandwich uncertainty; its ratings must not be relabeled as independent Persons.

MML assumptions and numerical approximation

MML integrates each Person's joint response probability under the declared ability distribution. Model-information intervals require the response and population models to hold for the actual rating assignments and covariates, including local independence given ability. An overall normal ability distribution does not by itself justify using the same distribution within rosters selected according to ability. Response-dependent assignment, stopping or missingness requires a separate justification of the likelihood.

The regular asymptotic interpretation requires independent Persons, an identifiable fixed-dimensional marginal model, a consistent interior solution, smooth likelihood derivatives and information growing in every free direction. With bounded ratings per Person and stable design proportions, a positive-definite limiting information per Person is a sufficient information-growth condition. Correctly specified MML can then support inference with fixed exposure per Person; increasing Person count alone cannot repair misspecification or an unidentified design. A positive observed Hessian at one fitted vector does not prove global identification.

The continuous-normal likelihood is evaluated numerically. A fixed finite quadrature order can retain approximation error as Person count grows. Optimizer and integration error must be negligible relative to sampling uncertainty for the continuous-model Wald interpretation. Agreement at two quadrature orders is a sensitivity diagnostic, not an error bound or a proof of asymptotic equivalence. Standardization does not remove either numerical approximation error or bias from a misspecified population model. The ordinary sandwich route can estimate variation around a working-model target under its own regularity conditions; that target need not be the true structural parameter.

Conditional likelihood

Native fit_mfrm(method = "CML") results use the inverse full conditional information, transformed through the fitted facet constraints and requested contrasts. Other facets and steps remain estimated. Only method = "model", clusters = NULL, adjust = FALSE and scale = "native" are supported. CML estimates no ability population SD: scale = "standardized" cannot be computed from that fit. For MML with an estimated intercept-only normal population, that option divides by the same fit's estimated SD and includes its covariance with the contrast; see "Population-SD-standardized contrasts". A full CML fit can retain finite numerical point estimates when its identification, finite-MLE and stationarity checks pass but conditional information fails. Its non-fixed targets have Status = "information_unavailable", missing SE/bounds, CIEligible = FALSE and the reason in InferenceReview. The fit's checks$information_status and checks$information_reason record the information decision. Constraint-fixed targets retain zero SE and no inferential interval. An information warning also limits confidence in the numerical accuracy of points along weak directions; retaining a point does not qualify it for inference. CML intervals are asymptotic, requiring the correct conditional response law, independent Persons and information growing in every free direction of a fixed-dimensional identified model. More Persons alone is insufficient; extreme totals add no information. Response-dependent assignment, stopping or missingness requires justification that the conditional law is preserved. See the sufficient regular regime in fit_mfrm(). Neither numerical acceptance nor a distribution-free ability treatment ensures small-sample coverage. plot(ci), summary(ci) and RDS reuse apply to CML interval objects; attaching them to ordinary mfrm_results() is not supported. Non-fixed CML targets require finite, strictly positive representable variance. Rescale contrast coefficients if their units cause variance underflow or overflow; numerical zero variance is not a constraint-fixed target.

For fit_scope = "comparisons", call mfrm_facet_intervals(fit) without new contrasts or comparison_review. The saved estimates, joint target covariance and qualifications are reused without optimization or information recalculation. level changes only the pointwise normal critical value. Unidentified, boundary or unqualified-stationarity comparisons keep missing estimates and intervals. If only information failed, identified numerical points remain, with missing uncertainty, their reasons and CIEligible = FALSE; no unavailable value is zero-filled. parameter_covariance is the covariance of the fit's identified subspace, not of separately identified cell effects. Use summary(ci), plot(ci), apa_table(ci) and saveRDS(ci); the saved review retains group order, contrast weights and observed/empty cells. JML/MML comparison-only fits remain descriptive and are not covered by this interval route.

Anchors and group means

The ordinary fixed-population RSM/PCM route supports location anchors and group means. Write the fitted facet locations as $b=b_0+J\eta$ and a requested comparison as $Cb$. Its covariance is $CJ V_\eta J^{\mathsf T}C^{\mathsf T}$, where $V_\eta$ is extracted after computing the full joint covariance. Other facets and thresholds remain estimated; inverting only the selected facet's information would omit their estimation uncertainty. The affine offset $b_0$ affects estimates, but has no variance when its supplied values are treated as exact.

A comparison with $CJ=0$ is fixed: its Status is "fixed", SE is zero, and interval bounds are missing. This includes values implied by centering, not only rows in an anchor table. For example, with four raters, whole-facet centering and a fixed mean of 0.10 for the first three, the fourth rater is fixed at -0.30. The three group members can still differ unless other constraints fix them; their mean has no inferential interval. A difference between a free rater and an anchored rater can remain uncertain. These are conditional statements, not evidence that the source calibration or group-mean assumption is error-free. Native-location anchors use only the fixed-SD RSM/PCM extension below; groups remain excluded there.

Native locations

For estimated normal RSM/PCM and intercept-only normal one-family GPCM MML, method = "model" supplies pointwise Wald intervals for locations or contrasts on the model's native ability scale. One-family GPCM also admits the explicit fixed-N(0,1) restriction (gpcm_mml_identification = "fixed_standard_normal"). Shared or separate slope/step owners are supported, with geometric-mean-one relative slopes. Use direct MML, fixed or adaptive integration, unit weights, additive centered facets and centered steps. RSM/PCM also accepts a fitted full-rank conditional-normal regression, retaining its coefficient and residual-variance cross-information when computing structural uncertainty. The saved population table identifies coefficients and residual variance; it does not label regression coefficients as marginal population means. GPCM latent regression and sandwich covariance remain unavailable here. The native location procedure performs its own qualification even when the source fit has InferenceReady = FALSE. Read CIEligible, Status, checks, numerical_checks and cautions on this result; interval availability does not change the source fit's overall readiness or qualify other procedures. Interactions use the separate comparison_review route below. Population-coefficient and variance intervals are separate targets and are not returned by this helper. RSM/PCM populations may fix residual SD with population_sd. For an intercept-only population (population_formula = ~1), a direct anchor on the selected non-Person noncenter_facet identifies the origin while free levels on that facet remain uncentered. These anchored intervals require no groups or dummy facets and condition on the supplied anchors and SD; their source-calibration uncertainty is not propagated. Other facets keep their declared centered constraints. Fixed SD has no variance coordinate in the joint information.

The covariance transforms the inverse full joint marginal information, including all estimated population, slope and step coordinates. A location-only inverse or independent printed SEs cannot replace it. The source, category support, local solution and unregularized information must pass checks; covariance and score changes are also checked at a finer integration order at the same fitted parameter vector. This model-information procedure has no empirical Person-score rank requirement. For RSM/PCM populations with estimated variance, a zero-residual-variance candidate at the retained nuisance estimates is compared with the fitted likelihood. A higher boundary likelihood beyond the numerical allowance prevents intervals. This check does not optimize or exclude the entire boundary. A numerical refusal preserves the native point and its reason with unavailable SE/bounds. Constraint-fixed targets have no inferential interval.

The asymptotic justification requires independent Persons, a correctly specified response and population model, an identified interior true parameter, a fixed number of structural parameters, nonsingular limiting information and sufficiently accurate integration and optimization. Information must grow in all target directions as Persons are added; connected facet labels alone do not establish this condition. These numerical checks do not establish finite-sample coverage, global identification or robustness to misspecification. Native locations are not divided by the estimated population SD unless scale = "standardized" is requested for a zero-sum contrast (see below). A location difference need not produce a uniform rating difference with unequal slopes or steps. The calculation does not refit or change the saved fit/readiness. Save and attach ci with mfrm_results(fit, intervals = list(locations = ci), compute = "never"); select plot(res, type = "facet_locations") for RSM/PCM or "gpcm_locations" for one-family GPCM. Reports and exports retain the native scale, inferential qualifications and unavailable outputs.

Population-SD-standardized contrasts

Set scale = "standardized" and supply explicit contrast rows whose coefficients sum to zero. For native contrast $d$, population SD $\sigma$ and free log-variance $\lambda = \log(\sigma^2)$, the target is $t = d/\sigma$. Its Jacobian has location entries $J_d/\sigma$ and log-variance entry $-t/2$. The covariance is transformed from the inverse full joint observed information, including the covariance between the contrast and log-variance. Dividing native interval endpoints by the fitted SD omits this uncertainty. In scalar form, $$\widehat{\mathrm{Var}}(\widehat t) = \frac{\widehat{\mathrm{Var}}(\widehat d) + \widehat d^2\widehat{\mathrm{Var}}(\widehat\lambda)/4 - \widehat d\widehat{\mathrm{Cov}}(\widehat d,\widehat\lambda)} {\widehat\sigma^2}.$$ The covariance term can increase or decrease the SE relative to dividing the native SE by the fitted SD. A zero estimated contrast can still have positive variance. With fixed N(0,1), the result equals the native contrast because the known SD is one. With an explicit fixed population_sd, divide the native contrast and SE by that SD; no estimated-SD derivative or uncertainty is introduced. Conditional-normal regression is excluded: its residual SD is not a marginal population SD. Standardization does not equate expected ratings across models with different slopes or category steps.

The existing source and numerical checks still apply. These are pointwise normal delta-method intervals, not profile intervals or finite-sample coverage guarantees. The regular delta method assumes a fixed true population SD strictly above zero and an identifiable, interior joint solution under the stated marginal model. A positive fitted SD alone does not establish these conditions. At zero SD the target is undefined; near zero the ratio and normal approximation can be unstable. No universal SD cutoff certifies validity. Standardization does not resolve weak identification or boundary problems; see the MML assumptions above.

MML interaction contrasts

Supply comparison_review to obtain native or population-SD-standardized intervals for its explicitly weighted facet-by-origin interaction contrasts. The review must match the fitted responses, group identities, constraints, model, PCM step owner, signs and population design. Its model has one interaction, zero origin main facet and centered remaining facets and interaction margins. This extension requires a full, unadjusted RSM/PCM direct MML fit, unit weights and preserved categories, without shrinkage or additional anchors. Comparison-only MML fits, JML and GPCM are not admitted. CML comparison intervals remain available through fit_mfrm() but cannot estimate a population SD.

Full conditional-map identification is necessary here. Source tables, categories, local solution, unregularized full information and finer-grid stability must pass the native-location checks above. A failed numerical check retains the point estimate with missing SE/bounds and a reason. Positive conditional rank and local information do not establish global marginal identification or finite-sample interval coverage; the output retains MarginalIdentificationState and CoverageQualified = FALSE. These are model-based asymptotic intervals, not a calibrated DFF classification.

For scale = "standardized", the denominator is the SD estimated in the same intercept-only normal MML fit (population_formula = ~1). Such a fit assumes one common ability distribution across the declared origin groups. It is a different model from ~Group, which allows group means. For ~Group, native interaction intervals are supported, but its residual SD is not substituted for a marginal population SD. A fixed N(0,1) fit has known SD one; no estimated-SD uncertainty is introduced in that case.

The joint delta method uses interaction derivatives divided by the fitted SD and derivative -Estimate/2 for log population variance. All population, step and structural nuisance coordinates remain in the full information inverse. It includes the covariance between the numerator and denominator; dividing the native endpoints or SE by the fitted SD is insufficient.

Read table with comparison_review: group subtraction order, panel weights, observed/empty cells and assumptions are retained. Use plot(ci), saveRDS(ci) to retain IdentificationBasis and its explanation with the intervals. A free-Person response-map basis does not establish distribution-free MML inference; population-dependent conditional identification does not establish global marginal identification. Display the result directly or attach it as mfrm_results(fit, intervals = list(school = ci), compute = "never"). Plotting and reports reuse the saved uncertainty; they do not refit or recalculate information. These results do not extend the separate mfrm_population_intervals() consumer to models with fitted interactions.

Two-family location targets

For native two-family GPCM MML, select a modeled non-Person facet. Locations sum to zero except in the declared noncenter_facet, whose locations are uncentered on the fixed N(0,1) ability scale. These locations are measured before multiplication by both slope components. Increasing a location lowers expected ratings when ability, slopes and other effects stay fixed. A contrast between actual raters need not imply a uniform difference in expected ratings: their slopes and category-step offsets may also differ. Neither zero nor a location difference defines rater quality or accuracy.

Let J expand the free location coordinates under their fitted constraints and C be the requested contrast matrix. The target covariance is C J V J' C', where V is the location block extracted after inverting the full joint marginal information. Slopes, steps and the other facet remain estimated nuisance parameters. Inverting a location-only Hessian, adding independent printed SEs, or using the EM conditional-objective Hessian would omit their joint estimation uncertainty.

The source fit must pass the same identity, category support, score derivative, stationarity, full-information and finer-quadrature checks used for experimental two-family slope intervals in confint.mfrm_fit(). The full covariance is checked; deficient empirical Person-score rank is a retained diagnostic and warning, not a veto on the inverse-observed-information method. Passing these numerical checks does not establish repeated-sample coverage for locations or contrasts. Evidence for component slope intervals does not qualify these different targets. These are reported separately as CIEligible (numerical interval availability) and CoverageQualified = FALSE in table. The latter means that nominal sampling coverage has not been established for the target and design; it is neither a coverage estimate of zero nor a numerical-failure flag. Saving, reporting or exporting the result retains both fields. These are experimental pointwise normal approximations, with no sandwich, profile, multiplicity or automatic significance decision. A failed check retains estimates and reasons with unavailable intervals, even if the coarse-grid information could be inverted. Constraint-fixed targets have no interval. The calculation does not refit or change quadrature in the saved fit.

For example, use ci <- mfrm_facet_intervals(fit, "Rater"), or supply a named zero-sum row of contrasts for a difference. Plot and save ci directly, or attach it with mfrm_results(fit, intervals = list(locations = ci)) and select plot(res, type = "gpcm_locations"). The model must actually contain the specified facet. Location intervals do not add category-step/curve bands, Person uncertainty, Wright maps or tests of rater competence. Individual two-family rater sheets currently display component-slope intervals only; use the analyst report for these location and contrast results.

What robustness means here

Under model misspecification, the sandwich describes sampling variation around the working model's limiting parameter (its pseudo-true target). That target need not equal the generating rater severity or criterion difficulty. Changing the SE does not correct a biased estimate, informative assignment, unmodeled population differences or MNAR nonresponse. Check the model and assignment before interpreting a severity contrast.

These intervals condition on the observed fixed facet levels. They do not generalize to replacement raters sampled from a rater population. They are not multiway crossed-cluster, G/D-study, variance-boundary, EAP or multiple-imputation intervals. Shared random rater/task effects spanning the declared clusters violate this one-way independence assumption. Numerical success is not a general coverage or rater-diagnosis guarantee. Review quadrature sensitivity separately; the helper reuses the fitted grid.

Bounded evaluation

Availability is not a coverage guarantee: sandwich intervals can describe a working-model target accurately while missing the generating contrast. The online simulation comparison gives the evaluated designs, results and Monte Carlo uncertainty. For a worked rater comparison, start at the beginning of that guide. The assumptions, small-sample cautions and unavailable states are retained in the sections above for offline use.

Save, display and report the selected intervals

Use plot(), as_ggplot() and plot_data() to display or extract the saved result, and apa_table() for tables. Set title = NULL, subtitle = NULL or caption = NULL in the plot to omit that text. For supported RSM/PCM MML fits, attach one result or a named list to mfrm_results(), for example mfrm_results(fit, intervals = list(raters = intervals), include = c("fit", "plots"), compute = "never"). The route plot(res, type = "facet_raters") shows the selected intervals. For two-family GPCM, the corresponding plot type is "gpcm_raters". mfrm_report() and export_mfrm_results() retain the method, level, contrast coefficients, cluster mapping and unavailable outcomes. The source fit must match; replay reloads the saved results without refitting or recomputing covariance. These intervals do not replace uncertainty in ordinary Wright maps, fit diagnostics or Person scores. For CML, keep the interval object itself: use summary(intervals), plot(intervals), apa_table(intervals) and saveRDS(intervals, file). Reopen it with readRDS(file). Ordinary mfrm_results(), mfrm_report() and export_mfrm_results() do not accept CML fits or their intervals.

Session plot defaults

See mfrmr_visual_diagnostics for options(mfrmr.plot_preset), argument precedence, and appearance when saving or reusing a plot.

References

White, H. (1982). Maximum likelihood estimation of misspecified models. Econometrica, 50(1), 1–25. Zeileis, A. (2006). Object-oriented computation of sandwich estimators. Journal of Statistical Software, 16(9), 1–16. doi:10.18637/jss.v016.i09 . Zeileis, A., Koell, S. and Graham, N. (2020). Various versatile variances: An object-oriented implementation of clustered covariances in R. Journal of Statistical Software, 95(1), 1–36. doi:10.18637/jss.v095.i01 .

Examples

# Long-running workflow; run explicitly when needed.
# example("mfrm_facet_intervals", run.dontrun = TRUE)
if (FALSE) { # \dontrun{
ratings <- load_mfrmr_data("example_core")
fit <- fit_mfrm(ratings, "Person", c("Rater", "Criterion"), "Score",
  model = "RSM", method = "MML", rating_min = 1, rating_max = 4,
  category_policy = "preserve")
intervals <- mfrm_facet_intervals(fit, "Rater", method = "sandwich")
summary(intervals)
plot(intervals)

# Compare two selected raters, rather than each rater against the origin.
# Here choose the first two example labels; prespecify your own comparison.
raters <- unique(as.character(ratings$Rater))
contrast <- matrix(0, 1, length(raters), dimnames = list(
  paste(raters[1], "minus", raters[2]), raters))
contrast[1, 1:2] <- c(1, -1)
difference <- mfrm_facet_intervals(fit, "Rater", contrasts = contrast)
summary(difference)[, c("Target", "Estimate", "SE", "Lower", "Upper", "Status")]
# With the default negative rater sign, a positive contrast means the first
# rater is more severe. Units are native logits. An interval containing zero
# leaves the direction uncertain; it does not establish equivalence.

# Estimate the population SD in this same model before standardizing.
# Even an intercept-only population model needs one row per person.
persons <- unique(ratings["Person"])
population_fit <- fit_mfrm(ratings, "Person", c("Rater", "Criterion"),
  "Score", model = "RSM", method = "MML", population_formula = ~ 1,
  person_data = persons, person_id = "Person",
  rating_min = 1, rating_max = 4, category_policy = "preserve", quad_points = 61)
standardized <- mfrm_facet_intervals(population_fit, "Rater",
  contrasts = contrast, scale = "standardized")
summary(standardized)
# Estimate is now in fitted ability-population SD units; its delta-method
# interval includes uncertainty in that SD. It is not a Wald z statistic.

# CML estimates the native contrast without an ability distribution.
if (requireNamespace("lpSolve", quietly = TRUE)) {
  cml_fit <- fit_mfrm(ratings, "Person", c("Rater", "Criterion"), "Score",
    model = "RSM", method = "CML", rating_min = 1, rating_max = 4,
    category_policy = "preserve")
  cml_ci <- mfrm_facet_intervals(cml_fit, "Rater", contrasts = contrast)
  summary(cml_ci)
  apa_table(cml_ci)
  file <- tempfile(fileext = ".rds") # Choose a lasting path for your analysis
  saveRDS(cml_ci, file)
  saved <- readRDS(file)
  summary(saved) # Same conditional-information result, without refitting
  # Use plot(saved) for a figure; ordinary mfrm_results() does not accept CML.
}
} # }