Pointwise intervals for a conditional-normal population model

Report coefficients, prespecified linear contrasts and residual variance from an estimated RSM/PCM MML population model. Coefficient intervals use a normal approximation; residual variance intervals exponentiate a normal interval for log residual variance. These experimental intervals have unqualified repeated-sample coverage. Method qualification is incomplete even when numerical checks pass. Saved cautions are also retained as an interpretation_notes table in mfrm_results(), reports and table exports.

Usage

mfrm_population_intervals(
  object,
  level = 0.95,
  contrasts = NULL,
  anchor_covariance = NULL,
  independent_anchors = FALSE
)

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

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

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

Arguments

object: A native RSM/PCM MML fit with an active population model, or saved native-location mfrm_facet_intervals() output from that fit. The latter reuses its numerically checked full joint covariance without differentiation. For summary() and vcov(), supply the population-interval result.
level: Pointwise confidence level, strictly between zero and one.
contrasts: Optional numeric matrix of prespecified linear combinations of population coefficients. Give each row a distinct nonempty target name and name every column with a fitted coefficient name, including zero-weight columns, as in names(fit$population$coefficients). Columns are matched by name. Each row must be finite and nonzero; dependent rows are allowed for pointwise intervals. NULL retains the coefficient/variance-only output.
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.
x: An interval result.
...: Unused.

Value

An mfrm_population_intervals object with the interval table, working-coordinate covariance, full parameter covariance, numerical checks, source identity and unchanged source fit. With contrasts, also contains the column-aligned contrasts matrix and contrast_covariance; targets in the table have type coefficient_contrast and prefix contrast:.

Details

For the bounded fixed-SD anchor-origin route, anchor_covariance with independent_anchors = TRUE propagates an independently estimated anchor covariance into the population coefficients and contrasts. Alternatively, pass a saved facet-interval result already containing source propagation and omit both arguments; its sensitivity and conditional information are reused once. See mfrm_facet_intervals() for the formula, required independence, supported scope and complete bundled-data example. Point estimates are unchanged; conditional SE/bounds remain in separate Conditional* columns. vcov() then returns source-propagated coefficient covariance. The fixed population variance still has no SE or interval. Drift, source-estimate bias, dependent samples and uncertainty in the supplied SD are not corrected.

The fitted model is $\theta_i\mid X_i\sim N(X_i\beta,\sigma_e^2)$. Residual variance is conditional on the fitted predictors; it is not the marginal variance of ability. Predictor transformations and coefficient coding are those retained by the fit. The covariance is taken from the inverse of the full information, including all structural nuisance terms. A contrast matrix $L$ gives estimates $L\widehat\beta$ and covariance $L V_\beta L^{\mathsf T}$, using the coefficient block of that inverse. Contrast rows are added to the interval table. These describe conditional population means or their linear combinations, not differences between two individual abilities or causal effects. For example, in ~ X, weights (0, 2) compare the means at X = 1 and X = -1. Interactions, factor reference levels and transformations must use the fitted coding. The helper accepts weights in coefficient coordinates; it does not infer raw covariate profiles or rebuild a transformation from a new sample.

Use direct MML with fixed or adaptive integration, unit observation weights, full-rank population coding, centered additive facets and centered steps, without constraint groups or fitted interactions. The fixed-SD anchor-origin extension described below also admits supplied anchors. Source consistency, category support, local solution, full joint information and finer-integration checks must pass as in mfrm_facet_intervals(). The procedure can return eligible intervals while the source fit retains InferenceReady = FALSE; inspect the interval's CIEligible, Status, checks and cautions. This does not change overall fit readiness or qualify other inference procedures. A recorded identification or boundary failure prevents intervals. Failed numerical checks retain point estimates and reasons with unavailable uncertainty. This is an interior, model-based approximation: it does not handle a zero-variance boundary, misspecified populations, dependent Persons or informative assignment. Passing numerical checks does not prove global optimality or coverage. When residual variance is estimated, a zero-residual-variance candidate at the retained nuisance estimates is also compared with the fitted likelihood at both integration orders. A higher boundary likelihood beyond the numerical allowance prevents intervals. This inexpensive comparison does not optimize the boundary or rule out other better boundary candidates. Requests using saved facet inference also perform this likelihood comparison, without refitting or differentiating again. A fit with explicit population_sd conditions on that positive fixed SD; zero variance is not a feasible candidate. Its residual-variance row retains the supplied value with Status = "fixed", missing SE/bounds and CIEligible = FALSE. Coefficients and their contrasts still have model-based intervals when numerical checks pass. With population_formula = ~1, direct anchors on the selected noncenter_facet can identify the population mean and free facet locations. By default, these intervals condition on the supplied anchors and SD. The optional anchor_covariance extension adds independent source-anchor uncertainty while still conditioning on the SD.

SE is on the reported estimate scale (delta-method SE for variance). WorkingEstimate and WorkingSE use coefficient/log-variance coordinates. WorkingLower and WorkingUpper retain the interval on that scale. Exponentiation can give displayed variance bounds of zero or infinity outside floating-point range; NaturalScaleStatus identifies that representation limit while the finite log-scale interval is retained. vcov() returns their covariance, with explicit coordinate names; its final coordinate is log residual variance, not natural variance, when variance is estimated. A fixed variance has no free covariance coordinate. It remains the coefficient/log-variance covariance when contrasts are supplied; contrast_covariance separately contains the covariance of the requested rows. No multiplicity adjustment or automatic significance decision is made.

Save the result directly, or attach it through mfrm_results(fit, intervals = list(population = ci), compute = "never"). Tables, reports and exports reuse saved intervals and numerical checks. Supplied contrast weights and their covariance are also exported. These tables prefix coordinate columns with beta:/contrast: to distinguish them from the Target identifier column. The saved result retains the unchanged fit, including its population formula, trained terms, factor levels and coding; save the raw reference profiles separately if you used them to construct the weights. The ordinary population summary remains a point-estimate table.

Reading the result

Start with summary(ci), then inspect ci$checks, ci$numerical_checks and ci$cautions. Keep these with the interval table when saving or sharing. plot(ci) displays saved coefficients; plot(ci, type = "contrasts") displays the supplied linear contrasts. Unavailable intervals remain visible, and as_ggplot() reuses the same data.

Status = "available" and CIEligible = TRUE mean that the numerical checks permit the experimental interval. They do not certify nominal coverage. InferenceReview records this distinction in each row.
Status = "fixed" retains a supplied residual variance without an inferential interval; missing uncertainty is not an estimated zero SE.
Status = "numerical_review_failed" retains the point estimate and failed-check reason, with unavailable SEs and bounds. nonpositive_variance or nonfinite_bounds instead identifies an unusable target-specific uncertainty calculation. Missing bounds are not infinite endpoints.
NaturalScaleStatus = "outside_floating_point_range" means the finite log-variance interval cannot be fully represented after exponentiation. Inspect WorkingLower and WorkingUpper; a displayed zero is not an interval qualified for a zero-variance boundary.

Coverage evidence is specific to the target, design and complete estimation procedure. In the retained RSM checks, mean-at-$X=1$ and log-variance intervals at 20 Persons remain inconclusive under the prespecified practical criteria. The independent 240-Person slope/mean-difference confirmation supports those targets in its tested design; it does not establish a minimum sample size or qualify arbitrary regressions, PCM designs or all contrasts. All outputs from this helper therefore retain their experimental status.

These intervals do not add calibration uncertainty to individual scoring. predict_mfrm_units() has separate source and scoring checks; its conditional posterior intervals hold the fitted calibration and population parameters fixed. Availability here does not authorize an otherwise refused scoring result. See mfrmr_interval_guide() for the different uncertainty targets.

Examples

# Long-running workflow; run explicitly when needed.
# example("mfrm_population_intervals", run.dontrun = TRUE)
if (FALSE) { # \dontrun{
ratings <- load_mfrmr_data("example_core")
persons <- data.frame(Person = unique(ratings$Person))
# Fictional predictor for this example; use your observed covariates.
persons$X <- seq(-1, 1, length.out = nrow(persons))
fit <- fit_mfrm(ratings, "Person", c("Rater", "Criterion"), "Score",
  model = "RSM", method = "MML", population_formula = ~ X,
  person_data = persons, quad_points = 61)
L <- matrix(c(0, 2), nrow = 1,
  dimnames = list("X = 1 minus X = -1", c("(Intercept)", "X")))
ci <- mfrm_population_intervals(fit, contrasts = L)
subset(summary(ci), Type == "coefficient_contrast")
plot(ci)
p <- plot(ci, type = "contrasts")
plot_data(p, "contrast_weights")
plot_data(p, "notes")
results <- mfrm_results(fit, intervals = list(population = ci),
  include = "fit", compute = "never")
apa_table(results$population_intervals$population)
} # }