Population-SD profiles and explicit shared-rater model intervals

Usage

# S3 method for class 'mfrm_random_rater'
confint(object, parm = "rater_sd", level = 0.95, ...)

Arguments

object: A result from fit_mfrm_random_rater(). Profiling requires estimated rater SD.
parm: "rater_sd" (default) profiles population variation. "raters" explicitly requests first-order normal prediction intervals for all observed raters. These are not supplied automatically because their nominal coverage is not established. "calibration" explicitly requests observed-information normal approximations for fixed-facet levels and steps, excluding population SDs.
level: Confidence level, between zero and one; default 0.95.
...: Unused.

Value

A one-row matrix with Lower and Upper, and attributes level, method, profile and note. The profile retains every evaluated SD, refitted likelihood, LR (twice the loss from the source fit), information checks and Detail. Unresolved numerical checks or endpoint searches stop with an mfrm_profile_error; its profile field retains the evaluations. A failed search does not establish that a statistical interval is absent. With parm = "raters", one row per rater, computed from saved estimates and PredictionSE without refitting or RTMB. Attributes retain the method, target, level and interpretation. Unresolved numerical/information checks, estimated variance boundaries and unavailable SEs give missing bounds. With parm = "calibration", one row per fixed-facet level or step, with the same level/method/target/note attributes and availability guards. Default calibration tables omit bounds; this explicit calculation uses saved SEs without refitting and does not establish finite-sample coverage. Both normal-interval routes retain row-level IntervalStatus, CIEligible and plain-language InferenceReview in attr(ci, "diagnostics"), together with saved estimates/SEs, and source checks in attr(ci, "checks"). Export the diagnostics table when saving CSV; a two-column numeric matrix alone loses the explanation. See confint.mfrm_testlet() for the status meanings. A missing interval does not turn a retained point into a reliable estimate, and more iterations cannot by themselves remove an estimated-variance boundary restriction.

Details

For parm = "rater_sd", fixed facets, steps and any estimated ability SD are refitted at each candidate rater SD, including zero. A specified known ability SD stays fixed. Earlier saved fits retain their known N(0,1) population. The search brackets a lower and upper crossing of twice the log-likelihood loss with qchisq(level, df = 1), around the fitted SD. It uses the same approximate marginal likelihood as the fit. The lower bound is zero when the zero-variance submodel belongs to the likelihood-ratio region. An estimated boundary can therefore have a positive upper limit even though ordinary Wald intervals are unavailable. Each nonzero endpoint must attain the likelihood-ratio cutoff to 1e-4 * min(1, cutoff) absolute accuracy. Insufficient floating-point resolution stops the search instead of reporting coincident limits. Profile refits require positive nuisance information. With known ability SD, nuisance curvature is checked at both integration orders. These local numerical checks do not prove a global optimum or exclude disconnected likelihood-ratio regions between sampled candidates.

This is an asymptotic likelihood-ratio interval, not a finite-sample coverage guarantee. The chi-square reference is nonregular at a true zero variance; the usual one-degree-of-freedom cutoff is conservative under the standard single-variance boundary asymptotics. Few raters, Laplace error and design misspecification can alter coverage. The numerical checks do not establish those asymptotic conditions. The calculation stops if the fitted or profiled ability variance is an estimated zero boundary; that additional nuisance boundary requires a different reference distribution for inference. No ability-SD interval is supplied. This interval does not quantify the predictive distribution of a replacement rater, whose variation is a different target.

The separate parm = "raters" calculation uses the conditional mode plus or minus qnorm((1 + level)/2) * PredictionSE. It targets realized, uncentered rater effects relative to the population mean. First-order calibration uncertainty does not ensure nominal coverage. In particular, few-rater coverage remains unresolved; see online random-rater guide. Requesting this approximation does not make it qualified for classification or exclusion of raters. It is not a profile interval, a bootstrap or an interval for a difference of raters.

References

Self, S. G. and Liang, K.-Y. (1987). Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions. Journal of the American Statistical Association, 82, 605–610. doi:10.1080/01621459.1987.10478472 .