Fit the extended frame of reference model for paired comparisons

Estimates object locations from paired comparisons when the unit of the latent scale differs across frames – judge-panel by object-set cells – an extension of Humphry's (2005) extended frame of reference model to the Bradley-Terry-Luce family. Objects are partitioned into sets and judges into panels. Each object has a common-scale value v_k = alpha_s beta_k + kappa_s, with beta_k the within-set calibration location, alpha_s > 0 the set unit and kappa_s the set origin; a comparison judged in panel g carries the panel unit phi_g. Within a set the comparison logit is phi_g (beta_a - beta_b) (the set origin cancels and the set unit is confounded with the spread of beta, so neither is identified within a set); across sets it is phi_g (v_a - v_b), which places the sets on one common scale.

Usage

btl_efrm(
  data,
  object_a,
  object_b,
  winner,
  judge,
  panels,
  object_sets,
  response = NULL,
  ties = c("drop", "error"),
  min_link = 20,
  se_method = c("bootstrap", "judge_bootstrap", "conditional"),
  boot_reps = 60,
  maxit = 60,
  tol = 1e-08
)

Arguments

data: A data frame with one comparison per row.
object_a, object_b: Names of the columns holding the two compared objects.
winner: Name of the column holding the winner; its value must equal the row's object_a or object_b entry, "tie" or "draw" marks a tie, and anything else is treated as missing.
judge: Name of the judge column (clusters the stage-one standard errors and defines the panels when panels is a judge attribute).
panels: Either the name of a judge-attribute column in data or a named vector mapping judge to panel.
object_sets: A named list mapping set names to character vectors of object names; every compared object must belong to exactly one set.
response: Not supported: this first implementation fits dichotomous winner data only. Supplying it raises an informative error.
ties: "drop" (default, removed with a note) or "error".
min_link: Minimum number of cross-set comparisons a set pair must supply to be used for linking; sets not reachable from the reference set through sufficient cross-set pairs raise an error.
se_method: "bootstrap" (the default): a parametric bootstrap; "judge_bootstrap": judges resampled with replacement within panels, carrying within-judge dependence; of the ENTIRE two-stage pipeline – winners resampled from the fitted probabilities, both stages refitted boot_reps times – so the reported standard errors carry every source of sampling variability, including the stage-one uncertainty that flows into the linking. "conditional": the fast analytic errors, exact for beta and phi but conditional on stage one for alpha and kappa, which they therefore UNDERSTATE (verified by simulation); for quick inspection only.
boot_reps: Bootstrap replicates for se_method = "bootstrap".
maxit, tol: Newton iteration cap and convergence tolerance.

Value

An object of class "rasch_btl_efrm": objects (object, set, beta_set and its standard error, common-scale v and its standard error), phi_table (panel units with Wald tests against log phi = 0), alpha_table and kappa_table (set units and origins with Wald tests; the reference set carries alpha = 1, kappa = 0 with no standard error), frames (panel by set: common-scale discrimination rho = phi / alpha (the within-set logit is phi (beta_a - beta_b) with beta = (v - kappa) / alpha), comparison count, pooled fit residual), equal_unit (the descriptive single-unit comparison), n_cross (cross-set comparison counts per set pair), notes and converged.

Details

One modelling convention deserves plain statement. Rewritten with a single latent value per object, the model says within-set-s contests are judged at discrimination phi_g / alpha_s while every cross-set contest is judged at exactly phi_g: the common scale is defined as the scale of between-frame judgement. That is an assumption about how discriminal dispersion behaves when unlike objects meet (a Thurstonian question with no assumption-free answer), not a consequence of the theory; the per-frame fit residuals in frames are where a violation would show.

Estimation follows rasch_efrm in two conditional stages. In stage one the within-set comparisons of each set, pooled over panels, fit the bilinear model logit = rho_{gs} (b_a - b_b) with b sum-zero and the most-used panel fixed at rho = 1; the ratios rho_{gs} = phi_g / phi_{ref(s)} estimate the panel units up to each set's reference panel and are reconciled across sets by a precision-weighted least squares over the panel-by-set linking graph, then normalised to geometric mean one. In stage two the cross-set comparisons are a low-dimensional maximum likelihood in (log alpha, kappa) for the non-reference sets, with alpha_1 = 1 and kappa_1 = 0 fixing the reference set (the first, alphabetically).

The set units are identified here WITHIN the conditional framework: the cross-set linking uses only comparison outcomes and makes no distributional assumption about the objects. This is the substantive difference from the persons-by-items EFRM, whose item-set units can only be identified from the person side – that is, from the distribution of the persons over the linked sets. The paired-comparison design supplies its own conditional link, so no such distributional step is needed. See Humphry (2005) and Humphry and Andrich (2008) for the theory of the unit, and Thurstone (1927) and David (1988) for the varying-discriminal-dispersion lineage from which the frame-dependent unit descends. The paired-comparison form is this package's extension of Humphry's model.

The ESTIMATES are always the staged conditional estimator: the within-frame calibrations are invariant to the linking data (the frame-of-reference property), a deliberate trade of some efficiency for invariance, exactly as anchored equating trades. Inference defaults to a parametric bootstrap of the whole pipeline (se_method = "bootstrap"): winners are resampled from the fitted probabilities and both stages refitted, so the standard errors of phi, alpha, kappa, beta and the common-scale v carry every stage's sampling variability – verified by simulation to restore nominal coverage where the conditional errors cover at a third of their nominal rate on linked designs. The "conditional" option keeps the fast analytic errors (judge-clustered sandwich for stage one, inverse observed information for stage two, conditional on stage one) for quick inspection; its alpha and kappa errors understate, and the fit says so.

Two honesty notes on the bootstrap. The default is model-based: replicates are drawn as independent Bernoulli outcomes at the fitted probabilities, which is self-consistent (the model has no judge parameter) but does not carry extra-model dependence within judges. When judges plausibly carry idiosyncratic preferences across their comparisons, use se_method = "judge_bootstrap": judges are redrawn with replacement within each panel and the whole pipeline is refitted, so the errors carry both the stage-one uncertainty and the within-judge dependence (it needs enough judges per panel to resample stably; failed replicates are counted and reported). And a parameter that reaches its boundary in some replicates (a set unit driven to zero when a resampled within-set order flips against the cross-set evidence, the signature of a two-object set with a near-even internal pair) has no normal sampling distribution: its standard error is reported as NA and a note names the parameter and the boundary count rather than manufacturing a number. Relatedly, a set whose within-set contests are all near-even (or all one-sided) carries no stable information about the panel-unit ratios; such sets are screened out of the phi reconciliation, refit with the panel units held at the reconciled phi (which the frame model says apply to them regardless), and named in a note.

A single set (S = 1) reduces the model to panel units alone; stage two is skipped and the print states the panel-units model. When additionally G = 1 the fit reduces exactly to btl on the same data. The equal-unit (single-unit) comparison refits plain btl on all comparisons pooled and reports the descriptive composite log-likelihood difference against the frames model; because that comparison is a composite likelihood, the inference on the units is carried by the Wald tests on log phi_g and log alpha_s in phi_table and alpha_table.

References

Bradley, R. A. and Terry, M. E. (1952). Rank analysis of incomplete block designs: I. The method of paired comparisons. Biometrika, 39, 324-345.

David, H. A. (1988). The Method of Paired Comparisons (2nd ed.). Griffin.

Humphry, S. M. (2005). Maintaining a common arbitrary unit in social measurement. PhD thesis, Murdoch University.

Humphry, S. M. and Andrich, D. (2008). Understanding the unit in the Rasch model. Journal of Applied Measurement, 9(3), 249-264.

Luce, R. D. (1959). Individual Choice Behavior. Wiley.

Thurstone, L. L. (1927). A law of comparative judgment. Psychological Review, 34, 273-286.

Examples

# \donttest{
d <- simulate_btl_efrm(n_objects_per_set = 6, n_sets = 2, n_panels = 2,
                       set_units = c(1, 1.4), set_origins = c(0, 0.8),
                       seed = 1)
fit <- btl_efrm(d, "object_a", "object_b", winner = "winner",
                judge = "judge", panels = "panel",
                object_sets = attr(d, "truth")$object_sets)
fit$alpha_table
#>    set    alpha se_log_alpha        z          p
#> 1 set1 1.000000           NA       NA         NA
#> 2 set2 1.606727    0.1908155 2.485119 0.01295081
# }