Rasch’s criterion of invariant comparison is stated within a frame of reference (Rasch 1961). The extended frame of reference model allows the unit to differ over linked combinations of item sets and person groups (Humphry 2005; Humphry and Andrich 2008). For item \(i\) in set \(s\) and person \(n\) in group \(g\),
\[ P(X_{ni}=x)= \frac{\exp\{\rho_{sg}[x\theta_n-\sum_{k=1}^{x}\delta_{ik}]\}} {\sum_{y=0}^{m_i}\exp\{\rho_{sg}[y\theta_n- \sum_{k=1}^{y}\delta_{ik}]\}},\qquad \rho_{sg}=\alpha_s\phi_g . \]
Following Humphry and Andrich (2008, eq. 15), the multiplicative parameter of a frame is the ratio of the common reference unit to the frame’s own unit: \(\alpha_s\) carries that ratio for item set \(s\), and \(\phi_g\) for person group \(g\). Their geometric means are fixed at one; no observed set or group is the reference. A set with \(\alpha_s = 1.3\) has a smaller natural unit than the geometric-mean set unit, so its response curves are steeper on the common scale. Two observed sets can be compared directly by the ratio \(\alpha_s/\alpha_t\). The partial credit model holds within each frame in its natural unit; dichotomous items are the one-threshold case.
Use this model when variation in the unit is part of the measurement account. Poor fit to an equal-unit model is not by itself evidence for a frame-dependent unit.
The distributed vignette uses a precomputed fit, including its bootstrap uncertainty. The code below reproduces the calculation.
d <- simulate_efrm(
n_per_group = 150,
items_per_set = 8,
set_unit_ratio = 1.30,
group_unit_ratio = 1.10,
seed = 25
)
truth <- attr(d, "truth")
d$site <- rep(c("A", "B"), length.out = nrow(d))
d
#> Simulated efrm data: 300 persons, 2 sets x 2 groups, 16 items (2 categories)
#> Generating values and departures:
#> - set-unit ratio 1.30 across 2 sets
#> - group-unit ratio 1.10 across 2 groupsThe simulated data carry the units the fit below has to recover: a set-unit ratio of 1.30 and a group-unit ratio of 1.10.
fit <- rasch_efrm(
d,
item_sets = truth$item_sets,
groups = "group",
id = "id",
factors = "site",
boot_reps = 50,
workers = 1,
seed = 25
)
fit#> rasch extended frame of reference analysis: 16 items in 2 set(s) x 2 group(s) = 4 frames, 300 persons
#> Within-frame pairwise conditional ML: converged in 10 iterations
#> PSI 0.781, separation quality: reasonable
#>
#> Person group units (phi):
#> group phi se_log_phi
#> g1 0.897 0.040
#> g2 1.115 0.040
#>
#> Item set units (alpha) and locations:
#> set alpha se_log_alpha mu n_items
#> set1 0.840 0.043 0.081 8
#> set2 1.190 0.043 -0.081 8
#>
#> Equal-unit comparison: 2(ll_EFRM - ll_equal) = 20.620 with 1 extra unit parameter(s)
#> (composite likelihood: descriptive; informative for group units (phi))
#> Omnibus Wald tests of equal units (Holm-adjusted family):
#> term df df2 wald f p p_adj significant
#> group units (phi) 1 Inf 7.442 7.442 0.006 0.006 *
#> set units (alpha) 1 49 16.034 16.034 < 0.001 < 0.001 *
#> Holm-adjusted exploratory unit contrasts (H0: unit = 1):
#> parameter estimate se z df p p_adj significant
#> log phi[g1] -0.109 0.040 -2.728 Inf 0.006 0.006 *
#> log phi[g2] 0.109 0.040 2.728 Inf 0.006
#> log alpha[set1] -0.174 0.043 -4.004 49 < 0.001 < 0.001 *
#> log alpha[set2] 0.174 0.043 4.004 49 < 0.001
#> (the units are centred: the second row of a two-row family restates the first, so its adjustment is withheld)
#>
#> Notes: person measures use the weighted score; per-group score curves replace the raw-score table (see score_curves); a universal raw-score conversion is not defined across the expanded frame response cells; use score_curves and design-specific information; two centred unit rows are one hypothesis: the adjusted probability and flag are reported on the first row of the pair and withheld on the second, which restates itThe principal model-specific tables are:
fit$phi_table # person-group units
#> group phi se_log_phi
#> g1 0.897 0.040
#> g2 1.115 0.040
fit$alpha_table # item-set unit ratios (reference unit / set unit)
#> set alpha se_log_alpha
#> set1 0.840 0.043
#> set2 1.190 0.043
fit$set_table # linked set locations
#> set mu alpha n_items
#> set1 0.081 0.840 8
#> set2 -0.081 1.190 8
fit$frames # complete frame units
#> set group n_persons n_items alpha phi rho se_log_rho origin infit_ms
#> set1 g1 150 8 0.840 0.897 0.754 0.062 0.081 1.031
#> set2 g1 150 8 1.190 0.897 1.067 0.056 -0.081 1.048
#> set1 g2 150 8 0.840 1.115 0.937 0.056 0.081 1.063
#> set2 g2 150 8 1.190 1.115 1.327 0.062 -0.081 1.012
#> outfit_ms fit_resid n_responses
#> 0.989 0.185 1168
#> 0.988 -0.316 1168
#> 1.008 0.422 1184
#> 0.865 -1.361 1184
fit$linking # set-linking design
#> $phi_edges
#> $phi_edges[[1]]
#> [1] 2 1
#>
#> $phi_edges[[2]]
#> [1] 2 1
#>
#>
#> $alpha_edges
#> set_a set_b n log_slope converged edge_mass loglik
#> set1 set2 294 0.348 1 0.000 -3522.116
#>
#> $boot_reps_requested
#> [1] 50
#>
#> $boot_reps_used
#> [1] 50
#>
#> $boot_reps_failed
#> [1] 0
#>
#> $alpha_method
#> [1] "finite-grid semiparametric maximum likelihood"
#>
#> $alpha_grid
#> lower upper points
#> -8 8 61The frame-defining group is part of the model and is not retested as
DIF. Other person factors can be examined with dif_anova().
Its bootstrap conditions on each person’s subtotal within each observed
item set, then repeats the Extended Frames fit and the complete DIF
analysis. Its familywise probabilities refer to the fitted global
invariant null.
Person-group units are estimated from common threshold patterns across groups. Set units are estimated from persons observed in more than one set. For a linked pair, the natural coordinates satisfy \(u_b=r u_a+c\). The package estimates \(r\), \(c\), and masses for the nuisance person distribution on a finite grid by semiparametric likelihood:
\[ \prod_n \int P(X_{na}\mid u)P(X_{nb}\mid ru+c)\,dF_{g(n)}(u). \]
The masses are estimated jointly with the link, separately for each observed person group, so neither a normal shape nor a common distribution across groups is prescribed. The link holds the conditional item thresholds and person-group units fixed while estimating only \(r\), \(c\), and the nuisance masses. The common-scale parameters are then
\[ \delta_{ik}=\widetilde\delta_{ik}/\alpha_s+\mu_s, \qquad \rho_{sg}=\alpha_s\phi_g. \]
The item thresholds and person-group units remain pairwise conditional estimates; the mixing distribution is used only for the relative set unit, which is not identified by that conditional stage. The discrete nonparametric margin follows Follmann (1988); its use for linked item-set units is an extension implemented here.
The group-by-set frame graph and the set-linking graph must each connect to a common scale. A linking response pattern must span a score range of at least four within a set. Overlapping item sets are not permitted.
Several group factors may define crossed frames:
fit_crossed <- rasch_efrm(
data,
item_sets = item_sets,
groups = c("language", "cohort"),
id = "id"
)
fit_crossed$phi_factorial
fit_crossed$phi_factorial_testsThe factorial tables give the GLS decomposition of the estimated log
group units. Use p_adj, which applies Holm’s correction
across the factorial terms, for decisions. The tables do not replace the
frame-level estimates.
The default hybrid covariance combines the pairwise Godambe
covariance with a person bootstrap for set linking. Within-frame
thresholds and group units are jointly redrawn, and the link is rebuilt
from each draw. The resulting covariance is retained for the
common-scale thresholds and complete frame units. Set
se_method = "bootstrap" to refit the complete model to each
person resample.
The linking calculations use a compiled numerical kernel. Bootstrap
replicates run on four workers by default, or fewer where the system
limit is lower. A fixed seed gives the same samples and
results for every worker count without altering the caller’s
random-number stream. In the Shiny application the fit runs in a
separate process. The analyst can select up to four workers, follow
completed batches, or cancel the process without replacing the current
analysis. R scripts can supply progress and
cancel callbacks for the same controls.
The fit records the requested, usable and failed bootstrap counts.
The requested number must exceed the free directions in the largest
covariance block used by the fit. The stored bootstrap row contains
several constrained blocks and is not treated as one covariance. Once
estimation begins, the linking covariance needs at least 30 usable
replicates and a majority of those requested; otherwise the fit stops. A
numerical failure or non-convergence on any supported link invalidates
the whole replicate. Pairs without enough informative common persons can
be omitted only if the remaining links connect all sets. Full-bootstrap
refits with unidentified group units are also discarded. The conditional
calibration checks the exact likelihood curvature as well as the score.
Stationary saddles and flat solutions are refused; the same check
applies to bootstrap refits. Saved frame fits without a current
likelihood-check record must be refitted before reopening in the app. So
must an extended frame fit whose stored score curves predate the shared
design enumeration, since their designs and labels are no longer the
ones test_information() and the curve plots would give.
Neither refusal alters the source file. The alpha–phi cross-covariance
has the same usable-draw rule and is withheld if it cannot be estimated.
A full bootstrap that falls below its minimum returns hybrid standard
errors instead, retaining its attempted, usable and failed counts in
full_boot_reps_*.
fit$efrm_vs_rasch$unit_omnibus
#> term df df2 wald f p p_adj significant
#> group units (phi) 1 Inf 7.442 7.442 0.006 0.006 *
#> set units (alpha) 1 49 16.034 16.034 < 0.001 < 0.001 *
fit$efrm_vs_rasch$unit_tests
#> parameter estimate se z df p p_adj significant
#> log phi[g1] -0.109 0.040 -2.728 Inf 0.006 0.006 *
#> log phi[g2] 0.109 0.040 2.728 Inf 0.006
#> log alpha[set1] -0.174 0.043 -4.004 49 < 0.001 < 0.001 *
#> log alpha[set2] 0.174 0.043 4.004 49 < 0.001The omnibus Wald tests assess the set- and group-unit families. Their
probabilities are adjusted together by Holm’s method. Individual unit
contrasts form a separate Holm-adjusted follow-up family. The raw
composite-likelihood difference is descriptive and compares the
group-unit stage only. Set units are identified in the linking stage and
are assessed by their Wald test. compare_fits() withholds
generic likelihood differences involving EFRM: its calibration
likelihood uses within-set pairs, whereas ordinary PCM also uses
cross-set pairs. The matched group-unit comparison is retained in
fit$efrm_vs_rasch. Probabilities require at least 50
persons or effective persons in every group and a path between each pair
of sets whose links each have at least 50 informative common persons. A
person at the same extreme tail in both sets remains in the likelihood
but does not count as link support. Sparse designs retain the unit
estimates without an inferential probability.
Two-set simulations under normal, bimodal and deliberately different group distributions gave set-unit bias within 0.004 log-units, empirical-to-reported SE ratios from 0.97 to 1.05, and null rejection from 4.0 to 5.0 per cent for the hybrid method. The complete person bootstrap was mildly conservative in the corresponding null design. Sets with only three dichotomous items were not identified; four-item sets were recovered without bias in adequately supported samples.
A separate null study checked the complete decision families. With 200 persons per group and six items per set, 489 of 500 fits were analysed. The Holm familywise rejection rates were 3.5 per cent for the omnibus tests and 1.6 per cent for the individual unit contrasts. A boundary design with 100 persons per group and four items per set refused 493 of 1,000 fits and did not converge in another 24. Among the 483 analysed fits, the corresponding rates were 5.2 and 2.5 per cent. A design should not be used merely because it is formally identifiable.
The fitted EFRM gives an item one location across its frames, scaled
by the frame unit. frame_invariance() examines that
restriction by calibrating each frame separately. Each compared set and
frame needs at least 50 distinct persons contributing an informative
item pair: not both zero or both at their item maxima after any item
removal or category recoding in that frame’s separate calibration. Items
with weak separate-frame standard errors are excluded from the
comparison.
For frame \(f\), the common-scale item location is \(\hat\delta_{if}^{*}=\hat\delta_{if}/\hat\rho_f\). The separate calibrations have independent origins, so differences are centred over the common thresholds before testing. Let \(w_i=m_i/\sum_jm_j\), where \(m_i\) is the number of thresholds for item \(i\). The default covariance conditions on the fitted frame units:
\[ C\{V_1/\hat\rho_1^2+V_2/\hat\rho_2^2\}C^{\mathsf T}, \qquad C=I-\mathbf 1\mathbf w^{\mathsf T} . \]
inv <- frame_invariance(fit, se_method = "conditional")
inv$summary
inv$locations
inv$discriminationA whole-person bootstrap within person group retains each sampled row’s item-set response pattern and includes uncertainty in the fitted units. A replicate is used only if it retains the same item-comparison family, and therefore the same centred frame origin, as the observed analysis:
The conditional method returns raw and Holm-adjusted probabilities for the location comparisons. Its standardised-infit and slope-ratio columns are descriptive. The bootstrap returns discrimination probabilities as well and applies Holm adjustment to the combined location and discrimination family.
In 300 null simulations with 500 persons per frame, the bootstrap SE ratios were 1.00 for locations and 1.03 for log discrimination ratios; 95 per cent coverage was 0.948 and 0.954, and combined Holm familywise error was 3.0 per cent. A separate 2,000-replicate check found that probabilities based only on the conditional standardised-infit comparison gave 7.1 per cent combined familywise error. The conditional method therefore leaves those probabilities undefined. At the same sample size, bootstrap power was 96.3 per cent for two items shifted by one logit, but 9.6 per cent for two items made 1.5 times as steep. Discrimination departures require appreciably more information.
The location contrasts are relative to the mean difference of the common items. If a small number of items is shifted, centring also gives the remaining items non-zero relative contrasts. The table identifies the pattern of departures; item content or external anchors are needed to decide which items provide the defensible reference.
resolve_frames() gives an item a separate location in
each frame. The item continues to contribute to person measurement but
no longer links those frames. drop_items() removes it. Both
functions refit the model and update the frame, item and person
estimates. Each set must retain common items linking its groups’
origins; a unit link through another set is not sufficient.
The remaining common items and linked sets must still identify the frame units. The functions preserve the original crossed factors, uncertainty method and fitting controls, and return a result only when the revised model converges.
Follmann, D. (1988). Consistent estimation in the Rasch model based on nonparametric margins. Psychometrika, 53, 553–562. https://doi.org/10.1007/BF02294407
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.
Rasch, G. (1961). On general laws and the meaning of measurement in psychology. In Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability (Vol. 4, pp. 321–333). University of California Press.