# The Extended Frame of Reference model

#### Josh McGrane

    library(rasch)

## Frames and units

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 \(g\),

\[ P(X_{ni}=x)= \frac{\exp\{\rho_{sg}[x\theta_n-\sum_{k=1}^{x}\delta_{ik}]\}} {\sum_{y=0}^{m_i}\exp\{\rho_{sg}[y\theta_n- \sum_{k=1}^{y}\delta_{ik}]\}},\qquad \rho_{sg}=\alpha_s\phi_g . \]

The item-set unit is \(\alpha_s\), and the person-group unit is \(\phi_g\). 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.

## Fit the model

    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")

    fit <- rasch_efrm(
      d,
      item_sets = truth$item_sets,
      groups = "group",
      id = "id",
      boot_reps = 30,
      workers = 1
    )
    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, power of fit: 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        0.049  0.081       8
    #>  set2 1.190        0.049 -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:
    #>               term df   wald       p
    #>  group units (phi)  1  7.442   0.006
    #>  set units (alpha)  1 12.711 < 0.001
    #> Holm-adjusted exploratory unit contrasts (H0: unit = 1):
    #>        parameter estimate    se      z       p p_adj significant
    #>      log phi[g1]   -0.109 0.040 -2.728   0.006 0.013           *
    #>      log phi[g2]    0.109 0.040  2.728   0.006 0.013           *
    #>  log alpha[set1]   -0.174 0.049 -3.565 < 0.001 0.001           *
    #>  log alpha[set2]    0.174 0.049  3.565 < 0.001 0.001           *
    #>
    #> 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

The 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 units
    #>   set alpha se_log_alpha
    #>  set1 0.840        0.049
    #>  set2 1.190        0.049
    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      0.061  0.081    1.031
    #>  set2    g1       150       8 1.190 0.897 1.067      0.065 -0.081    1.048
    #>  set1    g2       150       8 0.840 1.115 0.937      0.065  0.081    1.063
    #>  set2    g2       150       8 1.190 1.115 1.327      0.061 -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 300     0.348         1     0.009 -3522.108
    #>
    #> $alpha_method
    #> [1] "finite-grid semiparametric maximum likelihood"
    #>
    #> $alpha_grid
    #>  lower  upper points
    #>     -8      8     61

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_tests

The factorial tables give the GLS decomposition of the estimated log group units. They do not replace the frame-level estimates.

## Uncertainty and equal-unit comparisons

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.

    fit$efrm_vs_rasch$unit_omnibus
    #>               term df   wald       p
    #>  group units (phi)  1  7.442   0.006
    #>  set units (alpha)  1 12.711 < 0.001
    fit$efrm_vs_rasch$unit_tests
    #>        parameter estimate    se      z       p p_adj significant
    #>      log phi[g1]   -0.109 0.040 -2.728   0.006 0.013           *
    #>      log phi[g2]    0.109 0.040  2.728   0.006 0.013           *
    #>  log alpha[set1]   -0.174 0.049 -3.565 < 0.001 0.001           *
    #>  log alpha[set2]    0.174 0.049  3.565 < 0.001 0.001           *

The omnibus Wald tests assess the set- and group-unit families. Individual unit contrasts are Holm-adjusted follow-ups. 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. Probabilities require at least 50 persons or effective persons in every group and at least 50 common persons on every set-link edge. Sparse designs retain the unit estimates without an inferential probability.

Simulation 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.

## Item invariance across frames

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 frame needs at least 50 persons with two or more responses. 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$discrimination

A person-within-frame bootstrap includes uncertainty in the fitted units:

    inv_boot <- frame_invariance(
      fit,
      se_method = "bootstrap",
      boot_reps = 300,
      seed = 1
    )

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.

## Refit after a departure

`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.

    resolved <- resolve_frames(fit, "S1I02")
    removed <- drop_items(fit, "S1I02")

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.

## References

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.