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Compute design-weighted precision curves for ordered many-facet fits

Source: R/api-advanced.R
compute_information.Rd

Calculates design-weighted score-variance curves across the latent trait (theta) for a fitted ordered-category RSM, PCM, or bounded GPCM model. Returns both an overall precision curve ($tif) and per-facet-level contribution curves ($iif) based on the realized observation pattern.

Usage

compute_information(fit, theta_range = c(-6, 6), theta_points = 201L)

Arguments

fit

Output from fit_mfrm().

theta_range

Numeric vector of length 2 giving the range of theta values. Default c(-6, 6).

theta_points

Integer number of points at which to evaluate information. Default 201.

Value

An object of class mfrm_information (named list) with:

  • tif: tibble with columns Theta, Information, SE. The Information column stores the design-weighted precision value.

  • iif: tibble with columns Theta, Facet, Level, Information, and Exposure. Here too, Information stores a design-weighted contribution value retained under that column name for compatibility.

  • theta_range: the evaluated theta range.

Details

For RSM / PCM, the score variance at theta for one observed design cell is: $$I(\theta) = \sum_{k=0}^{K} P_k(\theta) \left(k - E(\theta)\right)^2$$ where \(P_k\) is the category probability and \(E(\theta)\) is the expected score at theta. In mfrmr, these cell-level variances are then aggregated with weights taken from the realized observation counts in fit$prep$data.

The resulting total curve is therefore a design-weighted precision screen rather than a pure textbook test-information function for an abstract fixed item set. The associated standard error summary is still \(SE(\theta) = 1 / \sqrt{I(\theta)}\) for positive information values.

In an ordered Rasch-family model, category discrimination is fixed at 1, so this score-variance representation is the natural conditional information identity rather than a separate approximation. For binary data it reduces to the familiar \(p(\theta)\{1 - p(\theta)\}\) form. For PCM, the package evaluates each observed design cell using the threshold vector associated with that cell's realized step_facet level. For bounded GPCM, the same design-weighted score variance is scaled by the squared discrimination attached to the realized slope_facet level, which is the \(a_j^2 \cdot \mathrm{Var}(T \mid \theta)\) item-information identity that Muraki (1993, Equation 10) derives by applying Samejima's (1974) polytomous information formula to the GPCM kernel of Muraki (1992).

What tif and iif mean here

In mfrmr, this helper supports ordered-category RSM, PCM, and the current bounded GPCM fit. The total curve ($tif) is the sum of design-weighted cell contributions across all non-person facet levels in the fitted model. The facet-level contribution curves ($iif) keep those weighted contributions separated, so you can see which observed rater levels, criteria, or other facet levels are driving precision at different parts of the scale. For PCM, step-facet-specific thresholds are respected when each observed design cell is evaluated. For bounded GPCM, those same cell-level variances are additionally scaled by the squared discrimination associated with the realized slope_facet level.

What this quantity does not justify

  • It is not a textbook many-facet test-information function for an abstract fixed item set.

  • It should not be used as if it were design-free evidence about a form's precision independent of the realized observation pattern.

  • It does not currently extend beyond the ordered-category RSM / PCM / bounded GPCM family implemented by fit_mfrm().

When to use this

Use compute_information() when you want a design-weighted precision screen for an RSM, PCM, or bounded GPCM fit along the latent continuum. In practice:

  • start with the total precision curve for overall targeting across the realized observation pattern

  • inspect facet-level contribution curves when you want to see which raters, criteria, or other facet levels account for more of that design-weighted precision

  • widen theta_range if you expect extreme measures and want to inspect the tails explicitly

Choosing the theta grid

The defaults (theta_range = c(-6, 6), theta_points = 201) work well for routine inspection. Expand the range if person or facet measures extend into the tails, and increase theta_points only when you need a smoother grid for reporting or custom graphics.

References

The ordered-category probability structures come from Andrich's RSM formulation and Masters' PCM. The bounded GPCM information identity \(a_j^2 \cdot \mathrm{Var}(T \mid \theta)\) is derived in Muraki (1993, Equation 10) by applying Samejima's (1974) general polytomous information formula \(I_j(\theta) = \sum_k P_{jk}(\theta) [-\partial^2 \ln P_{jk} / \partial \theta^2]\) to the GPCM probability kernel of Muraki (1992). For the integer scoring function \(T_k = k\) used by mfrmr, this reduces to \(a_j^2 \cdot \mathrm{Var}(K \mid \theta)\). In mfrmr, those formulas are applied to the realized many-facet observation design, so the output should be read as a design-weighted precision summary rather than as a design-free abstract test function.

  • Andrich, D. (1978). A rating formulation for ordered response categories. Psychometrika, 43(4), 561-573.

  • Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47(2), 149-174.

  • Muraki, E. (1992). A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16(2), 159-176. doi:10.1177/014662169201600206 (See Equations 6, 10, and 13 for the probability kernel and the \(\partial P_k / \partial \theta = a_j P_k (k - E[K])\) derivative used by all GPCM helpers in mfrmr.)

  • Muraki, E. (1993). Information functions of the generalized partial credit model. Applied Psychological Measurement, 17(4), 351-363. doi:10.1177/014662169301700403 (Equation 10 derives the item information function for the GPCM, \(I_j(\theta) = D^2 a_j^2 \mathrm{Var}(T \mid \theta)\), by applying Samejima's (1974) polytomous information formula to the GPCM kernel; this is the canonical reference for compute_information() under bounded GPCM.)

  • Samejima, F. (1974). Normal ogive model on the continuous response level in the multidimensional latent space. Psychometrika, 39, 111-121. (Source for the general polytomous information formula that Muraki 1993 specializes to the GPCM.)

Interpreting output

  • $tif: design-weighted precision curve data with theta, Information, and SE.

  • $iif: design-weighted facet-level contribution curves for the fitted non-person facets.

  • Higher information implies more precise measurement at that theta.

  • SE is inversely related to information.

  • Peaks in the total curve show the trait region where the realized calibration is most informative.

  • Facet-level curves help explain which observed facet levels contribute to those peaks; they are not standalone item-information curves and should be read as design contributions.

How to read the main columns

  • Theta: point on the latent continuum where the curve is evaluated.

  • Information: design-weighted precision value at that theta.

  • SE: approximate 1 / sqrt(Information) summary for positive values.

  • Exposure: total realized observation weight contributing to a facet-level curve in $iif.

Compare the precision peak with person/facet locations from a Wright map or related diagnostics. If you need to decide how strongly SE/CI language can be used in reporting, follow with precision_review_report().

Typical workflow

  1. Fit a model with fit_mfrm().

  2. Run compute_information(fit).

  3. Plot with plot_information(info, type = "tif").

  4. If needed, inspect facet contributions with plot_information(info, type = "iif", facet = "Rater").

See also

fit_mfrm(), plot_information()

Examples

toy <- load_mfrmr_data("example_core")
fit <- fit_mfrm(toy, "Person", c("Rater", "Criterion"), "Score",
                 method = "JML", model = "RSM", maxit = 30)
info <- compute_information(fit)
head(info$tif)
#> # A tibble: 6 × 3
#>   Theta Information    SE
#>   <dbl>       <dbl> <dbl>
#> 1 -6           7.61 0.363
#> 2 -5.94        8.07 0.352
#> 3 -5.88        8.56 0.342
#> 4 -5.82        9.09 0.332
#> 5 -5.76        9.64 0.322
#> 6 -5.7        10.2  0.313
info$tif$Theta[which.max(info$tif$Information)]
#> [1] -0.06