compute_information: Compute design-weighted precision curves for ordered...

In mfrmr: Estimation and Diagnostics for Many-Facet Measurement Models

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

Description

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.

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).

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.

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. \Sexpr[results=rd]{tools:::Rd_expr_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. \Sexpr[results=rd]{tools:::Rd_expr_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.

Recommended next step

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$tif)
info$tif$Theta[which.max(info$tif$Information)]

mfrmr documentation built on June 13, 2026, 1:07 a.m.