Compute design-weighted precision curves for RSM fits

Calculates design-weighted score-variance curves across the latent trait (theta) for a fitted RSM many-facet Rasch 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 a polytomous Rasch model with K+1 categories, 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.

What `tif` and `iif` mean here

In mfrmr, this helper currently supports only RSM fits. 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.

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 to PCM fits; the helper stops for model = "PCM".

When to use this

Use compute_information() when you want a design-weighted precision screen for an RSM 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.

Interpreting output

$tif: design-weighted precision curve data with theta, Information, and SE.
$iif: design-weighted facet-level contribution curves for an RSM fit.
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_audit_report().

Typical workflow

Fit a model with fit_mfrm().
Run compute_information(fit).
Plot with plot_information(info, type = "tif").
If needed, inspect facet contributions with plot_information(info, type = "iif", facet = "Rater").

Examples

toy <- load_mfrmr_data("example_core")
fit <- fit_mfrm(toy, "Person", c("Rater", "Criterion"), "Score",
                 method = "JML", model = "RSM", maxit = 25)
info <- compute_information(fit)
head(info$tif)
#> # A tibble: 6 × 3
#>   Theta Information    SE
#>   <dbl>       <dbl> <dbl>
#> 1 -6           7.60 0.363
#> 2 -5.94        8.07 0.352
#> 3 -5.88        8.56 0.342
#> 4 -5.82        9.09 0.332
#> 5 -5.76        9.64 0.322
#> 6 -5.7        10.2  0.313
info$tif$Theta[which.max(info$tif$Information)]
#> [1] -0.06