Bootstrap fit statistics

Refers fit statistics to replicated datasets fitted in the same way as the observed data. For a person-by-item Rasch model, the default generator conditions on each person's observed raw score and missingness pattern. The person parameter then cancels by sufficiency. Item parameters and person locations are re-estimated in every replicate. The original class-interval rule is repeated, including automatic per-item allocation with missing responses. An automatic count is resolved against the observed fit and held across replicates, so every replicate chi-square is scored on the intervals the observed one used. Tied locations remain in one interval. A result stored by an earlier version under theta = "resample", "fixed" or "normal" was built from a null that mixed interval counts, so it is refused and must be recomputed; results from the default generator are unchanged. The generator assumes independent response rows. A fit with repeated person IDs is therefore refused because this bootstrap does not reproduce within-person dependence.

Usage

fit_bootstrap(
  fit,
  B = 200,
  theta = c("conditional", "resample", "fixed", "normal"),
  workers = 4L,
  seed = NULL
)

Arguments

fit: A fitted object from rasch or btl. Extended-frame, many-facet and explanatory person-by-item fits are not supported. Explanatory paired-comparison fits are supported. Fully anchored scoring fits are not supported by this refitting procedure. An older anchored fit must retain its original anchor settings.
B: Positive whole number of bootstrap replicates.
theta: Generator for a person-by-item fit. "conditional" retains each observed raw score and missingness pattern. "resample" resamples estimated person locations, "fixed" reuses each location with the same response row and missingness pattern, and "normal" draws from a normal distribution with error-corrected variance. Fixed generation requires a finite location for each row with an observed response. This argument does not apply to paired comparisons.
workers: Number of parallel bootstrap workers. The default is four, reduced when fewer physical cores are available or the R process has a lower system limit. Per-replicate seeds and random-number generator settings are shared, so results do not depend on the worker count.
seed: Optional non-negative whole-number seed within the integer range. The caller's random stream is restored on exit. This calculation does not support Box–Muller, including when seed = NULL; see rasch_rng.

Value

An object of class rasch_fit_bootstrap. For a person-by-item fit, it contains items, persons, total, replicates, adjustment metadata and replicate counts, including separate non-convergence and other-failure counts. For a paired-comparison fit, the corresponding tables are pairs, objects, judges and total. Runs requesting at least 30 replicates are withheld unless at least 30 and 90 percent of the requested refits are usable.

Details

Item chi-squares use the upper tail. Fit residuals, infit and outfit use equal-tailed probabilities. Holm adjustment is applied separately to each predeclared item-statistic family; an unavailable item remains in the multiplicity count. Under the conditional generator, the same replicates also give person-specific null distributions. Person probabilities are adjusted with a single-step maximum-statistic distribution across persons for each statistic. A maximum-statistic adjustment is withheld for the complete family if any testable member lacks a usable joint null.

The adjustments describe the fitted global null. They do not guarantee familywise error control among otherwise fitting items, persons, objects or judges when another member departs from the model. Each fit statistic is a separate family. The marginal probabilities are retained.

For a btl fit, outcomes are generated on the fitted comparison design and the model is refitted. The result covers the total pairwise chi-square and pair, object and judge fit. Ordered response thresholds, judge allocation, counts, anchors, position effects and explanatory object restrictions are retained. History-dependent effects are generated in sequence. Half-weighted ties and frame-dependent paired-comparison fits are refused because the present generator does not reproduce those models.

Bootstrap probabilities use \((1+r)/(1+B)\). Their one-sided resolution is therefore \(1/(1+B)\) and their equal-tailed resolution is \(2/(1+B)\). For \(L\) items, Holm-adjusted inference at .05 requires at least \(20L\) replicates for the chi-square and \(40L\) for the two-sided statistics. Runs of at least 30 replicates also require at least 30 and 90 percent of the requested refits to be usable. Otherwise inference is withheld because failed refits can select a non-random subset of the bootstrap null.

The approach follows Wolfe (2013), who bootstrapped critical values for Rasch fit statistics by generating from the estimated person and item parameters and averaging replicate quantiles into cut points. This implementation conditions on the observed scores instead — generating at estimated locations carries their estimation error into the null, which his single 1,000-person design could not surface — and returns per-statistic probabilities under a declared multiplicity policy rather than averaged cut points.

The need for a replicated null is not a real-data artefact. Wu and Adams (2013) derive the mean squares' null variance as about \(2/N\) but conclude the standardised statistics have a sample-size-independent null, reading contrary reports as flawed simulation or as genuine misfit surfacing in large samples. Under data generated from the model — neither explanation applies — infit z beyond 1.96 still flags 11.5 percent of correctly fitting items at 250 persons and 68.4 percent at 4,000 with eight items: parameter estimation alone moves the null, and reproducing that estimation in every replicate is what calibrates it.

References

Andrich, D. and Marais, I. (2019) A Course in Rasch Measurement Theory. Springer.

Wolfe, E. W. (2013). A bootstrap approach to evaluating person and item fit to the Rasch model. Journal of Applied Measurement, 14(1), 1–9.

Wu, M. and Adams, R. J. (2013). Properties of Rasch residual fit statistics. Journal of Applied Measurement, 14(4), 339–355.

Molenaar, I. W. and Hoijtink, H. (1996). Person-fit test statistics for the Rasch model. Applied Measurement in Education, 9(1), 87–106.

Westfall, P. H. and Young, S. S. (1993). Resampling-Based Multiple Testing. Wiley.

Examples

set.seed(1)
d <- seq(-1.5, 1.5, length.out = 6)
X <- matrix(rbinom(300 * 6, 1, plogis(outer(rnorm(300), d, "-"))), 300, 6)
colnames(X) <- paste0("I", 1:6)
# an exploratory run, kept small for speed: raw probabilities are usable,
# and the warning says what Holm-adjusted flagging at .05 would need
bs <- suppressWarnings(fit_bootstrap(rasch(X), B = 49, seed = 1))
bs$items[c("item", "chisq", "chisq_p_boot", "fit_resid", "fit_resid_p_boot")]
#>  item chisq chisq_p_boot fit_resid fit_resid_p_boot
#>    I1 2.839        0.920    -0.090            0.800
#>    I2 3.867        0.680     0.679            0.640
#>    I3 5.085        0.660     0.313            0.760
#>    I4 6.088        0.500     1.172            0.520
#>    I5 3.258        0.820     0.023            0.800
#>    I6 7.682        0.160    -0.200            0.920