RMitemInfit: Conditional Item Infit MSQ

In easyRasch2: Psychometric Analysis with Rasch Measurement Theory

Conditional Item Infit MSQ

Description

Computes conditional infit mean-square (MSQ) statistics for each item using iarm::out_infit(), enriched with item locations relative to the sample mean person location.

Usage

RMitemInfit(
  data,
  cutoff = NULL,
  p_value = NULL,
  correction = c("fwer", "fdr_bh", "fdr_by", "none"),
  alpha = 0.05,
  output = "kable",
  sort,
  statistic = "infit"
)

Arguments

data	A data.frame or matrix of item responses. Items must be scored starting at 0 (non-negative integers). Missing values (NA) are allowed, but at least one complete case (row with no NA) must be present.
cutoff	Optional. Default NULL (no cutoff applied, behaviour is identical to the current version). Can be: The return value of RMitemInfitCutoff (a list with ⁠$item_cutoffs⁠): the data.frame is extracted automatically and the number of simulation iterations, cutoff_method, and hdci_width are included in the kable caption. The ⁠$item_cutoffs⁠ data.frame from RMitemInfitCutoff directly: must have columns Item, infit_low, and infit_high (or outfit_low and outfit_high when statistic = "outfit"). When provided, adds columns Infit_low, Infit_high, and Flagged to the result. Flagged labels the misfit direction: "overfit" (infit below the range), "underfit" (above the range), or "" (within range, no misfit).
p_value	Logical or NULL. Whether to compute bootstrap p-values from the simulated null distribution and flag on them. NULL (the default) means use them when they are available: TRUE when cutoff is the full RMitemInfitCutoff object, which carries the per-item simulated values in its ⁠$results⁠ element, and FALSE when cutoff is NULL or the summarised ⁠$item_cutoffs⁠ data.frame, neither of which can support a p-value. Explicit TRUE with an insufficient cutoff is an error rather than a silent downgrade. With FALSE, items are flagged against the interval instead, and a one-time message reports the family-wise error rate that implies. The interval describes where a fitting item's statistic is expected to fall. Using it as a decision rule tests all k items at once, so its width sets a family-wise error rate of 1 - width^k, which is 37% for the default 95% interval over nine items. The corrected p-value targets alpha directly and needs far fewer iterations to do it (Johansson, 2026). Up to and including version 1.1.1 the default was FALSE, so Flagged came from the interval. Existing scripts that pass the full cutoff object will now flag on the corrected p-value instead and may report different items.
correction	Character. Multiple-comparison correction applied across items when p_value = TRUE: "fwer" (default) for the Westfall-Young studentised-max step-down (family-wise error rate), "fdr_bh" / "fdr_by" for Benjamini-Hochberg / Benjamini-Yekutieli false discovery rate control, or "none" for uncorrected per-item p-values.
alpha	Numeric in (0, 1). Significance level for the Flagged column when p_value = TRUE (an item is flagged when its corrected p-value is below alpha). Default 0.05.
output	Character string controlling the return value. Either "kable" (default) for a formatted knitr::kable() table, or "dataframe" for the underlying data.frame.
sort	Optional character string. When sort matches statistic ("infit" by default), rows are sorted by the reported MSQ column in descending order before output.
statistic	Character string. Which conditional fit statistic the table reports: "infit" (default) or "outfit". Both come from the same iarm::out_infit() call and the same parametric bootstrap, so the choice only selects which one is displayed and tested. Column names, cutoff bounds, p-values and Flagged all follow it, and when p_value = TRUE the multiplicity correction is applied across items within the selected statistic (the family is the k items, not 2k item-by-statistic combinations). Testing both statistics on the same items and reporting whichever flags is a larger family than either call corrects for.

Details

Infit MSQ is a weighted fit statistic that emphasises deviations near the item location. Values close to 1.0 indicate good fit. Values substantially above 1.0 suggest underfit (unexpected responses), while values substantially below 1.0 suggest overfit (overly predictable responses). The definition of "substantially" depends on several factors such as sample size, and needs to be determined by simulation using RMitemInfitCutoff. There is no general rule-of-thumb value that is correct.

Conditional infit MSQ statistics are computed via iarm::out_infit(), which uses the conditional distribution of the sufficient statistics (Müller, 2020). Only complete cases (rows without any NA) are used in the conditional fit calculation.

Item parameters are estimated by conditional maximum likelihood via psychotools::pcmodel() (a dichotomous item is a 2-category PCM); the conditional infit/outfit MSQ comes from iarm::out_infit() and is invariant to the estimation engine. Per-item average locations are the means of the CML thresholds, and the person-location reference is the mean of the Warm WLE estimates.

Relative item location is defined as the item's average location minus the sample mean person location, providing a measure of item targeting.

The iarm package must be installed (it is in Suggests, not Imports).

Bootstrap p-values. When p_value = TRUE, each item's observed infit (or outfit, per statistic) is compared against its simulated null distribution (from cutoff$results). The per-item statistic is the residual studentised by the bootstrap mean and SD – deliberately the empirical SD rather than the Wilson-Hilferty / ZSTD transform, which is uninformative for conditional MSQ (Müller, 2020). The marginal p-value is the two-sided Monte-Carlo p-value ⁠(1 + #{|t*| >= |t|}) / (B + 1)⁠. For correction = "fwer" the family-wise adjustment uses the Westfall-Young studentised-max step-down, which exploits the bootstrap dependence among items and is more powerful than Bonferroni/Holm (Ferreira, 2024); its validity rests on subset pivotality. "fdr_bh"/"fdr_by" apply Benjamini-Hochberg / Benjamini- Yekutieli instead. These are model-conditional, sample-size-sensitive p-values and are reported alongside the simulated effect-size band, not in place of it. p-values can be no smaller than 1 / (B + 1). Below 400 iterations in RMitemInfitCutoff() the studentised-max (FWER) correction is mildly liberal, because the bootstrap mean and SD used for studentisation are too noisy, and a message says so. From 400 iterations the family-wise error rate is at the nominal level, and more iterations improve reproducibility instead: two analysts using different seeds disagree about at least one item roughly 10 percent of the time at 400 iterations against 4 percent at 2000 (Johansson, 2026). The table caption notes this below 1000 iterations, so use 1000 to 2000 for a final analysis. The marginal p-values are well calibrated even at a few hundred iterations.

Value

• If output = "kable": a knitr_kable object (plain text table via format = "pipe") with columns "Item", "Infit MSQ", and "Relative location", and a caption showing the number of complete cases. When cutoff is provided, columns "Infit low", "Infit high", and "Flagged" are also included, and the caption notes the simulation-based cutoffs.

• If output = "dataframe": a data.frame with columns Item, Infit_MSQ, and Relative_location. When cutoff is provided, columns Infit_low, Infit_high, and Flagged are also included (inserted after Infit_MSQ, before Relative_location). Flagged is a character column with values "overfit", "underfit", or "" (not the previous logical). When p_value = TRUE, columns p_infit (marginal two-sided p-value) and padj_infit (corrected p-value) are added and Flagged reflects items with padj_infit < alpha (direction from Infit_MSQ relative to 1).

With statistic = "outfit" the layout is identical and the column names follow the statistic: Outfit_MSQ, Outfit_low, Outfit_high, p_outfit, padj_outfit.

Multiple comparisons

The marginal p-value controls the error rate of a single comparison: for one item (or item pair) decided on in advance it is the relevant value. But scanning all k comparisons and flagging whichever fall below alpha tests k hypotheses at once, so the chance of at least one false flag inflates to roughly 1 - (1 - \alpha)^k (e.g. about 34% for k = 8 at alpha = 0.05) – even when every marginal p-value is correctly calibrated. The corrected (adjusted) p-value controls this: correction = "fwer" bounds the probability of any false flag (strict, lower power), while "fdr_bh" / "fdr_by" bound the expected proportion of false flags among those raised (a more lenient middle ground). Rule of thumb: use the marginal p-value for a single pre-specified comparison, and a corrected p-value when screening the whole table – the usual workflow.

Flags depend on the other items

Item infit and item-restscore both judge each item against expectations from a Rasch model fitted to all items, so misfitting items change what is expected of the others. A flag in one direction can therefore produce flags in the other. Overfitting items can push fitting items into the underfit range, and underfitting items, for example from a second dimension, can make fitting items look overfit. How large the effect is depends on how strong the misfit is and how many items share it. In simulation with seven polytomous items, two overfitting items led to about 6 percent of the fitting items being flagged as underfit, and one or two mildly multidimensional items to 1 to 2 percent being flagged as overfit. When items are flagged in both directions, consider removing the clearest misfit and testing again before interpreting the rest.

References

Müller, M. (2020). Item fit statistics for Rasch analysis: Can we trust them? Journal of Statistical Distributions and Applications, 7(5). \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1186/s40488-020-00108-7")}

Ferreira, J. A. (2024). Methods of testing a 'small' or 'moderate' number of hypotheses simultaneously. Journal of Statistical Theory and Practice, 19(6). \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/s42519-024-00412-4")}

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

Johansson, M. (2026). Simulation-based cutoffs for conditional item fit in Rasch models: Iterations, multiplicity correction, and decision stability. PsyArXiv. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.31234/osf.io/7pqz4_v2")}

See Also

RMitemInfitCutoff

Examples

if (requireNamespace("iarm", quietly = TRUE)) {
  # Simulate binary item response data (5 items, 40 persons)
  set.seed(42)
  sim_data <- as.data.frame(
    matrix(sample(0:1, 40 * 5, replace = TRUE), nrow = 40, ncol = 5)
  )
  colnames(sim_data) <- paste0("Item", 1:5)

  # Default kable output
  RMitemInfit(sim_data)

  # Sorted by infit MSQ descending
  RMitemInfit(sim_data, sort = "infit")

  # Return as data.frame for further processing
  df <- RMitemInfit(sim_data, output = "dataframe")

  # Simulation-based cutoffs (100 Monte-Carlo iterations)
  if (requireNamespace("ggdist", quietly = TRUE)) {
    cutoff_res <- RMitemInfitCutoff(sim_data, iterations = 100,
                                    parallel = FALSE, seed = 42)
    RMitemInfit(sim_data, cutoff = cutoff_res)
    RMitemInfit(sim_data, cutoff = cutoff_res, output = "dataframe")

    # Bootstrap p-values with family-wise (Westfall-Young) correction
    # (use iterations >= 1000 in real analyses for stable p-values)
    RMitemInfit(sim_data, cutoff = cutoff_res, p_value = TRUE,
                output = "dataframe")

    # The same for conditional outfit MSQ
    RMitemInfit(sim_data, cutoff = cutoff_res, p_value = TRUE,
                statistic = "outfit", output = "dataframe")
  }
}

easyRasch2 documentation built on Oct. 6, 2026, 5:06 p.m.