Simulation-based Cutoff for First-Contrast Eigenvalue

Parametric bootstrap producing an empirical upper percentile for the largest eigenvalue from a PCA of standardized residuals under a correctly fitting Rasch model. For each iteration the function resamples theta values from the data-based estimates, simulates response data, refits the appropriate Rasch model, and extracts the first-contrast eigenvalue. Several upper-tail percentiles of the resulting distribution are returned; the 99th percentile is reported as the suggested cutoff (matching the convention used by RMlocdepQ3Cutoff).

Usage

RMdimResidualPCACutoff(
  data,
  iterations = 250,
  parallel = TRUE,
  n_cores = NULL,
  verbose = FALSE,
  seed = NULL
)

Arguments

data

A data.frame or matrix of item responses (0-based, non-negative integers).

iterations

Integer. Number of simulation iterations. Default 250.

parallel

Logical. Use parallel processing via mirai. Default TRUE.

n_cores

Integer or NULL. Number of parallel workers. When NULL, getOption("mc.cores") is checked first; if neither is set, parallel = TRUE falls back to sequential with a warning.

verbose

Logical. Show a progress bar. Default FALSE.

seed

Integer or NULL. Random seed for reproducibility.

Value

A list with components:

results

data.frame: iteration, eigenvalue.

p95, p99, p995, p999

Empirical percentiles of eigenvalue.

max

The largest simulated eigenvalue.

suggested_cutoff

The 99th percentile (p99) — pass this list back into RMdimResidualPCA via cutoff = , or use suggested_cutoff directly.

suggested_cutoff_percentile

The percentile used for suggested_cutoff, currently always 99.

actual_iterations

Number of successful iterations.

sample_n

Number of complete cases used.

item_names

Character vector of item names.

Details

Rule-of-thumb cutoffs for the first-contrast eigenvalue depend strongly on sample size, test length, and item-parameter spread; simulation-based cutoffs tailored to the data are more defensible (Chou & Wang, 2010).

Per iteration: theta values are sampled with replacement from the WLE/MLE estimates derived from the fitted full-sample model; response data are simulated under the model (psychotools::rrm for dichotomous, partial-credit simulator for polytomous); the model is refitted with eRm::RM() / eRm::PCM(); standardized residuals are extracted via eRm::itemfit()$st.res; prcomp() is run; the first-contrast eigenvalue is recorded.

Iterations that fail (e.g., due to a degenerate simulated dataset where some category isn't represented) are silently dropped. The iarm package is not required.

References

Chou, Y.-T., & Wang, W.-C. (2010). Checking dimensionality in item response models with principal component analysis on standardized residuals. Educational and Psychological Measurement, 70(5), 717-731. doi:10.1177/0013164410379322

See also

RMdimResidualPCA

Examples

# \donttest{
set.seed(1)
dat <- as.data.frame(
  matrix(sample(0:1, 200 * 12, replace = TRUE), nrow = 200, ncol = 12)
)
colnames(dat) <- paste0("I", 1:12)

# Few iterations for a fast example; use 250+ in real analyses
bound <- RMdimResidualPCACutoff(dat, iterations = 50, parallel = FALSE, seed = 1)
bound$suggested_cutoff
#> [1] 1.656817

RMdimResidualPCA(dat, cutoff = bound)
#> 
#> 
#> Table: Rasch model (200 complete cases, 12 items). Total observed variance: 9.7% explained by measures, 90.3% unexplained
#> (basis for PCA; n = 200 non-extreme cases). First-contrast cutoff = 1.657 based on 50 simulation iterations (99th percentile).
#> 
#> |Component | Eigenvalue| Proportion of variance|Flagged |
#> |:---------|----------:|----------------------:|:-------|
#> |PC1       |      1.464|                  0.121|FALSE   |
#> |PC2       |      1.425|                  0.118|FALSE   |
#> |PC3       |      1.281|                  0.106|FALSE   |
#> |PC4       |      1.155|                  0.096|FALSE   |
#> |PC5       |      1.125|                  0.093|FALSE   |
# }