Fit and diagnose a Rasch model by pairwise conditional estimation

Runs a complete Rasch analysis: Andrich and Luo pairwise conditional maximum likelihood item estimation (see pcml), Warm weighted likelihood person estimates per missing-data pattern, item and person fit residuals (the log-of-mean-square statistic of Andrich and Marais 2019, ch. 23, with its untransformed natural form and degrees of freedom), infit and outfit, the item-trait interaction chi-square and the class-interval ANOVA item-fit F, the person separation index with and without extremes, Cronbach's alpha, targeting, threshold diagnostics, and the score-to-measure table.

Usage

rasch(
  data,
  model = c("PCM", "RSM"),
  id = NULL,
  factors = NULL,
  items = NULL,
  n_groups = NULL,
  adjust_N = NA,
  anchors = NULL,
  na_codes = -1,
  key = NULL,
  pc_components = NULL,
  maxit = 60,
  tol = 1e-08
)

Arguments

data: Persons-by-items integer score matrix (categories from 0), or a data frame also containing ID and person-factor columns. Missing values are allowed.
model: Either "PCM" (partial credit) or "RSM" (rating scale).
id: Optional name of an ID column in data, or a vector of IDs; carried through to the person estimates.
factors: Optional character vector of person-factor column names in data (for DIF analysis), or a data frame of factors.
items: Optional character vector naming the item columns; by default every column not named in id or factors.
n_groups: Number of class intervals for the item-trait chi-square and ANOVA item fit. The default NULL applies the rule of Andrich and Marais (2019, ch. 15): as many intervals of at least 50 non-extreme persons as the sample allows, at most 10, at least 2. The resolved value is stored in fit$n_groups.
adjust_N: Optional reference sample size; if supplied, item-trait chi-squares are rescaled to this size (a sample-size adjustment for the sensitivity of the chi-square to large samples). The scaling is proportional and global – every item's chi-square is multiplied by adjust_N over the number of classified persons – so an item answered by a subset of persons keeps its proportionally smaller share of the notional sample rather than being inflated to the full adjust_N.
anchors: Optional anchor table for equating: a data frame with columns item, k, and tau fixing nominated thresholds at known values; see pcml. With anchors in place the scale origin comes from the anchors, so person measures are directly comparable across separately analysed datasets.
na_codes: Values to read as missing. Defaults to -1, the conventional missing-response code; any negative score is also treated as missing, since valid category scores start at zero.
key: Optional multiple-choice scoring key, in any of three forms. (1) A named vector or data frame with columns item and key naming each item's correct option: scored 0/1 (case-insensitive after trimming; blanks become missing). (2) Double keying: several correct options separated by "/" (for example "A/C"), all scoring 1. (3) Polytomous option scoring (Andrich and Styles 2011): a data frame with columns item, option, and score assigning an integer score to every credited option (unlisted options score 0), so informative distractors receive partial credit and the item is fitted as polytomous; see distractor_rescore for an evidence-based proposal. Raw responses are retained in fit$mc for distractor_analysis and plot_distractors.
pc_components: NULL (default) estimates every PCM threshold freely. An integer 1 to 4 instead estimates each item's thresholds through the Andrich principal-components reparameterisation (see pcml_pc): 1 = location only, 2 = + spread (the dispersion model of Andrich 1982), 3 = + skewness, 4 = + kurtosis (the full principal-components model; Pedler 1987). Useful when some categories are sparsely populated; the component estimates are returned in fit$est$components. PCM only, and not combinable with anchors.
maxit, tol: Newton-Raphson iteration cap and convergence tolerance of the pairwise conditional estimation.

Value

An object of class "rasch": a list with the item summary (items), thresholds (with standard errors), the person table (person, including ID and factors), the score table, residuals, reliability (psi, psi_noext, the item separation index isi, alpha), targeting, item-trait statistics (item_trait, item_anova), the summary distribution block (summary_stats: location and fit residual mean/SD/skewness/kurtosis, fit-location correlations, and the cell degrees-of-freedom factor), threshold diagnostics, and estimation details (est).

Details

The fit residual follows Andrich and Marais (2019, ch. 23) exactly: standardised residuals are squared and summed over each item's (person's) observed cells among non-extreme persons, compared with the summed cell degrees of freedom (the model-testing degrees of freedom, cells minus estimated parameters, apportioned equally over cells), and symmetrised by the log-of-mean-square transform $f (\ln Y^2 - \ln f)/\sqrt{V[Y^2]}$ with model-based variance $V[Y^2] = \sum (C_4/V^2 - 1)$. Values are approximately N(0,1) under fit; the conventional flagging value is 2.5 (Andrich and Marais 2019, ch. 15). Negative values indicate over-discrimination (Guttman-like responses), positive values under-discrimination.

A calibration note that applies to the whole test-of-fit suite: the item-trait chi-square evaluates unconditional residuals at estimated person measures and refers the sum of correlated item statistics to a chi-square with summed degrees of freedom, following the convention of Andrich and Marais (2019); the class-interval ANOVA F and the Wilson-Hilferty-style transformations are approximations of the same kind. Null simulation with this package shows rejection rates near but not exactly at the nominal level (conservative in the settings examined), and the direction can vary with design. These statistics are therefore approximate, convention-faithful diagnostics for ordering and flagging misfit – not exactly calibrated hypothesis tests; where exact calibration matters, use the simulation tools (sim_replicate) to build parametric-bootstrap reference distributions for the observed design.

Examples

set.seed(1)
d <- seq(-2, 2, length.out = 8)
X <- matrix(rbinom(500 * 8, 1, plogis(outer(rnorm(500), d, "-"))), 500, 8)
colnames(X) <- paste0("I", 1:8)
fit <- rasch(X, model = "PCM")
fit$items
#>   item max   location         se  fit_resid df_fit natural_resid  infit_ms
#> 1   I1   1 -2.1604709 0.14430225  0.1666750  423.5     0.1695416 1.0175589
#> 2   I2   1 -1.3518390 0.11268892 -0.5756998  423.5    -0.5548927 1.0232863
#> 3   I3   1 -0.8944690 0.10569675 -1.4271840  423.5    -1.3314617 0.9991795
#> 4   I4   1 -0.2525755 0.09842604 -0.6843318  423.5    -0.6675975 1.0560293
#> 5   I5   1  0.3908159 0.09808533  0.8675845  423.5     0.8959149 1.1551913
#> 6   I6   1  0.8900794 0.10034984 -0.5631943  423.5    -0.5486780 1.0677814
#> 7   I7   1  1.4679700 0.11257588  0.1864529  423.5     0.1887381 1.0962765
#> 8   I8   1  1.9104891 0.12794128  0.2287863  423.5     0.2332503 0.9877807
#>   outfit_ms       infit_z    outfit_z     chisq df          p     p_adj
#> 1 0.9656714  0.2292642388 -0.10652343  5.367357  6 0.49763112 0.5687213
#> 2 0.8940697  0.3978631415 -0.80938097  8.360771  6 0.21284560 0.5687213
#> 3 0.8499775  0.0008857027 -1.53389144 13.543617  6 0.03517114 0.2813691
#> 4 0.9443024  1.3119157175 -0.70309136  7.579412  6 0.27056336 0.5687213
#> 5 1.0590841  3.5516550010  0.76045287  2.836605  6 0.82905758 0.8290576
#> 6 0.9345140  1.4295369932 -0.66157031  5.525811  6 0.47834263 0.5687213
#> 7 0.9906992  1.6184015308 -0.02736870  5.760869  6 0.45050380 0.5687213
#> 8 0.9880019 -0.1486702923 -0.01628983  6.873008  6 0.33275124 0.5687213
#>      p_bonf   F_anova    p_anova
#> 1 1.0000000 0.5500185 0.77003278
#> 2 1.0000000 1.3872863 0.21785564
#> 3 0.2813691 2.7706872 0.01171615
#> 4 1.0000000 1.4928791 0.17862223
#> 5 1.0000000 0.4990124 0.80919806
#> 6 1.0000000 0.9406784 0.46536016
#> 7 1.0000000 0.7079669 0.64333086
#> 8 1.0000000 0.9560240 0.45459628
fit$psi$PSI
#> [1] 0.4907417