Fit a many-facet Rasch model — rasch

Estimates the many-facet Rasch model (Linacre 1989) for long-format data in which each row is one scored response carrying a person, an item, a score, and one or more facet levels (for example the rater). Every item-by-facet combination becomes a virtual item whose thresholds are the item's thresholds shifted by the facet severities, and the whole structure is estimated in one pass of the pairwise conditional likelihood, in which the person parameter cancels. Facet severities are reported with standard errors and pooled fit statistics; the returned object is also a full rasch fit at the virtual-item level, so every diagnostic table and plot in the package applies to it.

Usage

rasch_mfrm(
  data,
  person,
  item = NULL,
  score = NULL,
  facets,
  items = NULL,
  n_groups = NULL,
  adjust_N = NA,
  na_codes = -1,
  interaction = NULL,
  factors = NULL,
  maxit = 60,
  tol = 1e-08
)

Arguments

data: Long-format data frame.
person: Name of the person identifier column.
item: Name of the item column.
score: Name of the integer score column (categories from 0; gaps are collapsed per item with a note).
facets: Character vector naming one or more facet columns (for example a rater column).
items: Optional character vector of item score columns for data in wide format: one row per person-by-facet combination (for example one row per script per rater) with one column per item or criterion. The long form (item + score) remains available for data where the facet varies within items.
n_groups: Number of class intervals for the item-trait chi-square; NULL (the default) applies the class-interval rule of Andrich and Marais (2019, ch. 15) (at least 50 non-extreme persons per interval, at most 10 intervals, at least 2).
adjust_N: Optional reference sample size for the chi-square.
na_codes: Score values to read as missing (default -1); any negative score is also treated as missing.
interaction: Optional name of one facet to interact with the items (interactive facet mode). Adds item-by-facet terms gamma[item, level] with double sum-to-zero constraints on top of the additive severities, so each level may be more or less severe on particular items; estimates are returned in interaction_effects. The interactive model remains in the Rasch class (all discriminations equal one and the parameters are additive), but a significant interaction qualifies specific objectivity in practice: comparisons of the interacting facet's levels become item-dependent, which is itself the substantive finding.
factors: Optional person factors for DIF analysis: a character vector naming columns of data that are constant within person, or a data frame with one row per data row or per unique person. They are carried into the fit so dif_anova works on an MFRM fit directly. Facets are not person factors: facet DIF is an item-by-facet interaction (interaction=).
maxit, tol: Newton-Raphson iteration cap and convergence tolerance.

Value

An object of classes "rasch_mfrm" and "rasch". In addition to every component of a rasch fit (computed over the virtual items), it carries facet_effects (per facet: level, severity, standard error, observation count, pooled fit), item_effects (underlying item locations and pooled fit), item_thresholds (the structural delta_ik with standard errors), and facet_spec. Two fit residuals are reported per facet level and per underlying item. fit_resid is the facet-margin statistic of the published three-facet fit tables (Andrich and Marais 2019, ch. 26 and app. C), the mean of the constituent virtual items' fit residuals; it weighs each virtual item equally, so an erratic level shows the average of its per-item misfit. fit_resid_pooled is the log-of-mean-square statistic summed over the margin's observed cells of non-extreme persons, with its degrees of freedom in df_fit; it weighs each response equally and is the more powerful statistic when misfit is spread evenly over the level's cells.

Examples

set.seed(1)
simP <- function(th, tau) { x <- 0:length(tau); p <- exp(x * th - c(0, cumsum(tau))); p / sum(p) }
persons <- sprintf("P%03d", 1:120); raters <- paste0("R", 1:4)
th <- setNames(rnorm(120, 0, 1.3), persons)
rho <- setNames(c(-0.6, -0.2, 0.2, 0.6), raters)
tau <- list(A = c(-1, 1), B = c(-0.5, 1.2), C = c(-1.2, 0.4))
d <- expand.grid(person = persons, item = names(tau), rater = raters,
                 stringsAsFactors = FALSE)
d$score <- mapply(function(p, i, r)
  sample(0:2, 1, prob = simP(th[p], tau[[i]] + rho[r])), d$person, d$item, d$rater)
fit <- rasch_mfrm(d, person = "person", item = "item", score = "score",
                  facets = "rater")
fit$facet_effects$rater
#>    level   severity         se   n  infit_ms outfit_ms    fit_resid
#> R1    R1 -0.6741794 0.08866656 354 1.0748808 1.0555789  0.352198069
#> R2    R2 -0.1984195 0.07726631 354 1.0559605 1.0389336  0.337676916
#> R3    R3  0.1850839 0.07784122 354 1.0009494 0.9972743 -0.007416974
#> R4    R4  0.6875150 0.07293532 354 0.9666351 0.9568306 -0.336987303
#>    fit_resid_pooled df_fit
#> R1      0.736766065  322.5
#> R2      0.548246161  322.5
#> R3      0.008326603  322.5
#> R4     -0.508911000  322.5