Fit a many-facet Rasch model

Fits an additive many-facet Rasch model (Linacre 1989) to scored responses indexed by person, item, and one or more facets such as rater, task, or occasion. Facet severities, item thresholds, person locations, and fit statistics are reported on a common logit scale.

Usage

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

Arguments

data: Long-format data frame, or a wide data frame when items is supplied.
person: Name of the person identifier column. Person, item, score, facet, and person-factor columns must define distinct roles.
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).
na_codes: Numeric or character score values to read as missing. They are matched before scores are converted to numbers, including numerically equivalent labels (for example, "09" matches a score of 9). The default is -1; any negative score is also treated as missing.
interaction: Optional name of one facet to interact with the items (interactive facet mode). See Details.
factors: Optional person factors for DIF analysis: a character vector naming columns constant within person, or a data frame with one row per data row or unique person. Within each person, observed factor values must agree; missing entries do not override an observed value. Facets belong in facets, not here.
maxit, tol: Newton-Raphson iteration cap and convergence tolerance.

Value

An object of classes "rasch_mfrm" and "rasch". Model-specific components describe the facets, items, thresholds, and facet specification. Interactive fits also contain an omnibus test and the corresponding item-by-facet effects, whose t, df, p, and Holm-adjusted p_adj columns use the finite-person reference described in Details. The component fit_resid averages virtual-item residuals within a margin. Its response-weighted counterpart is fit_resid_pooled; its degrees of freedom are in df_fit. A non-converged fit retains estimates and residual patterns for diagnosis but withholds standard errors and inferential probabilities.

Details

For person $n$, item $i$, and facet levels $f_1,\ldots,f_Q$, the additive model is $$P(X_{ni\mathbf{f}}=x)=\frac{\exp\{x\theta_n- \sum_{k=1}^{x}[\delta_{ik}+\sum_{q=1}^{Q}\rho_{qf_q}]\}} {\sum_{y=0}^{m_i}\exp\{y\theta_n- \sum_{k=1}^{y}[\delta_{ik}+\sum_{q=1}^{Q}\rho_{qf_q}]\}}.$$ Positive facet values therefore denote greater severity. The item thresholds have a common sum-zero origin and the levels of each facet sum to zero. If interaction is requested, an item-by-level term is added with both its item and facet margins constrained to sum to zero.

Estimation represents each observed item-by-facet combination as a virtual item and imposes the additive structure in the pairwise conditional likelihood. The person parameter cancels before calibration. The covariance of the structural parameters is the transformed Godambe sandwich covariance.

Facet levels must be connected through common persons and items. A facet nested within an item or a person-disjoint block can be confounded with the item location. The function checks the structural rank and response graph before fitting the model.

An item-by-facet interaction retains equal discrimination but allows facet differences to vary by item. The omnibus Wald test in interaction_test is the primary test; cell tests are Holm-adjusted follow-ups. Each cell table reports a Wald t statistic and its denominator degrees of freedom, using the least effective item-by-level person support minus one. Interaction probabilities require at least $\max\{30,q+2\}$ persons and effective persons in every observed item-by-level cell, where $q$ is the omnibus degrees of freedom. The interaction covariance must also identify the omnibus contrast and leave positive denominator degrees of freedom. Estimates remain descriptive when these conditions are not met.

References

Andrich, D. and Marais, I. (2019). A Course in Rasch Measurement Theory: Measuring in the Educational, Social and Health Sciences. Springer.

Linacre, J. M. (1989). Many-Facet Rasch Measurement. Chicago: MESA Press.

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 fit_resid_pooled  df_fit
#>     R1   -0.674 0.089 354    1.075     1.056     0.352            0.737 322.500
#>     R2   -0.198 0.077 354    1.056     1.039     0.338            0.548 322.500
#>     R3    0.185 0.078 354    1.001     0.997    -0.007            0.008 322.500
#>     R4    0.688 0.073 354    0.967     0.957    -0.337           -0.509 322.500