Fit an explanatory Rasch model

Fits the linear logistic test model (LLTM) for dichotomous responses or the linear partial credit model (LPCM) for polytomous responses. Item or threshold locations are linear functions of observed predictors. The response model remains Rasch and is estimated by pairwise conditional maximum likelihood.

Usage

rasch_explanatory(
  data,
  predictors,
  formula,
  items = NULL,
  level = c("item", "threshold"),
  id = NULL,
  factors = NULL,
  n_groups = NULL,
  na_codes = -1,
  key = NULL,
  maxit = 60,
  tol = 1e-08
)

Arguments

data, items, id, factors, n_groups, na_codes, key, maxit, tol: As in rasch.
predictors: Data frame containing an item column and the predictors named in formula. With level = "threshold", it must also contain threshold, with one row for every fitted item threshold. Column names must be unique.
formula: One-sided explanatory formula. For example, ~ format + operation + format:operation. The reserved threshold factor permits threshold-specific effects. Formula offsets (offset()) are not supported.
level: Whether predictors contains one row per "item" or per "threshold". Item rows are expanded over their thresholds.

Value

An object of class "rasch_explanatory" inheriting from "rasch". Standard item, person, fit and diagnostic components use the explanatory thresholds. The explanatory component contains the formula, metadata and design matrices; reference_fit is the free PCM calibration. est$coefficients reports the estimates, standard errors, $t$ statistics, reference degrees of freedom, raw probabilities and Holm-adjusted probabilities.

Details

For threshold $k$ of item $i$, $$\delta_{ik}=z_{ik}^{T}\gamma.$$ The adjacent-category log odds are $$\log\{P(X_{ni}=k)/P(X_{ni}=k-1)\}=\theta_n-\delta_{ik}.$$ The threshold origin is fixed to the same mean-item-location zero used by rasch. An intercept therefore sets the arbitrary origin and is not separately estimated. Numeric predictors are continuous, unordered factors are categorical, and ordered factors use successive contrasts between adjacent levels. Character predictors are converted to unordered factors. The reserved factor threshold identifies the within-item threshold number; threshold_number supplies its integer value. Design columns are centred and rescaled internally for numerical stability; reported coefficients and standard errors use the supplied predictor units. Coincident coefficient labels receive numeric suffixes; this does not change the predictor design.

A free PCM reference is fitted to the same prepared responses and retained on the object. explanatory_test applies the first-order Kent calibration required for the pairwise composite likelihood. When an identifier occurs on more than one response row, coefficient covariance is clustered by person. A linearised delete-one-person correction accounts for finite-cluster leverage without refitting the model once per person. Supported fits use a $t$ reference with degrees of freedom equal to the number of independent person units contributing conditional information minus one, whether or not identifiers repeat; inference is withheld when the calibration lacks enough independent information. Holm adjustment covers the coefficient family. With few persons and unequal numbers of response rows, these approximate tests can still be mildly liberal; the correction does not guarantee nominal coverage in small samples.

References

Fischer, G. H. (1973). The linear logistic test model as an instrument in educational research. Acta Psychologica, 37, 359–374.

Fischer, G. H. and Ponocny, I. (1994). An extension of the partial credit model with an application to the measurement of change. Psychometrika, 59, 177–192.

Examples

set.seed(1)
q <- data.frame(item = paste0("I", 1:8),
                operation = rep(0:1, each = 4),
                format = rep(c("A", "B"), 4))
difficulty <- -1 + 0.7 * q$operation + 0.4 * (q$format == "B")
X <- matrix(rbinom(500 * 8, 1,
  plogis(outer(rnorm(500), difficulty, "-"))), 500, 8)
colnames(X) <- q$item
fit <- rasch_explanatory(X, predictors = q,
                         formula = ~ operation + format)
fit$est$coefficients
#>       term estimate    se     t  df       p   p_adj
#>  operation    0.697 0.070 9.996 449 < 0.001 < 0.001
#>    formatB    0.423 0.071 5.925 449 < 0.001 < 0.001
explanatory_test(fit)
#>  model parameters free_parameters r_squared r_squared_adj              r2_basis
#>   LLTM          2               7     0.945         0.923 threshold calibration
#>   chisq df p_naive chisq_kent     p p_kent
#>  29.999  5 < 0.001      7.585 0.181  0.181