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,
  adjust_N = NA,
  na_codes = -1,
  key = NULL,
  maxit = 60,
  tol = 1e-08
)

Arguments

data, items, id, factors, n_groups, adjust_N, 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.
formula: One-sided explanatory formula. For example, ~ format + operation + format:operation. The reserved threshold factor permits threshold-specific effects.
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.

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.

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.

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     z       p   p_adj
#>  operation    0.697 0.070 9.996 < 0.001 < 0.001
#>    formatB    0.423 0.071 5.925 < 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