GUM {GGUM}

Fit the graded unfolding model (GUM)

Description

GUM estimates all item parameters for the GUM.

Usage

GUM(
  data,
  C,
  SE = TRUE,
  precision = 4,
  N.nodes = 30,
  max.outer = 60,
  max.inner = 60,
  tol = 0.001
)

Arguments

data

The N\times I data matrix. The item scores are coded 0, 1, \ldots, C for an item with (C+1) observable response categories.

C

C is the number of observable response categories minus 1 (i.e., the item scores will be in the set \{0, 1, ..., C\}). It should be a scalar since the GUM expects all items to be based on the same number of observable response categories.

SE

Logical value: Estimate the standard errors of the item parameter estimates? Default is TRUE.

precision

Number of decimal places of the results (default = 4).

N.nodes

Number of nodes for numerical integration (default = 30).

max.outer

Maximum number of outer iterations (default = 60).

max.inner

Maximum number of inner iterations (default = 60).

tol

Convergence tolerance (default = .001).

Value

The function returns a list (an object of class GGUM) with 12 elements:

data

Data matrix.

C

Vector C.

alpha

In case of the GUM this is simply a vector of 1s.

delta

The estimated difficulty parameters.

taus

The estimated threshold parameters.

SE

The standard errors of the item parameters estimates.

rows.rm

Indices of rows removed from the data before fitting the model, due to complete disagreement.

N.nodes

Number of nodes for numerical integration.

tol.conv

Loss function value at convergence (it is smaller than tol upon convergence).

iter.inner

Number of inner iterations (it is equal to 1 upon convergence).

model

Model fitted.

InformationCrit

Loglikelihood, number of model parameters, AIC, BIC, CAIC.

Details

The graded unfolding model (GUM; Roberts & Laughlin, 1996) is a constrained version of the GGUM (Roberts et al., 2000; see GGUM). GUM is constrained in two ways: All discrimination parameters are fixed to unity and the threshold parameters are shared across items. In particular, the last constraint implies that only data with the same response categories across items should be used (i.e., C is constant for all items).

Estimated GUM parameters are used as the second step of fitting the more general GGUM. Since under the GGUM data may include items with different number of response categories, the code to fitting the GUM was internally extended to accommodate for this.

The marginal maximum likelihood algorithm of Roberts et al. (2000) was implemented.

Author(s)

Jorge N. Tendeiro, [e-mail]

References

Roberts JS, Laughlin JE (1996). “A unidimensional item response theory model for unfolding responses from a graded disagree-agree response scale.” Applied Psychological Measurement, 20, 231-255.

Roberts JS, Donoghue JR, Laughlin JE (2000). “A general item response theory model for unfolding unidimensional polytomous responses.” Applied Psychological Measurement, 24, 3-32.

Examples

# Generate data:
gen <- GenData.GGUM(400, 5, 3, "GUM", seed = 139)
# Fit the GUM:
fit <- GUM(gen$data, 3)
# Compare true and estimated item parameters:
cbind(gen$delta, fit$delta)
cbind(c(gen$taus[, 5:7]), c(fit$taus[,  5:7]))