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Extract Person Parameters from a Hurdle Partial Credit Model

Source: R/brms_hpcm.R
person_parameters_hpcm.Rd

Extracts person trait estimates from a fitted hurdle Rasch partial credit model (family = hurdle_acat()), returning a per-submodel breakdown: a presence trait \(\theta_{hurdle}\) (susceptibility) on the Bernoulli gate and a severity trait \(\theta_{pcm}\) on the partial credit submodel. Each submodel's output mirrors the shape of person_parameters so res$hurdle and res$pcm can be passed to existing downstream functions (RMUreliability, plot_targeting, etc.).

Usage

person_parameters_hpcm(
  model,
  item_var = item,
  person_var = id,
  draws = FALSE,
  center = TRUE,
  theta_range = c(-7, 7)
)

Arguments

model

A fitted brmsfit object using the hurdle_acat custom family.

item_var

An unquoted variable name identifying the item grouping variable in the model data. Default is item.

person_var

An unquoted variable name identifying the person grouping variable in the model data. Default is id.

draws

Logical. If TRUE, a matrix of full posterior draws (persons x draws) is included in each submodel's output. Default is FALSE.

center

Logical. If TRUE (the default), item parameters and person parameters are shifted within each submodel so that the mean item parameter is zero. Person traits are shifted by the same constant as the items in their submodel.

theta_range

A numeric vector of length 2 giving the range for the Newton-Raphson WLE search on the hurdle submodel. Default is c(-7, 7).

Value

A list with three elements:

hurdle

A list with the same structure as person_parameters for a dichotomous Rasch model: person_estimates (one row per person; sum_score is the number of items with \(Y > 0\); both EAP and Warm's WLE are provided), score_table, and optionally draws_matrix. The hurdle EAP is the sign-flipped brms random effect on hu, so higher values mean greater presence / susceptibility.

pcm

A list with person_estimates (columns id, sum_score, n_active, eap, eap_se), score_table, and optionally draws_matrix. sum_score is the sum of \(Y\) across items with \(Y_{vi} > 0\) for person \(v\); n_active is the count of such items. WLE columns are omitted intentionally (see Details).

correlation

A tibble summarising the posterior of \(\rho(\theta_{hurdle}, \theta_{pcm})\), identical to the correlation element returned by item_parameters_hpcm.

Details

Hurdle person trait. Defined as \(\theta_{hurdle, v} = -r_{id}(v, \texttt{hu\_Intercept})\), i.e., the brms random effect on hu with its sign flipped. Under this convention, the hurdle submodel reads \(P(Y > 0) = \mathrm{plogis}(\theta_{hurdle, v} - \delta_{hurdle, i})\), so higher \(\theta_{hurdle}\) means greater presence / susceptibility. WLE is computed by treating the gate as a dichotomous Rasch model on \(1[Y > 0]\) with item difficulties \(\delta_{hurdle, i}\) from item_parameters_hpcm.

Partial credit person trait. Defined as \(\theta_{pcm, v} = r_{id}(v, \texttt{Intercept})\). Higher values mean greater severity given presence.

Why no WLE on the partial credit submodel. Conditional on the hurdle, the severity submodel is a PCM, but the set of items contributing to person \(v\)'s severity score depends on \(v\)'s own pattern of \(1[Y_{vi} > 0]\): only items with \(Y_{vi} > 0\) provide PCM information about \(\theta_{pcm, v}\). The sum of partial credit scores is therefore not a sufficient statistic across the sample, and a standard score-table WLE is not well defined. The Bayesian EAP, which integrates over the (correlated) multivariate prior on \((\theta_{hurdle}, \theta_{pcm})\), remains well defined for all persons (including those with all-zero patterns) and is therefore the recommended point estimate; see Magnus and Garnier-Villarreal (2022, p. 272ff) for discussion in the closely-related MZI-GRM.

Centering. When center = TRUE, the same shifts that item_parameters_hpcm applies to the item parameters are applied here to the corresponding person traits. The likelihood is unchanged.

Row ordering. person_estimates and draws_matrix preserve the order of first appearance of each person ID in the model data, matching person_parameters.

References

Magnus, B. E. & Garnier-Villarreal, M. (2022). A multidimensional zero-inflated graded response model for ordinal symptom data. Psychological Methods, 27(2), 261-279. doi:10.1037/met0000395

Warm, T. A. (1989). Weighted likelihood estimation of ability in item response theory. Psychometrika, 54(3), 427–450. doi:10.1007/BF02294627

See also

person_parameters for the single-submodel version, item_parameters_hpcm for the item-side counterpart, hurdle_acat for the custom brms family.