Extract Item Parameters from a Hurdle Partial Credit Model

Extracts item parameters from a fitted hurdle Rasch partial credit model (family = hurdle_acat()), returning a per-submodel breakdown: hurdle item difficulties (Bernoulli logit on $P(Y > 0)$) and partial credit thresholds. Each submodel's output mirrors the shape of item_parameters so existing plotting and downstream functions can be applied directly to res$hurdle or res$pcm.

Usage

item_parameters_hpcm(
  model,
  item_var = item,
  person_var = id,
  draws = FALSE,
  center = TRUE,
  prob = 0.95
)

Arguments

model

A fitted brmsfit object using the hurdle_acat custom family. The recommended formula is


bf(
  response | thres(gr = item) ~ 1 + (1 |g| id),
  hu ~ 0 + factor(item) + (1 |g| id)
)

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 draws matrix of full posterior draws is included in each submodel's output. Default is FALSE.

center

Logical. If TRUE (the default), item parameters are shifted within each submodel so their mean is zero, matching the convention in item_parameters. Person parameters reported by person_parameters_hpcm use the same shifts.

prob

Numeric in $(0, 1)$. Width of the highest density continuous interval (HDCI) reported in the summary. Default is 0.95.

Value

A list with three elements:

hurdle: A list with the same structure as the output of item_parameters applied to a dichotomous Rasch model: locations, locations_wide, summary, item_information, person_sd, optionally draws_matrix. location is the brms posterior mean of b_hu_factoritem<i>; higher values mean more zeros (a harder hurdle to cross).
pcm: A list with the same structure as item_parameters applied to a PCM: locations (long), locations_wide (t1, t2, ..., location), summary, item_information, threshold_order, person_sd, optionally draws_matrix. location columns are posterior means of b_Intercept[<item>, <k>].
correlation: A tibble with mean, sd, hdci_lower, hdci_upper summarising the posterior of $\rho(\theta_{hurdle}, \theta_{pcm})$. This is the brms correlation cor_id__Intercept__hu_Intercept with its sign flipped to match the "higher = more presence" convention used for the hurdle person trait (see Details).

Details

Hurdle submodel. The hurdle linear predictor is $logit(hu) = \delta_{hurdle, i} + \tilde{r}_v$ with $hu = P(Y = 0)$; higher $\delta_{hurdle, i}$ means more zeros, so this is reported directly as the item "location" (harder = higher value, standard Rasch convention for the Bernoulli on $P(Y > 0)$). Hurdle item information is the Bernoulli variance $p(1-p)$ evaluated at $\theta = \delta_{hurdle, i}$, which is exactly $1/4$.

Partial credit submodel. Thresholds $\tau_{ik}$ are the per-item PCM threshold parameters. Item information uses the standard PCM formula; threshold ordering diagnostics are the same as for item_parameters.

Sign of the trait correlation. The brms model reports cor_id__Intercept__hu_Intercept, which is the correlation between the brms random effects r_id[, Intercept] and r_id[, hu_Intercept]. The latter has the opposite sign of the conventional "susceptibility" person trait $\theta_{hurdle}$ (because higher values of the brms random effect mean more zeros, i.e., lower susceptibility). The correlation reported here is therefore $-\text{cor}_{brms}$, so that positive values mean higher susceptibility goes with higher severity. The marginal SDs are unchanged by the sign flip.

Centering. When center = TRUE, the hurdle item difficulties are shifted by their mean and the PCM thresholds are shifted by the grand mean of all PCM thresholds. The same shifts are applied to the corresponding person traits in person_parameters_hpcm, preserving the underlying likelihood.

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

Bürkner, P.-C. (2021). Bayesian Item Response Modeling in R with brms and Stan. Journal of Statistical Software, 100, 1–54. doi:10.18637/jss.v100.i05

Usage

Arguments

Value

Details

References

See also