Extract Item Parameters from a Bayesian Rasch Model

Extracts item difficulty (threshold) parameters from a fitted Bayesian Rasch model. Returns a simple location table in both long and wide formats, a full summary with posterior SEs and HDCIs, item-level information, threshold ordering diagnostics, and optionally the full posterior draws matrix.

Usage

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

Arguments

model

A fitted brmsfit object. Supported parameterisations:

Polytomous ordinal (PCM)

e.g., family = acat with thres(gr = item), producing item-specific thresholds.

Dichotomous Rasch (random items)

e.g., response ~ 1 + (1 | item) + (1 | id) with family = bernoulli().

Dichotomous 1PL (fixed items)

e.g., response ~ 0 + item + (1 | id) with family = bernoulli().

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 the output. Default is FALSE.

center

Logical. If TRUE (the default), item parameters are shifted so that the grand mean of all threshold locations is zero, matching the frequentist CML convention and the centering used in plot_targeting.

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 the following elements:

locations

A tibble in long format with one row per item (dichotomous) or per item-threshold (polytomous), containing item, threshold (integer, always 1 for dichotomous), and location (posterior mean on the logit scale).

locations_wide

A tibble in wide format with one row per item, sorted by mean location. For polytomous models, threshold columns are named t1, t2, etc., and a location column gives the mean across thresholds. For dichotomous models, only item and location columns are present.

summary

A tibble extending the long locations with: se (posterior SD), hdci_lower and hdci_upper (highest density continuous interval bounds at the level specified by prob), and n_eff (effective sample size for the parameter).

item_information

A tibble with one row per item containing: item, location (mean item location, i.e. mean of thresholds for polytomous items), info_at_location (Fisher information at the item's own location), and max_info (maximum Fisher information across theta). For Rasch dichotomous items, both are 0.25. For polytomous items, information depends on the number and spacing of thresholds.

threshold_order

(Polytomous models only) A tibble with one row per item containing: item, n_thresholds, ordered (logical: are all thresholds in ascending order?), and prob_disordered (posterior probability that at least one pair of adjacent thresholds is disordered, i.e. \(\tau_{k+1} \le \tau_k\)). NULL for dichotomous models.

person_sd

A tibble with one row containing: mean, sd, hdci_lower, hdci_upper — the posterior summary of the person-level standard deviation parameter \(\sigma_\theta\).

draws_matrix

(Only if draws = TRUE) A numeric matrix with rows = thresholds (named "item[threshold]") and columns = posterior draws. For dichotomous models, row names are item labels only.

Details

Dichotomous models with random item effects (response ~ 1 + (1 | item) + (1 | id)) parameterise item difficulty as \(\delta_i = -(b_0 + r_i)\) where \(b_0\) is the global intercept and \(r_i\) is the item random effect. Models with fixed item effects (response ~ 0 + item + (1 | id)) parameterise difficulty as \(\delta_i = -b_i\).

Polytomous (acat/PCM) models with grouped thresholds (thres(gr = item)) directly estimate item-specific threshold parameters. Each row in the long output represents one threshold within one item; each row in the wide output represents one item.

Item information is computed from the posterior mean item parameters using the standard Rasch/PCM information formulae:

Dichotomous

\(I_i(\theta) = P_i(\theta) Q_i(\theta)\) where \(P_i = \text{logistic}(\theta - \delta_i)\).

Polytomous (PCM)

\(I_i(\theta) = \sum_c (c - E_i)^2 P_{ic}(\theta)\) where \(E_i = \sum_c c \cdot P_{ic}\) is the expected score for item \(i\).

Threshold ordering: In the partial credit model, disordered thresholds (\(\tau_{k+1} \le \tau_k\)) indicate that the probability of responding in the intermediate category never exceeds both adjacent categories — the category is empirically "absorbed". This does not necessarily indicate misfit (see Adams et al., 2012), but may suggest response categories should be collapsed. The prob_disordered column reports the posterior probability of at least one disordered pair per item, providing a Bayesian alternative to post-hoc threshold checks.

References

Adams, R. J., Wu, M. L., & Wilson, M. (2012). The Rasch rating model and the disordered threshold controversy. Educational and Psychological Measurement, 72(4), 547–573. doi:10.1177/0013164411432166

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

See also

person_parameters, plot_targeting, plot_ipf, posterior_to_prior.

Examples

# \donttest{
library(brms)
library(dplyr)
library(tidyr)
library(tibble)

# --- Partial Credit Model ---

df_pcm <- eRm::pcmdat2 %>%
  mutate(across(everything(), ~ .x + 1)) %>%
  rownames_to_column("id") %>%
  pivot_longer(!id, names_to = "item", values_to = "response")

fit_pcm <- brm(
  response | thres(gr = item) ~ 1 + (1 | id),
  data   = df_pcm,
  family = acat,
  chains = 4, cores = 2, iter = 1000 # use more iter and cores
)
#> Compiling Stan program...
#> Error in .fun(model_code = .x1): Boost not found; call install.packages('BH')

ip <- item_parameters(fit_pcm)
#> Error: object 'fit_pcm' not found

# Long format: one row per threshold
ip$locations
#> Error: object 'ip' not found

# Wide format: one row per item, easy to scan
ip$locations_wide
#> Error: object 'ip' not found

# Full summary with SE and HDCI
ip$summary
#> Error: object 'ip' not found

# Item information
ip$item_information
#> Error: object 'ip' not found

# Threshold ordering diagnostic
ip$threshold_order
#> Error: object 'ip' not found

# Person SD
ip$person_sd
#> Error: object 'ip' not found

# With full posterior draws
ip_draws <- item_parameters(fit_pcm, draws = TRUE)
#> Error: object 'fit_pcm' not found
dim(ip_draws$draws_matrix)
#> Error: object 'ip_draws' not found

# --- Dichotomous Rasch ---

df_rm <- eRm::raschdat3 %>%
  as.data.frame() %>%
  rownames_to_column("id") %>%
  pivot_longer(!id, names_to = "item", values_to = "response")

fit_rm <- brm(
  response ~ 1 + (1 | item) + (1 | id),
  data   = df_rm,
  family = bernoulli(),
  chains = 4, cores = 2, iter = 1000 # use more iter and cores
)
#> Compiling Stan program...
#> Error in .fun(model_code = .x1): Boost not found; call install.packages('BH')

ip_rm <- item_parameters(fit_rm)
#> Error: object 'fit_rm' not found
# Wide and long are equivalent for dichotomous:
ip_rm$locations
#> Error: object 'ip_rm' not found
ip_rm$locations_wide
#> Error: object 'ip_rm' not found
ip_rm$summary
#> Error: object 'ip_rm' not found
# }