q3_statistic: Posterior Predictive Q3 Residual Correlations for Bayesian...

In easyRaschBayes: Bayesian Rasch Analysis Using 'brms'

Posterior Predictive Q3 Residual Correlations for Bayesian IRT Models

Description

Computes a Bayesian analogue of Yen's Q3 statistic (Yen, 1984) for detecting local dependence between item pairs in Rasch-family models fitted with brms. For each posterior draw, residual correlations are computed for both observed and replicated data, yielding draw-level Q3 values that can be summarized and visualized via q3_post.

Usage

q3_statistic(model, item_var = item, person_var = id, ndraws_use = NULL)

Arguments

model	A fitted brmsfit object from an ordinal IRT model (e.g., family = acat for a partial credit model) or a dichotomous model (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.
ndraws_use	Optional positive integer. If specified, a random subset of posterior draws of this size is used. If NULL (the default), all draws are used.

Details

The procedure works as follows for each posterior draw s:

1. Compute expected values E^{(s)}_{vi} from the category probabilities returned by posterior_epred. For ordinal models: E^{(s)}_{vi} = \sum_c c \cdot P^{(s)}(X_{vi} = c). For binary models: E^{(s)}_{vi} = P^{(s)}(X_{vi} = 1).

2. Compute observed residuals: d^{(s)}_{vi} = X_{vi} - E^{(s)}_{vi}.

3. Compute replicated residuals: d^{rep(s)}_{vi} = Y^{rep(s)}_{vi} - E^{(s)}_{vi}, where Y^{rep} is drawn via posterior_predict.

4. For each item pair (i, j), compute Q3 as the Pearson correlation of residuals across all persons who responded to both items.

Value

A tibble in long format with one row per draw per item pair, containing:

draw

Integer draw index.

item_pair

Character label of the item pair ("item1 : item2").

item_1

First item in the pair.

item_2

Second item in the pair.

q3

Observed Q3 residual correlation for this draw.

q3_rep

Replicated Q3 residual correlation for this draw.

This long-format output parallels the structure of infit_statistic and can be passed directly to q3_post for summary tables and plots.

References

Yen, W. M. (1984). Effects of local item dependence on the fit and equating performance of the three-parameter logistic model. Applied Psychological Measurement, 8(2), 125–145. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1177/014662168400800201")}

Christensen, K. B., Makransky, G. & Horton, M. (2017). Critical values for Yen's Q3: Identification of local dependence in the Rasch model using residual correlations. Applied Psychological Measurement, 41(3), 178–194. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1177/0146621616677520")}

Bürkner, P.-C. (2021). Bayesian Item Response Modeling in R with brms and Stan. Journal of Statistical Software, 100, 1–54. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.18637/jss.v100.i05")}

See Also

q3_post for postprocessing summaries and plots, infit_statistic for Bayesian infit/outfit, posterior_epred, posterior_predict.

Examples

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  = 1, # use more cores if you have
  iter   = 500 # use at least 2000 
)

# Q3 residual correlations
q3_draws <- q3_statistic(fit_pcm, ndraws_use = 500)

# Postprocess
result <- q3_post(q3_draws)
result$summary
result$hdi
result$plot

# --- Dichotomous Rasch Model ---

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  = 1, # use more cores if you have
  iter   = 500 # use at least 2000 
)

q3_draws <- q3_statistic(fit_rm, ndraws_use = 500)

# Postprocess
result <- q3_post(q3_draws)
result$summary
result$hdi
result$plot

easyRaschBayes documentation built on March 28, 2026, 5:07 p.m.