infit_statistic: Posterior Predictive Infit Statistic for Bayesian IRT Models

In easyRaschBayes: Bayesian Rasch Analysis Using 'brms'

Posterior Predictive Infit Statistic for Bayesian IRT Models

Description

Computes a Bayesian analogue of the conditional item infit statistic (as described in Christensen, Kreiner & Mesbah, 2013) for Rasch-family models fitted with brms. For each posterior draw, expected values and variances are derived from the category probabilities returned by posterior_epred, and variance-weighted standardised residuals are computed for both observed and replicated data. The result can be summarised into posterior predictive p-values to assess item fit.

Usage

infit_statistic(
  model,
  item_var = item,
  person_var = id,
  ndraws_use = NULL,
  outfit = FALSE
)

Arguments

model	A fitted brmsfit object from an ordinal IRT model (e.g., family = acat for a partial credit model or family = bernoulli() for a dichotomous Rasch model).
item_var	An unquoted variable name identifying the item grouping variable in the model data (e.g., item).
person_var	An unquoted variable name identifying the person grouping variable in the model data (e.g., 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.
outfit	Logical. If TRUE, outfit statistics are computed alongside infit. Default is FALSE (infit only), since outfit is highly sensitive to outliers and rarely recommended for Rasch diagnostics.

Details

The procedure adapts the conditional infit/outfit statistics (Christensen et al., 2013; Kreiner & Christensen, 2011; Müller, 2020) to the Bayesian framework:

1. For each posterior draw s, category probabilities P^{(s)}(X_{vi} = c) are obtained from posterior_epred.

2. The conditional expected value and variance for each observation are computed as:

   E^{(s)}_{vi} = \sum_c c \cdot P^{(s)}(X_{vi} = c)

   Var^{(s)}_{vi} = \sum_c (c - E^{(s)}_{vi})^2 \cdot P^{(s)}(X_{vi} = c)

3. Standardised squared residuals are:

   Z^{2(s)}_{vi} = (X_{vi} - E^{(s)}_{vi})^2 / Var^{(s)}_{vi}

4. The infit statistic for item i is the variance-weighted mean of Z^2 across persons:

   Infit_i^{(s)} = \frac{\sum_v Var_{vi}^{(s)} Z^{2(s)}_{vi}} {\sum_v Var_{vi}^{(s)}}

5. If requested, the outfit is the unweighted mean of Z^2.

Value

A tibble with the following columns:

item

The item identifier.

draw

Integer index of the posterior draw.

infit

The observed infit statistic for that item and draw.

infit_rep

The replicated infit statistic (based on posterior predicted data) for that item and draw.

outfit

(Only if outfit = TRUE) The observed outfit statistic for that item and draw.

outfit_rep

(Only if outfit = TRUE) The replicated outfit statistic for that item and draw.

The output is grouped by the item variable. Posterior predictive p-values can be obtained by computing, e.g., mean(infit_rep > infit) within each item.

References

Christensen, K. B., Kreiner, S. & Mesbah, M. (Eds.) (2013). Rasch Models in Health. Iste and Wiley, pp. 86–90.

Kreiner, S. & Christensen, K. B. (2011). Exact evaluation of Bias in Rasch model residuals. Advances in Mathematics Research, 12, 19–40.

Müller, M. (2020). Item fit statistics for Rasch analysis: can we trust them? Journal of Statistical Distributions and Applications, 7(1). \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1186/s40488-020-00108-7")}

See Also

fit_statistic_pcm for a general-purpose posterior predictive fit statistic with user-supplied criterion functions, fit_statistic_rm for a general-purpose posterior predictive fit statistic with user-supplied criterion functions, posterior_epred, posterior_predict.

Examples

library(brms)
library(dplyr)
library(tidyr)
library(tibble)

# --- Partial Credit Model (polytomous) ---

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 
)

# Compute infit per item
item_infit <- infit_statistic(
  model      = fit_pcm,
  ndraws_use = 100 # use at least 500
)

# Post-process draws
infit_results <- infit_post(item_infit)
infit_results$summary
infit_results$hdi
infit_results$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 
)

item_infit_rm <- infit_statistic(
  model      = fit_rm,
  ndraws_use = 100 # use at least 500
)

# Post-process draws
infit_results <- infit_post(item_infit_rm)
infit_results$summary
infit_results$hdi
infit_results$plot

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