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birt

Bayesian Item Response Theory models in R, powered by Stan.

birt (Bayesian IRT) fits Item Response Theory models using full Bayesian estimation via CmdStan. It gives you posterior distributions, credible intervals, and convergence diagnostics for every parameter — not just point estimates. Stan models are pre-compiled during package installation using the instantiate package, so there is no compilation delay at runtime.

Supported Models

Model Function Item Parameters Minimum N
Rasch (1PL) rasch_fit() difficulty ~100
2PL twopl_fit() difficulty + discrimination ~200
3PL threepl_fit() difficulty + discrimination + guessing ~500

Installation 

# 1. Install cmdstanr
install.packages("cmdstanr", repos = c(
  "https://stan-dev.r-universe.dev/",
  getOption("repos")
))

# 2. Install CmdStan (~5 minutes, one-time setup)
cmdstanr::install_cmdstan()

# 3. Install birt (from source to trigger Stan compilation)
remotes::install_github("Ndukaboika/birt")

Quick Example 

library(birt)

# Simulate test data: 300 students, 10 items
sim <- rasch_simulate(J = 300, K = 10, seed = 42)

# Fit a Rasch model
fit <- rasch_fit(sim$data, seed = 123)

# View results
summary(fit)

# Extract parameters
item_params(fit)       # item difficulties with credible intervals
person_params(fit)     # student abilities
delta_param(fit)       # overall mean ability

# Diagnostics
item_fit(fit)          # outfit and infit statistics

# Plots
plot(fit, type = "icc")     # Item Characteristic Curves
plot(fit, type = "wright")  # Wright Map
plot(fit, type = "info")    # Test Information Function

Fitting a 2PL Model 

fit2 <- twopl_fit(sim$data, seed = 123)
summary(fit2)
discrim_params(fit2)     # item discrimination estimates
plot(fit2, type = "icc") # ICCs with varying slopes

Fitting a 3PL Model 

sim_large <- rasch_simulate(J = 500, K = 10, seed = 42)
fit3 <- threepl_fit(sim_large$data, seed = 123)
guessing_params(fit3)    # guessing parameter estimates

Custom Priors

Default priors are weakly informative and work for most tests. Override them when you have domain knowledge.

Class-Level Priors

Set the same prior for all items of a parameter type:

# Wider difficulty prior for all items
fit <- rasch_fit(data,
  prior_delta    = c(0, 2),       # Normal(0, 2) for mean ability
  prior_alpha_sd = 2,             # Normal(0, 2) for ability deviations
  prior_beta     = c(0, 2),       # Normal(0, 2) for all item difficulties
  seed = 123
)

# 4-option MC guessing for all items
fit3 <- threepl_fit(data,
  prior_c = c(5, 15),             # Beta(5, 15), mean = 0.25
  seed = 123
)

Per-Item Priors

Set a different prior for each individual item:

K <- ncol(data)

# Item 3 is known to be hard, item 7 is known to be easy
b_mean <- rep(0, K)
b_mean[3] <- 2.0
b_mean[7] <- -1.5

b_sd <- rep(1.5, K)
b_sd[c(3, 7)] <- 0.5    # tighter prior for items we know about

fit <- rasch_fit(data,
  prior_beta_mean = b_mean,
  prior_beta_sd = b_sd,
  seed = 123
)

# Mixed item formats: items 1-5 are 4-option MC, items 6-10 are 5-option MC
c_alpha <- rep(5, 10)
c_beta  <- c(rep(15, 5), rep(20, 5))   # mean 0.25 vs 0.20

fit3 <- threepl_fit(data,
  prior_c_alpha = c_alpha,
  prior_c_beta = c_beta,
  seed = 123
)

# Check what priors were used
fit$priors

Default Prior Summary

Parameter Default Class-level Per-item
Mean ability (δ) Normal(0, 1) prior_delta = c(0, 1) —
Ability deviation (α) Normal(0, 1.5) prior_alpha_sd = 1.5 —
Item difficulty (β) Normal(0, 1.5) prior_beta = c(0, 1.5) prior_beta_mean, prior_beta_sd
Discrimination (a) LogNormal(0, 0.5) prior_a = c(0, 0.5) prior_a_meanlog, prior_a_sdlog
Guessing (c) Beta(2, 8) prior_c = c(2, 8) prior_c_alpha, prior_c_beta

Real Data 

data(algebra)
dim(algebra)  # 1382 students, 12 items

fit_alg <- rasch_fit(algebra, seed = 123)
summary(fit_alg)
plot(fit_alg, type = "wright")

Learn More

Citation

If you use birt in your research, please cite:

Boika, N. (2026). birt: Bayesian Item Response Theory models in R.
R package. https://github.com/Ndukaboika/birt

References

  • Luo, Y., & Jiao, H. (2018). Using the Stan program for Bayesian item response theory. Educational and Psychological Measurement, 78(3), 384–408.
  • Stan Development Team (2024). Stan User’s Guide, §1.11: Item-Response Theory Models.
  • Gelman, A., Jakulin, A., Pittau, M. G., & Su, Y. S. (2008). A weakly informative default prior distribution for logistic and other regression models. Annals of Applied Statistics, 2(4), 1360–1383.