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Use this Skill for educational measurement with IRT: 2PL/3PL calibration, item information function, DIF detection, and test equating (Stocking-Lord).
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| name | irt-psychometrics-edu |
| description | Use this Skill for educational measurement with IRT: 2PL/3PL calibration, item information function, DIF detection, and test equating (Stocking-Lord). |
| tags | ["education","IRT","item-response-theory","DIF","test-equating","psychometrics"] |
| version | 1.0.0 |
| authors | [{"name":"awesome-rosetta-skills contributors","github":"@xjtulyc"}] |
| license | MIT |
| platforms | ["claude-code","codex","gemini-cli","cursor"] |
| dependencies | {"python":["numpy>=1.23","scipy>=1.9","pandas>=1.5","matplotlib>=3.6"]} |
| last_updated | 2026-03-18 |
| status | stable |
TL;DR — Implement 2PL/3PL IRT models, compute item characteristic curves, item information functions, detect differential item functioning (DIF) with Mantel-Haenszel, and understand test equating concepts.
Use this Skill when you need to:
Do NOT use this Skill for attitude/personality measurement (use CFA-based approaches) or when sample size is below ~200 per group.
| Model | Parameters | ICC Formula |
|---|---|---|
| 1PL (Rasch) | b (difficulty) | P(θ) = 1 / (1 + exp(-(θ - b))) |
| 2PL | a, b | P(θ) = 1 / (1 + exp(-a*(θ - b))) |
| 3PL | a, b, c | P(θ) = c + (1-c) / (1 + exp(-a*(θ - b))) |
I(θ) = [P'(θ)]² / [P(θ) * (1 - P(θ))]
For 3PL: I(θ) = a² * (P(θ) - c)² * (1 - P(θ)) / [(1 - c)² * P(θ)]
Test Information Function (TIF) = Σ_i I_i(θ) Standard Error of Measurement = 1 / √TIF(θ)
An item shows DIF if examinees from different groups (e.g., gender, ethnicity) with the same ability have different probabilities of answering correctly.
Mantel-Haenszel statistic classifies DIF as:
conda create -n irt_edu python=3.11 -y
conda activate irt_edu
pip install numpy>=1.23 scipy>=1.9 pandas>=1.5 matplotlib>=3.6
# Optional R packages for validation
# install.packages(c("mirt", "difR"))
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from scipy.optimize import minimize
from scipy.stats import chi2
def icc_3pl(theta: np.ndarray, a: float, b: float, c: float) -> np.ndarray:
"""
Three-parameter logistic (3PL) item characteristic curve.
Args:
theta: Array of ability values.
a: Discrimination parameter (a > 0, typically 0.5–2.5).
b: Difficulty parameter (typically -3 to +3).
c: Guessing parameter (typically 0.0–0.35).
Returns:
Array of P(correct | theta) values in [c, 1].
"""
return c + (1 - c) / (1 + np.exp(-a * (theta - b)))
def icc_2pl(theta: np.ndarray, a: float, b: float) -> np.ndarray:
"""Two-parameter logistic (2PL) ICC — special case of 3PL with c=0."""
return icc_3pl(theta, a, b, c=0.0)
def icc_1pl(theta: np.ndarray, b: float) -> np.ndarray:
"""One-parameter logistic / Rasch ICC — special case with a=1, c=0."""
return icc_3pl(theta, a=1.0, b=b, c=0.0)
def iif_3pl(theta: np.ndarray, a: float, b: float, c: float) -> np.ndarray:
"""
Item information function for the 3PL model.
Args:
theta: Ability values.
a: Discrimination.
b: Difficulty.
c: Guessing.
Returns:
Array of item information values I(θ).
"""
p = icc_3pl(theta, a, b, c)
q = 1 - p
info = (a ** 2) * ((p - c) ** 2) * q / ((1 - c) ** 2 * p)
return info
def tif(theta: np.ndarray, params: list[tuple]) -> np.ndarray:
"""
Test information function — sum of item information functions.
Args:
theta: Ability values.
params: List of (a, b, c) tuples for each item.
Returns:
Array of test information TIF(θ).
"""
total = np.zeros_like(theta)
for a, b, c in params:
total += iif_3pl(theta, a, b, c)
return total
def sem_from_tif(theta: np.ndarray, params: list[tuple]) -> np.ndarray:
"""Standard error of measurement from test information: SEM(θ) = 1/√TIF(θ)."""
test_info = tif(theta, params)
return 1.0 / np.sqrt(np.maximum(test_info, 1e-6))
def estimate_2pl_jml(
response_matrix: np.ndarray,
n_iter: int = 100,
lr_theta: float = 0.1,
lr_params: float = 0.05,
) -> tuple[np.ndarray, np.ndarray, np.ndarray]:
"""
Joint Maximum Likelihood Estimation for 2PL IRT model.
Alternates between updating ability (θ) estimates and item parameters (a, b)
using gradient ascent on the log-likelihood. For production use, prefer
marginal MLE via the R `mirt` package.
Args:
response_matrix: Binary matrix of shape (n_persons, n_items).
1 = correct, 0 = incorrect.
n_iter: Number of EM/JML iterations.
lr_theta: Learning rate for ability updates.
lr_params: Learning rate for item parameter updates.
Returns:
Tuple (theta_hat, a_hat, b_hat) — estimated abilities and item parameters.
"""
n_persons, n_items = response_matrix.shape
theta = np.zeros(n_persons)
a = np.ones(n_items)
b = np.zeros(n_items)
for iteration in range(n_iter):
# E-step equivalent: update theta for fixed item params
for s in range(n_persons):
p = icc_2pl(np.array([theta[s]] * n_items), a, b)
p = np.clip(p, 1e-6, 1 - 1e-6)
y = response_matrix[s]
grad_theta = np.sum(a * (y - p))
theta[s] += lr_theta * grad_theta
# M-step equivalent: update item params for fixed theta
for i in range(n_items):
p = icc_2pl(theta, a[i], b[i])
p = np.clip(p, 1e-6, 1 - 1e-6)
y = response_matrix[:, i]
residuals = y - p
grad_a = np.sum(residuals * (theta - b[i]))
grad_b = np.sum(residuals * (-a[i]))
a[i] += lr_params * grad_a
b[i] += lr_params * grad_b
a[i] = max(a[i], 0.05) # Discrimination must be positive
if (iteration + 1) % 20 == 0:
p_mat = np.array([icc_2pl(theta, a[i], b[i]) for i in range(n_items)]).T
p_mat = np.clip(p_mat, 1e-6, 1 - 1e-6)
ll = np.sum(response_matrix * np.log(p_mat) + (1 - response_matrix) * np.log(1 - p_mat))
print(f" Iteration {iteration + 1:3d}: log-likelihood = {ll:.2f}")
return theta, a, b
def mantel_haenszel_dif(
responses: np.ndarray,
group: np.ndarray,
score_bins: int = 10,
) -> pd.DataFrame:
"""
Detect Differential Item Functioning using the Mantel-Haenszel chi-square test.
Stratifies examinees by total score and computes the MH odds ratio and
ΔMHD statistic for each item.
Args:
responses: Binary response matrix (n_persons × n_items).
group: Binary group membership array (0 = reference, 1 = focal).
score_bins: Number of score strata (default 10).
Returns:
DataFrame with columns: item, MH_OR, MH_chi2, p_value, delta_MHD,
DIF_category (A/B/C).
"""
n_persons, n_items = responses.shape
total_scores = responses.sum(axis=1)
score_levels = pd.cut(total_scores, bins=score_bins, labels=False)
records = []
for item_idx in range(n_items):
y = responses[:, item_idx]
a_tbl_num = 0.0 # Σ (a_j * n_1j / n_j) numerator for OR
a_tbl_den = 0.0 # Σ (b_j * n_0j / n_j) denominator for OR
mh_num = 0.0
mh_den = 0.0
for level in range(score_bins):
mask = score_levels == level
if mask.sum() < 4:
continue
ref_mask = mask & (group == 0)
foc_mask = mask & (group == 1)
n_ref = ref_mask.sum()
n_foc = foc_mask.sum()
n_total = n_ref + n_foc
if n_total == 0:
continue
n_correct_ref = y[ref_mask].sum()
n_correct_foc = y[foc_mask].sum()
n_correct_tot = n_correct_ref + n_correct_foc
n_wrong_tot = n_total - n_correct_tot
if n_correct_tot == 0 or n_wrong_tot == 0:
continue
a_j = n_correct_ref * (n_total - n_ref) / n_total
b_j = (n_ref - n_correct_ref) * n_foc / n_total
a_tbl_num += a_j
a_tbl_den += b_j
mh_or = a_tbl_num / max(a_tbl_den, 1e-9)
# MH chi-square (simplified)
delta_mhd = -2.35 * np.log(mh_or)
# Approximate chi-square from OR
mh_chi2 = (np.log(mh_or)) ** 2 / (1 / max(a_tbl_num, 1e-9) + 1 / max(a_tbl_den, 1e-9))
p_value = 1 - chi2.cdf(mh_chi2, df=1)
adm = abs(delta_mhd)
category = "A" if adm < 1.0 else ("B" if adm < 1.5 else "C")
records.append({
"item": item_idx + 1,
"MH_OR": round(mh_or, 4),
"MH_chi2": round(mh_chi2, 4),
"p_value": round(p_value, 4),
"delta_MHD": round(delta_mhd, 4),
"DIF_category": category,
})
return pd.DataFrame(records)
def plot_icc_and_iif(
item_params: list[tuple],
item_labels: list[str] = None,
output_path: str = "icc_iif.png",
) -> None:
"""
Plot ICC (left) and IIF (right) for multiple items.
Args:
item_params: List of (a, b, c) tuples.
item_labels: Display labels for each item.
output_path: Output PNG path.
"""
theta = np.linspace(-4, 4, 200)
labels = item_labels or [f"Item {i+1}" for i in range(len(item_params))]
fig, axes = plt.subplots(1, 2, figsize=(12, 5))
for (a, b, c), label in zip(item_params, labels):
p = icc_3pl(theta, a, b, c)
inf = iif_3pl(theta, a, b, c)
axes[0].plot(theta, p, label=label)
axes[1].plot(theta, inf, label=label)
axes[0].set_xlabel("θ (Ability)")
axes[0].set_ylabel("P(correct | θ)")
axes[0].set_title("Item Characteristic Curves (3PL)")
axes[0].legend(fontsize=8)
axes[0].grid(alpha=0.3)
axes[1].set_xlabel("θ (Ability)")
axes[1].set_ylabel("Item Information I(θ)")
axes[1].set_title("Item Information Functions (3PL)")
axes[1].legend(fontsize=8)
axes[1].grid(alpha=0.3)
plt.tight_layout()
plt.savefig(output_path, dpi=150)
print(f"ICC and IIF plots saved to {output_path}")
plt.close()
def plot_tif_and_sem(
item_params: list[tuple],
output_path: str = "tif_sem.png",
) -> None:
"""Plot Test Information Function and SEM across θ range."""
theta = np.linspace(-4, 4, 200)
test_info = tif(theta, item_params)
sem_values = sem_from_tif(theta, item_params)
fig, ax1 = plt.subplots(figsize=(9, 4))
ax2 = ax1.twinx()
ax1.plot(theta, test_info, color="#1f77b4", label="TIF")
ax2.plot(theta, sem_values, color="#d62728", linestyle="--", label="SEM")
ax1.set_xlabel("θ (Ability)")
ax1.set_ylabel("Test Information", color="#1f77b4")
ax2.set_ylabel("SEM (1/√TIF)", color="#d62728")
ax1.set_title("Test Information Function and Standard Error of Measurement")
lines1, labels1 = ax1.get_legend_handles_labels()
lines2, labels2 = ax2.get_legend_handles_labels()
ax1.legend(lines1 + lines2, labels1 + labels2)
ax1.grid(alpha=0.3)
plt.tight_layout()
plt.savefig(output_path, dpi=150)
print(f"TIF/SEM plot saved to {output_path}")
plt.close()
| Problem | Likely Cause | Solution |
|---|---|---|
| JML estimates not converging | Learning rate too high or items with extreme difficulty | Reduce lr_params; check item p-values |
| Negative discrimination estimates | Items with reversed scoring or JML instability | Constrain a > 0; check item keys |
| DIF for most items | Groups differ in mean ability (impact, not DIF) | Condition on ability strata correctly |
| IIF peaks at wrong θ | Item difficulty mismatch for target population | Select items with b near target θ range |
| c parameter > 0.5 | Sample too small or item is poorly written | Fix c at 0.25 for 4-option MCQ |
| p_value all NaN in DIF table | Empty strata (all correct or all wrong) | Reduce score_bins to 5–8 |
mirt package: https://cran.r-project.org/package=mirtdifR package: https://cran.r-project.org/package=difRdef example_icc_iif_plot():
"""Plot ICC and IIF for a 5-item bank with varied parameters."""
# (a=discrimination, b=difficulty, c=guessing)
item_params = [
(0.8, -1.5, 0.0), # Easy, low discrimination
(1.2, -0.5, 0.2), # Moderate, guessing
(1.8, 0.0, 0.0), # Average difficulty, high discrimination
(2.0, 0.8, 0.25), # Hard, high discrimination, guessing
(0.6, 1.5, 0.0), # Very hard, low discrimination
]
labels = [f"Item {i+1} (a={p[0]}, b={p[1]}, c={p[2]})" for i, p in enumerate(item_params)]
plot_icc_and_iif(item_params, labels, output_path="icc_iif_5items.png")
plot_tif_and_sem(item_params, output_path="tif_sem_5items.png")
return item_params
if __name__ == "__main__":
example_icc_iif_plot()
def example_mh_dif():
"""Simulate item response data with planted DIF items and detect them."""
rng = np.random.default_rng(42)
n_ref, n_foc = 500, 500
n_items = 20
# Item parameters (same for both groups — no DIF)
a = rng.uniform(0.8, 1.8, n_items)
b = rng.uniform(-1.5, 1.5, n_items)
# Generate responses
theta_ref = rng.normal(0.0, 1.0, n_ref)
theta_foc = rng.normal(-0.3, 1.0, n_foc) # Focal group slightly lower ability
def simulate_responses(theta, a, b, c=0.0):
prob = icc_3pl(theta[:, None], a[None, :], b[None, :], c)
return (rng.uniform(size=prob.shape) < prob).astype(int)
resp_ref = simulate_responses(theta_ref, a, b)
resp_foc = simulate_responses(theta_foc, a, b)
# Plant DIF on item 5: focal group has b shifted by +0.8
b_dif = b.copy()
b_dif[4] += 0.8
resp_foc[:, 4] = simulate_responses(theta_foc, a, b_dif)[:, 4]
responses = np.vstack([resp_ref, resp_foc])
group = np.array([0] * n_ref + [1] * n_foc)
dif_results = mantel_haenszel_dif(responses, group, score_bins=8)
print("DIF Detection Results (item 5 should show DIF):")
print(dif_results.to_string(index=False))
flagged = dif_results[dif_results["DIF_category"] != "A"]
print(f"\nFlagged items (B or C DIF): {list(flagged['item'])}")
return dif_results
if __name__ == "__main__":
example_mh_dif()
def example_tcc_reliability():
"""Compute test characteristic curve and IRT-based reliability."""
rng = np.random.default_rng(7)
n_items = 40
# Simulate a realistic item bank for a 40-item test
a_params = rng.uniform(0.7, 2.0, n_items)
b_params = rng.uniform(-2.0, 2.0, n_items)
c_params = rng.uniform(0.05, 0.25, n_items)
params = list(zip(a_params, b_params, c_params))
theta = np.linspace(-4, 4, 300)
# Test characteristic curve: expected score E[X|θ] = Σ P_i(θ)
tcc = sum(icc_3pl(theta, a, b, c) for a, b, c in params)
# IRT reliability: ρ_XX' = 1 - E[1/TIF(θ)] / Var(θ)
# Approximate with standard normal ability distribution
test_info = tif(theta, params)
expected_error_var = np.trapz(1 / np.maximum(test_info, 0.01) * np.exp(-theta**2/2) / np.sqrt(2*np.pi), theta)
irt_reliability = max(0, 1 - expected_error_var / 1.0)
print(f"Test Characteristic Curve: Expected score range [{tcc.min():.1f}, {tcc.max():.1f}]")
print(f"Marginal reliability (IRT): ρ = {irt_reliability:.4f}")
fig, axes = plt.subplots(1, 2, figsize=(12, 4))
axes[0].plot(theta, tcc, color="#1f77b4")
axes[0].set_xlabel("θ (Ability)")
axes[0].set_ylabel("Expected Score")
axes[0].set_title(f"Test Characteristic Curve (n_items={n_items})")
axes[0].grid(alpha=0.3)
axes[1].plot(theta, test_info, color="#2ca02c")
axes[1].fill_between(theta, 0, test_info, alpha=0.2, color="#2ca02c")
axes[1].set_xlabel("θ (Ability)")
axes[1].set_ylabel("Test Information")
axes[1].set_title(f"TIF — Marginal Reliability = {irt_reliability:.3f}")
axes[1].grid(alpha=0.3)
plt.tight_layout()
plt.savefig("tcc_reliability.png", dpi=150)
print("TCC and reliability plot saved to tcc_reliability.png")
return irt_reliability
if __name__ == "__main__":
example_tcc_reliability()
| Version | Date | Change |
|---|---|---|
| 1.0.0 | 2026-03-18 | Initial release — 2PL/3PL ICC, IIF, JML estimation, MH DIF, TCC reliability |