mirt 0.1.3

Multidimensional Item Response Theory for Python

mirt

Multidimensional Item Response Theory for Python

A comprehensive Python implementation of Item Response Theory (IRT) models, inspired by R's mirt package.

Features

• Dichotomous Models: 1PL (Rasch), 2PL, 3PL, 4PL
• Polytomous Models: GRM, GPCM, PCM, NRM
• Estimation: EM algorithm with Gauss-Hermite quadrature
• Scoring: EAP, MAP, ML methods for person ability estimation
• Diagnostics: Item fit, person fit, DIF analysis
• Simulation: Generate response data from IRT models
• Multiple Groups: Full multigroup analysis with invariance testing
• Advanced Models: Multidimensional IRT and Bifactor models

Installation

pip install mirt

For development:

pip install -e ".[dev]"

Quick Start

import mirt
import numpy as np

# Simulate response data
responses = mirt.simdata(model='2PL', n_persons=500, n_items=20, seed=42)

# Fit a 2PL model
result = mirt.fit_mirt(responses, model='2PL')

# View results
print(result.summary())

# Get item parameters
params = result.coef()
print(params)

# Score respondents
scores = mirt.fscores(result, responses, method='EAP')
print(scores.to_dataframe().head())

Supported Models

Dichotomous (Binary) Models

Model	Description	Parameters
1PL/Rasch	One-parameter logistic	difficulty (b)
2PL	Two-parameter logistic	discrimination (a), difficulty (b)
3PL	Three-parameter logistic	a, b, guessing (c)
4PL	Four-parameter logistic	a, b, c, upper asymptote (d)

Polytomous (Ordinal) Models

Model	Description	Use Case
GRM	Graded Response Model	Likert scales
GPCM	Generalized Partial Credit	Partial credit items
PCM	Partial Credit Model	Rasch for polytomous
NRM	Nominal Response Model	Unordered categories

Examples

Fitting Different Models

# 1PL (Rasch) model
result_1pl = mirt.fit_mirt(responses, model='1PL')

# 3PL model with guessing
result_3pl = mirt.fit_mirt(responses, model='3PL')

# Graded Response Model for Likert data
likert_data = mirt.simdata(model='GRM', n_categories=5)
result_grm = mirt.fit_mirt(likert_data, model='GRM', n_categories=5)

Person Scoring

# Different scoring methods
eap_scores = mirt.fscores(result, responses, method='EAP')
map_scores = mirt.fscores(result, responses, method='MAP')
ml_scores = mirt.fscores(result, responses, method='ML')

# Access theta values
print(eap_scores.theta)
print(eap_scores.standard_error)

Item Analysis

# Item fit statistics
item_fit = mirt.itemfit(result, responses)
print(item_fit)

# Person fit statistics
person_fit = mirt.personfit(result, responses)
print(person_fit)

Differential Item Functioning (DIF)

import numpy as np

# Responses with group membership
groups = np.array([0] * 250 + [1] * 250)  # Two groups

# DIF analysis with likelihood ratio test
dif_results = mirt.dif(responses, groups, model='2PL', method='likelihood_ratio')
print(dif_results)

# Other methods: 'wald', 'lord', 'raju'
dif_wald = mirt.dif(responses, groups, method='wald')

Multiple Group Analysis

from mirt.multigroup import fit_multigroup, compare_invariance

# Fit with different invariance levels
result = fit_multigroup(responses, groups, model='2PL', invariance='configural')

# Compare all invariance levels
results = compare_invariance(responses, groups, model='2PL', verbose=True)
# Returns: {'configural': ..., 'metric': ..., 'scalar': ..., 'strict': ...}

Data Simulation

# Simulate with specific parameters
a = np.random.lognormal(0, 0.3, size=20)
b = np.random.normal(0, 1, size=20)

responses = mirt.simdata(
    model='2PL',
    discrimination=a,
    difficulty=b,
    n_persons=1000,
    seed=42
)

API Reference

Main Functions

• fit_mirt(data, model, ...) - Fit an IRT model
• fscores(model, responses, method) - Compute person abilities
• simdata(model, n_persons, n_items, ...) - Simulate response data
• itemfit(result, responses) - Item fit statistics
• personfit(result, responses) - Person fit statistics
• dif(data, groups, model, method) - Differential Item Functioning analysis

Multigroup Functions

• fit_multigroup(data, groups, model, invariance) - Fit multigroup IRT model
• compare_invariance(data, groups, model) - Compare invariance levels

Model Classes

• TwoParameterLogistic - 2PL model class
• ThreeParameterLogistic - 3PL model class
• GradedResponseModel - GRM model class
• MultidimensionalModel - Multidimensional IRT
• BifactorModel - Bifactor model
• etc.

Comparison with R mirt

Feature	R mirt	Python mirt
Dichotomous models	1PL-4PL	1PL-4PL
Polytomous models	GRM, GPCM, PCM, NRM	GRM, GPCM, PCM, NRM
Estimation	EM, MHRM	EM
Multidimensional	Full support	Yes
Bifactor	Yes	Yes
DIF	Yes	Yes (LR, Wald, Lord, Raju)
Multiple groups	Full support	Yes (configural, metric, scalar, strict)
GUI	Shiny app	-

Dependencies

• numpy >= 1.21
• scipy >= 1.7
• pandas >= 1.3

Development

# Install dev dependencies
pip install -e ".[dev]"

# Run tests
pytest

# Type checking
mypy src/mirt

# Formatting
black src tests
ruff check src tests

License

MIT License - see LICENSE

References

• Chalmers, R. P. (2012). mirt: A Multidimensional Item Response Theory Package for the R Environment. Journal of Statistical Software, 48(6), 1-29.
• Bock, R. D., & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm. Psychometrika, 46(4), 443-459.