GIRTH: Item Response Theory

Girth is a python package for estimating item response theory (IRT) parameters. In addition, synthetic IRT data generation is supported. Below is a list of available functions, for more information visit the GIRTH homepage.

Interested in Bayesian Models? Check out girth_mcmc. It provides markov chain and variational inference estimation methods.

Need general statistical support? Download my other project RyStats which implements commonly used statistical functions. These functions are also implemented in an interactive webapp GoFactr.com without the need to download or install software.

Item Response Theory

Unidimensional Models

Dichotomous Models

1. Rasch Model
   • Joint Maximum Likelihood
   • Conditional Likelihood
   • Marginal Maximum Likelihood
2. One Parameter Logistic Models
   • Joint Maximum Likelihood
   • Marginal Maximum Likelihood
3. Two Parameter Logistic Models
   • Joint Maximum Likelihood
   • Marginal Maximum Likelihood
   • Mixed Expected A Prior / Marginal Maximum Likelihood
4. Three Parameter Logistic Models
   • Marginal Maximum Likelihood (No Optimization and Minimal Support)

Polytomous Models

1. Graded Response Model
   • Joint Maximum Likelihood
   • Marginal Maximum Likelihood
   • Mixed Expected A Prior / Marginal Maximum Likelihood
2. Partial Credit Model
   • Joint Maximum Likelihood
   • Marginal Maximum Likelihood
3. Graded Unfolding Model
   • Marginal Maximum Likelihood

Ablity Estimation

1. Dichotomous
   • Maximum Likelihood Estimation
   • Maximum a Posteriori Estimation
   • Expected a Posteriori Estimation
2. Polytomous
   • Expected a Posteriori Estimation

Multidimensional Models

1. Two Parameter Logistic Models
   • Marginal Maximum Likelihood
2. Graded Response Model
   • Marginal Maximum Likelihood

Ablity Estimation

1. Dichotomous
   • Maximum a Posteriori Estimation
   • Expected a Posteriori Estimation
2. Polytomous
   • Maximum a Posteriori Estimation
   • Expected a Posteriori Estimation

Supported Synthetic Data Generation

Unidimensional

1. Rasch / 1PL Models Dichotomous Models
2. 2 PL Dichotomous Models
3. 3 PL Dichotomous Models
4. Graded Response Model Polytomous
5. Partial Credit Model Polytomous
6. Graded Unfolding Model Polytomous

Multidimensional

1. Two Parameters Logisitic Models Dichotomous
2. Graded Response Models Polytomous

Usage

Standard Estimation

To run girth with unidimensional models.

import numpy as np

from girth.synthetic import create_synthetic_irt_dichotomous
from girth import twopl_mml

# Create Synthetic Data
difficulty = np.linspace(-2.5, 2.5, 10)
discrimination = np.random.rand(10) + 0.5
theta = np.random.randn(500)

syn_data = create_synthetic_irt_dichotomous(difficulty, discrimination, theta)

# Solve for parameters
estimates = twopl_mml(syn_data)

# Unpack estimates
discrimination_estimates = estimates['Discrimination']
difficulty_estimates = estimates['Difficulty']

Missing Data

Missing data is supported with the tag_missing_data function.

from girth import tag_missing_data, twopl_mml

# import data (you supply this function)
my_data = import_data(filename)

# Assume its dichotomous data with True -> 1 and False -> 0
tagged_data = tag_missing_data(my_data, [0, 1])

# Run Estimation
results = twopl_mml(tagged_data)

Multidimensional Estimation

GIRTH supports multidimensional estimation but these estimation methods suffer from the curse of dimensionality, using more than 3 factors takes a considerable amount of time

import numpy as np

from girth.synthetic import create_synthetic_irt_dichotomous
from girth import multidimensional_twopl_mml

# Create Synthetic Data
discrimination = np.random.uniform(-2, 2, (20, 2))
thetas = np.random.randn(2, 1000)
difficulty = np.linspace(-1.5, 1, 20)

syn_data = create_synthetic_irt_dichotomous(difficulty, discrimination, thetas)

# Solve for parameters
estimates = multidimensional_twopl_mml(syn_data, 2, {'quadrature_n': 21})

# Unpack estimates
discrimination_estimates = estimates['Discrimination']
difficulty_estimates = estimates['Difficulty']

Standard Errors

GIRTH does not use typical hessian based optimization routines and, therefore, currently has limited support for standard errors. Confidence Intervals based on bootstrapping are supported but take longer to run. Missing Data is supported in the bootstrap function as well.

The bootstrap does not support the 3PL IRT Model or the GGUM.

from girth import twopl_mml, standard_errors_bootstrap

# import data (you supply this function)
my_data = import_data(filename)

results = standard_errors_bootstrap(my_data, twopl_mml, n_processors=4,
                                    bootstrap_iterations=1000)

print(results['95th CI']['Discrimination'])                                    

Factor Analysis

Factor analysis is another common method for latent variable exploration and estimation. These tools are helpful for understanding dimensionality or finding initial estimates of item parameters.

Factor Analysis Extraction Methods

1. Principal Component Analysis
2. Principal Axis Factor
3. Minimum Rank Factor Analysis
4. Maximum Likelihood Factor Analysis

Example

import girth.factoranalysis as gfa

# Assume you have converted data into correlation matrix
n_factors = 3
results = gfa.maximum_likelihood_factor_analysis(corrleation, n_factors)

print(results)

Polychoric Correlation Estimation

When collected data is ordinal, Pearson's correlation will provide biased estimates of the correlation. Polychoric correlations estimate the correlation given that the data is ordinal and normally distributed.

import girth.synthetic as gsyn
import girth.factoranalysis as gfa
import girth.common as gcm

discrimination = np.random.uniform(-2, 2, (20, 2))
thetas = np.random.randn(2, 1000)
difficulty = np.linspace(-1.5, 1, 20)

syn_data = gsyn.create_synthetic_irt_dichotomous(difficulty, discrimination, thetas)

polychoric_corr = gcm.polychoric_correlation(syn_data, start_val=0, stop_val=1)

results_fa = gfa.maximum_likelihood_factor_analysis(polychoric_corr, 2)

Support

Installation

Via pip

pip install girth --upgrade

From Source

pip install . -t $PYTHONPATH --upgrade

Dependencies

We recommend the anaconda environment which can be installed here

• Python ≥ 3.8
• Numpy
• Scipy

Unittests

pytest with coverage.py module

pytest --cov=girth --cov-report term

Contact

Please contact me with any questions or feature requests. Thank you!

Ryan Sanchez
ryan.sanchez@gofactr.com

License

MIT License

Copyright (c) 2021 Ryan C. Sanchez

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.