ClassicalTestTheory.jl

A high-performance, robust, and modern Julia package for Classical Test Theory (CTT) and Psychometrics.

Inspired by the widely used R packages CTT and psych, ClassicalTestTheory.jl provides an extensive suite of tools for item analysis, reliability and validity estimation, advanced correlations (e.g., polychoric), and exploratory factor analysis (EFA). Designed with real-world data in mind, it safely handles missing values through pairwise deletion and correlation matrix smoothing.

Installation

You can install ClassicalTestTheory.jl from the Julia REPL. Press ] to enter the Pkg REPL mode and run:

pkg> add ClassicalTestTheory

(Note: If the package is not yet registered, you may need to add it via its GitHub URL.)

Key Features

• Item Analysis: Difficulty, point-biserial, biserial correlation, and Cronbach's alpha if deleted.
• Reliability:
  • Cronbach's Alpha
  • Guttman's Lambda (1 through 6)
  • McDonald's Omega (powered by Principal Axis Factoring)
• Validity & Scaling
• Criterion Validity (criterion_validity)
• Scale Correlations (scale_cor)
• Disattenuated Correlation (disattenuated_cor)
• Standard Scores: zscore, tscore
• Advanced Correlations: Fast and pure-Julia implementation of Biserial, Polyserial, Tetrachoric, and Polychoric correlations using 2-step Maximum Likelihood Estimation (MLE).
• Exploratory Factor Analysis (EFA):
  • Principal Axis Factoring (PAF) to avoid Heywood cases.
  • Seamless integration with FactorRotations.jl for orthogonal (e.g., Varimax) and oblique (e.g., Promax) rotations.
  • Parallel Analysis via Monte Carlo simulations to suggest the optimal number of factors.
  • Kaiser-Meyer-Olkin (KMO) test for sampling adequacy.
• Robust Missing Data Handling: Built-in pairwise deletion and automatic correlation matrix smoothing (cor_smooth) to ensure non-positive definite matrices don't crash your analysis.

Quick Start

1. Basic Item Analysis and Reliability

using ClassicalTestTheory

# Prepare a response matrix (Rows = Subjects, Columns = Items)
responses = [
    1 1 0 1;
    1 0 1 1;
    0 0 1 0
]

# 1. Item Analysis
item_res = item_analysis(responses)
println("Difficulty: ", item_res.difficulty)
println("Point-biserial: ", item_res.pbis)

# 2. Reliability (Cronbach's Alpha & McDonald's Omega)
alpha = cronbach_alpha(responses)
omega = mcdonald_omega(responses)
println("Cronbach's Alpha: ", alpha)
println("McDonald's Omega: ", omega)

# 3. Exploratory Factor Analysis (EFA) - Principal Axis Factoring (PAF)
efa_res = efa(responses, 1, rotation="varimax")
println("Factor Loadings:\n", efa_res.loadings)

2. Advanced Correlations (Polychoric)

If you are dealing with Likert scale data, Pearson correlation assumes continuous variables and often underestimates the true relationship. You can easily compute the Polychoric correlation matrix:

# Compute the pairwise polychoric correlation matrix
R_poly = polychoric_matrix(X)

3. Exploratory Factor Analysis (EFA)

Stop guessing the number of factors. Use Parallel Analysis and run a robust EFA:

using FactorRotations

# 1. Determine the number of factors using Parallel Analysis
n_factors = parallel_analysis(X; cor_type=:polychoric)

# 2. Check Sampling Adequacy
overall_kmo, item_kmo = kmo(pairwise_cor(X))
println("Overall KMO: ", overall_kmo)

# 3. Run EFA with Varimax rotation (automatically smooths bad matrices)
efa_res = efa(X, n_factors; cor_type=:polychoric, rotation=Varimax())

# Extract factor loadings and communality
println(efa_res.loadings)
println(efa_res.communality)

Supported Types

Functions in ClassicalTestTheory.jl accept AbstractMatrix inputs, meaning they seamlessly support matrices of type Matrix{Float64}, Matrix{Int}, or arrays containing missing values like Matrix{Union{Missing, Float64}}.

License

This project is licensed under the MIT License.