latentmetrics 0.1.0

Minimalistic latent correlation estimators for discretized data.

Latent Correlation Estimation Package

This package provides minimalistic implementations of latent correlation estimators between pairs of continuous variables when one or both variables are discretized.

The main goal is to offer both value-based and rank-based correlation estimates in one place, with a simple and easy-to-understand implementation.

Note: Binary data is a special case of ordinal data. For clarity, we implemented binary correlations separately as a simpler example.

Supported Estimators

1. Value-Based Correlations

• Tetrachoric
• Polychoric
• Biserial
• Polyserial

These estimators assume a bivariate normal underlying distribution and estimate the correlation parameter (rho) by maximizing the likelihood of the observed discretized data.

2. Rank-Based Correlations

• Suitable for binary, ordinal, and mixed data, and also supports continuous data through Greiner's formula
• Based on the assumption that data arises from an arbitrary monotonic transformation of underlying bivariate Gaussian variables (Gaussian copula model)
• The correlation parameter (rho) is estimated by matching the observed Kendall's tau (rank correlation, invariant to monotonic transformations) to its expected value as a function of rho
• Can also handle continuous data (via Greiner's formula)

Usage

import numpy as np
from latentmetrics import make_corr_fn, VariableType, EstimateMethod

# Example latent data
x_latent = np.random.normal(size=100)
y_latent = 0.5 * x_latent + np.sqrt(1 - 0.5**2) * np.random.normal(size=100)

# Discretize to ordinal/binary
x_obs = np.digitize(x_latent, np.quantile(x_latent, [0.25, 0.5, 0.75]))
y_obs = np.digitize(y_latent, np.quantile(y_latent, [0.5]))

# Create correlation function and compute
corr_fn = make_corr_fn(VariableType.ORDINAL, VariableType.BINARY, method=EstimateMethod.VALUE)
result = corr_fn(x_obs, y_obs)

print("Estimated correlation:", result.estimate)

Literature & References

Value-Based Correlations

• Polychoric correlation
  Olsson, U. (1979). Maximum likelihood estimation of the polychoric correlation coefficient. Psychometrika, 44(4), 443–460.

• Polyserial correlation
  Olsson, U., Drasgow, F., & Dorans, N. J. (1982). The polyserial correlation coefficient. Psychometrika, 47(3), 337–347.

Rank-Based Correlations

• Dey, D., & Zipunnikov, V. (2022). "Semiparametric Gaussian Copula Regression Modeling for Mixed Data Types (SGCRM)." arXiv preprint arXiv:2205.06868.

Copula Background

• Hofert, M., Kojadinovic, I., Mächler, M., & Yan, J. (2018). Elements of copula modeling with R. Springer.

Available Packages

R Packages

• polycor – Polychoric and Polyserial correlations
• latentcor – Efficient implementations of rank-based correlations

Python Packages

• latentcor – Efficient rank-based correlations from the authors of the R package
• semopy – Structural Equation Modeling (SEM) package; includes polychoric and polyserial correlations