Horns: Horn's parallel analysis in Python

Horns is a Python implementation of Horn's (1965) parallel analysis, the most widely accepted method for determining the number of components or factors to retain in principal component analysis (PCA) or common factor analysis (FA). The functionality of this package is similar to that of the paran package in R.

Background

Parallel analysis involves simulating a large number of random datasets with the same shape as the original dataset but with no underlying correlation structure. We calculate the eigenvalues of the random datasets and the qth quantile of the distribution of each eigenvalue, as well as the eigenvalues of the original dataset. The original eigenvalues are then compared to the quantiles. The number of components/factors to retain is the number of original eigenvalues that are greater than their corresponding quantile until we encounter the first eigenvalue that is not greater than its quantile.

Horn (1965) originally proposed using the median at the selection criterion (i.e., $q=0.5$), but Glorfeld (1995) recommended q = 0.95 (and a large number of simulations) to reduce the chances of retaining too many components or factors. As in paran, the user can choose q and the number of simulations, allowing them to follow Glorfeld's recommendations or not.

There has been some debate about the best way to simulate random data for parallel analysis. Hayton et al.(2004) originally claimed it is necessary to simulate data with the same values of the original data, but later Dinno (2009) demonstrated that parallel analysis is robust to a wide range of distributional forms of the random data, and therefore recommended using the most computationally efficient method available. This may be good advice when one is performing parallel analysis on Pearson correlation or covariance matrices, but I'm not sure it makes sense for other kinds of matrices (i.e., polyserial or polychoric correlations). Therefore, I have included several methods of simulating random data, including shuffling and bootstrapping the original data.

PCA and FA, and therefore parallel analysis, are often performed on Pearson correlation matrices. However, a Pearson correlation matrix is not the correct choice in all cases. For example, Tran et al. (2008) showed that parallel analysis on binary data is more accurate when using polychoric correlation matrices. This package will select the appropriate correlation estimate per pair of variables based on the number of unique values a each variable.

As pointed out by Dinno (2014), some implementations of parallel analysis do not correctly calculate the eigenvalues for FA, which are different from those for PCA. This package uses the correct eigenvalues for both PCA and FA, like the paran package in R.

Horns optionally produces a figure showing the eigenvalues and the quantiles via Matplotlib.

Performance

Since there are apparently no other Python packages that perform parallel analysis, I didn't profile or benchmark my code extensively. However, the package does perform just-in-time (JIT) compilation of many of its functions via Numba, and parallelises where possible, so it should be reasonably fast. Parallel analysis with polychoric correlations does take much longer than with Pearson and/or polyserial correlations because each correlation is found iteratively.

Installation

You can install Horns directly from PyPI using pip:

pip install horns

Quick Start

Here's a quick example to get you started:

import pandas as pd  # <- not required by Horns, but you need to load your data somehow
from horns import parallel_analysis

# load your dataset
data = pd.read_csv("path/to/your/data.csv")

# perform parallel analysis to determine the optimal number of components for PCA
m = parallel_analysis(data)

print(f"Optimal number of components: {m}")

There should be no need to call anything other than parallel_analysis, but you may find some of the ancillary functions useful for other applications.

Contributing

Contributions to Horns are welcome! Submit an issue or pull request if you have any suggestions or would like to contribute.

License

This project is licensed under the MIT License.

Citation

If you use Horns in your research, please consider citing it:

@misc{horns2024,
  title={Horns: Horn's parallel analysis in Python},
  author={Samuel R. Mathias},
  year={2024},
  howpublished={\url{https://github.com/sammosummo/Horns}},
}

Thanks

Thanks for choosing Horns for your factor analysis needs!