2. Basis Function Likelihood Ratio Test

2.1. Summary

The Basis Function Likelihood Ratio Test is a flexible nonlinear conditional-independence test using finite basis expansions. It supports continuous, categorical, and mixed data by expanding:

  • continuous variables using polynomial/orthogonal basis functions, and

  • discrete variables using indicator basis functions.

It compares a full model (X depends on Y and S) to a reduced model (X depends only on S) using a likelihood ratio statistic.

2.2. When to use

  • Relationships may be nonlinear or smooth but non-Gaussian.

  • Dataset includes mixed variable types.

  • You want a parametric and scalable test for PC, PC-Max, BOSS-FCI, FCIT, or other nonlinear workflows.

2.3. Assumptions

  • Conditional means can be approximated with a finite set of basis functions.

  • Errors have finite variance (typically treated as independent Gaussian for the LRT).

  • Sufficient sample size to fit full and reduced models.

2.4. Test details (conceptual)

To test X ⟂ Y | S:

  1. Expand all variables (continuous + discrete) into basis functions up to the truncation limit.

  2. Fit:

    • full model: X ~ basis(Y, S)

    • reduced model: X ~ basis(S)

  3. Compute the likelihood ratio statistic.

  4. Use a chi-square approximation (difference in number of basis terms) to obtain a p-value.

  5. Compare p-value to alpha.

2.5. Parameters

Parameter (camelCase)

Description

alpha

Significance level for rejecting conditional independence.

truncationLimit

Maximum degree/order for continuous-variable basis functions.

singularityLambda

Ridge parameter for handling nearly singular basis regression matrices (or negative for pseudoinverse).

2.6. Strengths

  • Works with mixed continuous/discrete data without pre-discretization.

  • Captures nonlinear relationships missed by Fisher-Z or discrete tests.

  • More scalable than kernel methods (e.g., KCI) for large N.

2.7. Limitations

  • Requires selecting basis type and truncation level.

  • May underperform if true relationships require richer bases than provided.

2.8. References

  • Ramsey, J., Andrews, B., & Spirtes, P. (2025). Scalable causal discovery from recursive nonlinear data via truncated basis function scores and tests. arXiv:2510.04276.