Choosing Tests & Scores

Selecting the right conditional-independence test or score is one of the most important modeling decisions in Tetrad. Many search algorithms (PC, FGES, BOSS, GFCI, etc.) allow you to choose a test/score tailored to your data type and modeling assumptions.

This page gives practical guidance on which tests and scores to use for:

  • Continuous (Gaussian / nearly Gaussian)

  • Discrete

  • Mixed continuous/discrete

  • Non-Gaussian linear models

  • Nonlinear models (KCI, basis expansions)

  • Latent-variable block-based searches

The goal is not to list every option, but to highlight the ones that work well in practice in real causal-discovery workflows.

1. Continuous, Approximately Gaussian Data

When variables are continuous, roughly symmetric, and well-modelled by linear relationships:

Best-Fit Algorithms

  • PC, PC-Max, CPC

  • FGES (high-dimensional)

  • BOSS (small–medium dimensional; often gives better precision than FGES)

  • GRaSP

  • GFCI / RFCI

Rule of thumb:
If nothing fancy is required, use Fisher Z and Sem BIC.

2. Discrete Data (Binary / Ordinal / Categorical)

Discrete CI testing behaves very differently from continuous modeling. Tetrad offers fast standard tests plus Bayesian alternatives.

Recommended Tests

  • G² / Chi-Square Test
    Best for moderate–large samples.

  • “BDeu-style” Bayesian tests
    (Logically aligned with the BDeu Score, especially in small-sample / sparse-table settings.)

Recommended Scores

  • Discrete BDeu Score
    Works well for small–medium models.
    Note: For large numbers of discrete variables, CPTs explode.

Best-Fit Algorithms

  • BOSS (preferred for discrete; handles small–medium models extremely well)

  • FGES (works, but CPT blowup can be prohibitive; use only when p is modest)

  • PC / GFCI / RFCI with G² or BDeu-type tests

Rule of thumb:
Finite-state models are rarely high-dimensional; for score-based search on discrete data:
BOSS > FGES.

3. Mixed Continuous/Discrete Data

Mixed data requires special handling. In Tetrad you have three practically useful options:

A. Conditional Gaussian (CG)

Works when continuous variables are approximately Gaussian within levels of discrete parents.
Statistically principled but slower.

B. Degenerate Gaussian (DGC)

Treats discrete variables as “degenerate Gaussians.”
Much faster than full CG for large data.

C. Basis Function (BF) Tests/Scores

This approach expands continuous variables using orthogonal polynomials (Legendre/Chebyshev):

  • works for mixed data

  • handles nonlinear relationships

  • extremely fast compared to kernel methods

  • sample-size scaling ~ constant

4. Non-Gaussian Linear Models

If relationships are linear but noise terms are non-Gaussian (LiNGAM-type settings or visibly skewed residuals):

Recommended Tests

For LiNGAM-style algorithms (DirectLiNGAM, ICA-LiNGAM, etc.), independence is handled internally via ICA or related objective functions; there is no separate CI test to choose in the GUI.

If you still run “ordinary” DAG search (PC, FGES, BOSS, GRaSP) on linear non-Gaussian data:

  • It is usually fine to keep using Fisher Z as the CI test.
    The structure is still identifiable from second-order moments under many conditions, even if the noise is non-Gaussian.

Recommended Scores

  • Sem BIC Score (heuristic but empirically strong)
    Although derived under a Gaussian assumption, Sem BIC often works very well in linear non-Gaussian settings. In particular, the BOSS paper

    Andrews, B., Ramsey, J., Sanchez-Romero, R., Camchong, J., & Kummerfeld, E. (2023).
    Fast scalable and accurate discovery of DAGs using the best order score search and grow shrink trees.
    NeurIPS 36, 63945–63956.

    shows that BOSS + Sem BIC performs comparably to DirectLiNGAM in linear non-Gaussian simulations (see Figure 4a).

  • PoissonPriorScore (optional structural prior)
    This is a structural sparsity prior (Poisson on edges/parents), not a noise model. You can combine it with Sem BIC if you want an explicit probabilistic prior over graph complexity, but it is not specific to linear non-Gaussian noise.

Best-Fit Algorithms

  • DirectLiNGAM, ICA-LiNGAM, ICA-LiNG-D (internal ICA-type objective)

  • FASK / FASK-Vote

  • BOSS or FGES with Sem BIC (heuristic but well supported by experiments)

  • Pairwise-skewness-based orientation methods

Rule of thumb:
If residuals are visibly skewed or heavy-tailed, it is still quite reasonable to use Sem BIC with BOSS/FGES for structure learning, and to compare against dedicated LiNGAM-style methods when possible.

5. Nonlinear Models

Tetrad provides three practically useful nonlinear CI tests.

A. Kernel Conditional Independence Test (KCI)

Captures arbitrary nonlinear dependencies.
Very powerful but computationally expensive for large N.
Best for small–medium datasets.

B. Random Conditional Independence Test (RCIT)

Captures arbitrary nonlinear dependencies.
Faster than KCI.
Useful for larger datasets.

Summary Table (Practical Defaults)

Setting

Test

Score

Algorithms

Continuous linear

Fisher Z

Sem BIC

PC, FGES, BOSS*, GFCI

Discrete

G² or BDeu-style tests

BDeu

BOSS*, FGES, PC

Mixed

Degenerate Gaussian Test

Degenerate Gaussian BIC

PC, FGES, BOSS, GFCI, FCIT

Linear non-Gaussian

Internal ICA criteria or Fisher Z when using PC/BOSS

Sem BIC

DirectLiNGAM, FASK, BOSS (heuristic), FGES

Nonlinear

KCI, RCIT

(none / kernel-based)

PC+KCI, CAM

Nonlinear scalable

Basis Function Test

Basis Function BIC

PC+BF, GFCI+BF

Latent blocks

Blocks-Test-TS

Blocks-BIC

PC, GFCI, FGES, BOSS

* BOSS is recommended over FGES unless the number of variables is very large. GRaSP is a similar algorithm that should be considered as well.

Next Steps