36. Pairwise Orientation Methods — FaskPw & RSkew

Type: Non-Gaussian, pairwise orientation algorithms
Output: Directed graph (using a fixed skeleton provided as input)
Implements: Pairwise orientation rules from FASK and LOFS
References:

Sánchez-Romero et al. (2019) — FASK supplement
Hyvärinen & Smith (2013). Pairwise likelihood ratios for estimation of non-Gaussian structural equation models. JMLR 14(1):111–152.

Pairwise orientation methods take a fixed adjacency graph and assign edge directions using non-Gaussianity, typically skewness or likelihood ratios. They avoid conditional independence tests and do not alter the skeleton; instead, they apply a simple, fast, directional scoring rule to each adjacent pair.

This page documents the two most practically useful pairwise methods in Tetrad:

FaskPw — The pairwise left–right rule used in FASK
RSkew — A robust skewness-based pairwise rule from Hyvärinen & Smith (2013)

Both are implemented within LOFS (Lofs.java) and support background knowledge (forbidden/required edges, tiers).

36.1. Overview

Pairwise methods assume a linear, non-Gaussian structural equation model.
For an adjacency X—Y, they evaluate a direction score such as:

Which direction produces more skewed residuals?
Which model has larger likelihood under a non-Gaussian SEM?
Which direction better aligns with conditional or marginal skewness patterns?

Because these methods are pairwise, they scale extremely well and require no CI tests beyond the initial skeleton.

36.2. FaskPw — FASK Pairwise Left–Right Orientation

Type: Pairwise skewness
Origin: Supplementary material of Sánchez-Romero et al. (2019)
Goal: Provide a fast, lightweight version of the FASK orientation step.

FaskPw starts with a given skeleton (usually obtained via FAS, PC-like pruning, or IMaGES) and orients each edge X—Y using the left–right skewness heuristic:

36.3. Key Idea

For an edge X—Y:

Regress each variable on the other:
- Y = aX + e₁
- X = bY + e₂
Compare the skewness of residuals e₁ vs e₂.
The direction with more Gaussian, less skewed residuals is the effect;
the direction with more skewed residuals is the cause.

Formally (but heuristically):

If skew(e₁) < skew(e₂) → X → Y
If skew(e₂) < skew(e₁) → Y → X
If approximately equal → leave undirected

This rule is the pairwise version of the full FASK method used inside the FASK algorithm.

36.4. When to Use

You want FASK-like orientation but:
- You already have a skeleton
- You need a much faster method than full FASK
Non-Gaussianity (especially skewness) is expected
Large graphs where full FASK may be expensive

36.5. Strengths

Extremely fast (purely pairwise)
Captures the main orientation behavior of FASK
Works well on large high-dimensional datasets
Respects prior knowledge (forbidden/required edges)

36.6. Limitations

Uses only pairwise information—no collider/propagation rules
Requires non-Gaussianity (especially skewness)
Can be unstable when skewness is weak or sample size is small

36.7. Parameters in Tetrad

Mainly inherited from LOFS:

Parameter	Description
`score = LEFT_RIGHT`	Pairwise left-right skewness score
`rule = ORIENT_EACH`	Apply orientation independently to each edge
Knowledge constraints	Forbidden/required edges, tiers

36.8. RSkew — Robust Skewness Orientation (Hyvärinen & Smith, 2013)

Type: Pairwise likelihood / skewness method
Origin: Hyvärinen & Smith (2013), JMLR
Implements: One of the LOFS scores based on robust non-Gaussian likelihood ratios

RSkew implements a robust direction rule derived from the pairwise likelihood-ratio family proposed by Hyvärinen & Smith:

Fit linear SEMs X → Y and Y → X
Compute non-Gaussian likelihood approximations (robustified for outliers)
Prefer the direction with the higher likelihood (or lower penalized score)

The resulting rule often performs better than naive skewness comparison when data contain:

Heavy tails
Outliers
Nonlinear distortions that affect skewness estimation

36.9. Key Idea (informal)

For a pair X—Y,

Compute a robust estimate of the non-Gaussian log-likelihood for models
X → Y and Y → X
Choose the direction with the greater log-likelihood
If scores are similar, keep undirected

This score is implemented in LOFS as Score.RSkew.

36.10. When to Use

Skeleton is known and fixed
Strong non-Gaussian signals are present
Data contain outliers or are heavy-tailed
You want Hyvärinen-style likelihood ratio orientation

36.11. Strengths

More robust than plain skewness heuristics
Based on a well-studied likelihood approximation
Often gives better orientations with noisy/non-ideal data
Works edge-by-edge, so scales extremely well

36.12. Limitations

Requires non-Gaussianity to be effective
Purely pairwise—cannot detect colliders or propagate orientations
Sensitive to regression model mis-specification if nonlinearities are strong

36.13. Parameters in Tetrad

Parameter	Description
`score = RSKEW`	Hyvärinen & Smith robust skewness likelihood score
`rule = ORIENT_EACH`	Apply to each edge independently
Knowledge	Forbidden/required edges, tiers

36.14. Prior Knowledge Support

Both FaskPw and RSkew respect Tetrad’s standard Knowledge constraints:

Required edges
Forbidden edges
Tiers / temporal ordering
Partial ordering constraints

Since orientations are performed pairwise after the skeleton is fixed, knowledge constraints apply directly and cleanly.

36.15. Summary

Pairwise orientation methods offer extremely fast, purely non-Gaussian direction estimation on a fixed skeleton:

FaskPw: FASK’s left–right skewness rule; fast, simple, effective on many datasets.
RSkew: Hyvärinen–Smith robust likelihood-ratio orientation; more stable under outliers or heavy tails.

These are the two most useful LOFS-based pairwise options for practical work in Tetrad, and they provide complementary trade-offs in robustness vs simplicity.