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Core Concepts

What is RTHOR?

RTHOR (Randomization Test of Hypothesized Order Relations) is a statistical test for evaluating whether correlation matrices conform to theoretically predicted patterns, particularly circumplex and circular structures.

Developed by Hubert & Arabie (1987) 1, RTHOR is widely used in personality, emotion, and interpersonal research to validate circumplex models.

The Circumplex Problem

Many psychological constructs organize in circular patterns (circumplexes):

  • Interpersonal behavior: Dominance-submission and affiliation dimensions
  • Emotions: Valence and arousal dimensions
  • Personality traits: Various two-dimensional models

In a circumplex:

  • Variables are arranged in a circle
  • Adjacent variables have strong positive correlations
  • Opposite variables have weak or negative correlations
  • Intermediate distances have intermediate correlations

Traditional methods like exploratory factor analysis don't directly test whether data follow a hypothesized circular ordering. RTHOR provides a formal statistical test for this.

How RTHOR Works

1. Specify Order Hypotheses

You define predictions about relative correlation magnitudes. For example, in a 6-variable circumplex:

  • Adjacent pairs (1-2, 2-3, ..., 6-1) should have highest correlations
  • Alternate pairs (1-3, 2-4, ...) should have intermediate correlations
  • Opposite pairs (1-4, 2-5, 3-6) should have lowest correlations

This creates a set of order predictions comparing all pairs of variable-pairs.

2. Count Agreements

RTHOR counts how many predictions are satisfied by observed data:

  • Agreement: Predicted order matches observed order
  • Disagreement: Predicted order contradicts observed order
  • Tie: Correlations are equal

3. Compute Correspondence Index (CI)

The CI summarizes fit (Hubert & Arabie, 1987, Eq. 3, p. 176) 1:

\[ CI = \frac{A - D}{A + D + T} \]

Where:

  • A = agreements
  • D = disagreements
  • T = ties

Range: -1 (perfect disagreement) to +1 (perfect agreement)

4. Permutation Test

Statistical significance is determined by randomization (Hubert & Arabie, 1987, p. 175) 1:

  1. Generate all permutations of variable labels (or sample when n! > 50,000)
  2. For each permutation, recompute CI
  3. p-value = proportion of permutations with CI ≥ observed

This tests: "Is the observed fit better than chance?"

The key insight: By permuting object labels (not individual predictions), the test preserves the structural integrity of order relations while providing a valid null distribution.

Interpreting Results

Correspondence Index (CI)

General guidelines:

  • CI > 0.7: Strong support for hypothesis
  • CI = 0.4-0.7: Moderate support
  • CI < 0.4: Weak support
  • CI ≈ 0: No better than chance
  • CI < 0: Data contradicts hypothesis

Important: CI values depend on number of variables, correlation strength, and hypothesis complexity.

p-values

Conventional thresholds:

  • p < 0.05: Significant support
  • p < 0.01: Strong support
  • p < 0.001: Very strong support

Interpretation: Probability of obtaining CI this high (or higher) by chance alone.

The Ordering Vector

The ordering vector encodes your hypothesis. For n variables, it has length \(n(n-1)/2\) (one value per unique pair).

Example: Circular6

For a 6-variable circumplex:

        1
      6   2
     5     3
        4

The preset circular6 = [1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 2, 3, 1, 2, 1] encodes:

  • Pairs with value 1: Adjacent pairs (strongest correlations)
  • Pairs with value 2: Alternate pairs (intermediate)
  • Pairs with value 3: Opposite pairs (weakest)

Lower numbers = predicted higher correlations.

Common Applications

Interpersonal Circumplex (IPC)

Test if interpersonal scales follow the classic two-dimensional circular structure:

result = rthor.test(ipc_matrix, order="circular8")

Affect Circumplex

Test Russell's (1980) circumplex model of emotions:

result = rthor.test(emotion_matrix, order="circular8")

Custom Models

Test any hypothesized ordering:

# Linear ordering: 1 < 2 < 3 < 4
custom_order = [1, 2, 3, 2, 3, 3]
result = rthor.test(matrix, order=custom_order)

Method Advantages

  1. Theory-driven: Directly tests theoretical predictions
  2. Distribution-free: No parametric assumptions (permutation test)
  3. Flexible: Works with any hypothesized ordering
  4. Interpretable: CI provides intuitive effect size
  5. Validated: Exact parity with R implementation

References

Original R implementation: RTHORR 2.

Next Steps

  1. Lawrence Hubert and Phipps Arabie. Evaluating order hypotheses within proximity matrices. Psychological Bulletin, 102:172–178, 07 1987. doi:10.1037/0033-2909.102.1.172. ↩↩↩

  2. Terence J. G. Tracey and Michael L. Morris. Rthorr: randomization test of hypothesized order relations (rthor) and comparisons. 2025. R package version 0.1.3, commit c3edb36287c77733ec0a23236b478cc53c1cac0f. URL: https://github.com/michaellynnmorris/RTHORR. ↩

  3. Terence J. G. Tracey. Randall: a microsoft fortran program for a randomization test of hypothesized order relations. Educational and Psychological Measurement, 57(1):164–168, February 1997. doi:10.1177/0013164497057001012. ↩

  4. Andrew Mitchell. rthor: Python implementation of RTHOR (Randomization test of hypothesized order relations). URL: https://github.com/MitchellAcoustics/rthor. ↩