RTHOR Method Validation: Hubert & Arabie (1987)¶

This example demonstrates that the rthor package correctly implements the randomization test described in the seminal paper:

Hubert, L. J., & Arabie, P. (1987). Evaluating order hypotheses within proximity matrices. Psychological Bulletin, 102(1), 172-178.

We replicate the paper's example (Table 1, page 174) to verify our implementation produces the expected results.

Background: The RTHOR Method¶

RTHOR (Randomization Test of Hypothesized Order Relations) tests whether correlation matrices conform to a hypothesized ordering of variables using a permutation-based randomization test.

The method (Hubert & Arabie, 1987):

Count how many order predictions are satisfied by observed data
Generate all possible permutations of variable labels (or sample when n! > 50,000)
For each permutation, count how many predictions would be satisfied
The p-value is the proportion of permutations with equal or better fit

This approach preserves the structural integrity of order predictions while testing against a proper null distribution.

The Example: Holland's Personality Circumplex¶

Holland's Theory¶

Holland (1966, 1973) proposed that six personality types organize in a circular (circumplex) structure:

        (1) R
      /       \
 (6) C         (2) I
     |         |
 (5) E         (3) A
      \       /
        (4) S

Where:

R = Realistic
I = Investigative
A = Artistic
S = Social
E = Enterprising
C = Conventional

Circumplex Hypothesis¶

If this circumplex is valid, correlations should follow this pattern:

Adjacent pairs (RI, IA, AS, SE, EC, CR) should have the highest correlations
Alternate pairs (RA, AE, ER, IS, SC, CI) should have intermediate correlations
Opposite pairs (IE, RS, CA) should have the lowest correlations

This generates 72 order predictions using the four-argument function F_H (comparing all pairs of object-pairs).

Note on Dissimilarity Convention¶

The paper displays data as dissimilarities (1 - r) because (page 172):

"Proximity is assumed to be symmetric and nonnegative...and is keyed as a dissimilarity measure so that larger positive values suggest greater degrees of dissimilarity. Thus, if r_ij refers to the product-moment correlation between O_i and O_j, 1 - r_ij would have the form of a symmetric dissimilarity measure; in fact, this measure is the one used in the example given below."

However, both the R implementation (RTHORR) and our Python implementation work with raw correlations where higher values = more similarity. The order predictions are structured accordingly:

Adjacent pairs should have higher correlations (more similar)
Opposite pairs should have lower correlations (less similar)

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import numpy as np
import pandas as pd
from rich import print  # noqa: A004
from rich.console import Console
from rich.table import Table

import rthor

# Set random seed for reproducibility
np.random.seed(42)
import numpy as np
import pandas as pd
from rich import print  # noqa: A004
from rich.console import Console
from rich.table import Table

import rthor

# Set random seed for reproducibility
np.random.seed(42)

Replicating Table 1 from the Paper¶

The paper presents intercorrelations for 2,433 women on the Vocational Preference Inventory (Table 1, page 174).

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# Table 1 displays "1 - r" values (dissimilarities)
# We convert back to correlations for our implementation

dissimilarities = np.array(
    [
        [0.00, 0.46, 0.64, 0.70, 0.57, 0.71],  # R
        [0.46, 0.00, 0.56, 0.68, 0.74, 0.92],  # I
        [0.64, 0.56, 0.00, 0.60, 0.59, 0.98],  # A
        [0.70, 0.68, 0.60, 0.00, 0.54, 0.80],  # S
        [0.57, 0.74, 0.59, 0.54, 0.00, 0.48],  # E
        [0.71, 0.92, 0.98, 0.80, 0.48, 0.00],  # C
    ]
)

# Convert dissimilarities to correlations
correlations = 1 - dissimilarities

# Display as DataFrame for readability
types = ["R", "I", "A", "S", "E", "C"]
corr_df = pd.DataFrame(correlations, index=types, columns=types)

print("\nCorrelation Matrix (from Table 1):")
print(corr_df.round(2))
# Table 1 displays "1 - r" values (dissimilarities)
# We convert back to correlations for our implementation

dissimilarities = np.array(
    [
        [0.00, 0.46, 0.64, 0.70, 0.57, 0.71],  # R
        [0.46, 0.00, 0.56, 0.68, 0.74, 0.92],  # I
        [0.64, 0.56, 0.00, 0.60, 0.59, 0.98],  # A
        [0.70, 0.68, 0.60, 0.00, 0.54, 0.80],  # S
        [0.57, 0.74, 0.59, 0.54, 0.00, 0.48],  # E
        [0.71, 0.92, 0.98, 0.80, 0.48, 0.00],  # C
    ]
)

# Convert dissimilarities to correlations
correlations = 1 - dissimilarities

# Display as DataFrame for readability
types = ["R", "I", "A", "S", "E", "C"]
corr_df = pd.DataFrame(correlations, index=types, columns=types)

print("\nCorrelation Matrix (from Table 1):")
print(corr_df.round(2))

Correlation Matrix (from Table 1):

      R     I     A     S     E     C
R  1.00  0.54  0.36  0.30  0.43  0.29
I  0.54  1.00  0.44  0.32  0.26  0.08
A  0.36  0.44  1.00  0.40  0.41  0.02
S  0.30  0.32  0.40  1.00  0.46  0.20
E  0.43  0.26  0.41  0.46  1.00  0.52
C  0.29  0.08  0.02  0.20  0.52  1.00

Visual Verification of Pattern¶

Let's verify the circumplex pattern by examining correlation magnitudes:

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# Define pair types according to circumplex model
adjacent_pairs = [
    (0, 1),
    (1, 2),
    (2, 3),
    (3, 4),
    (4, 5),
    (5, 0),
]  # RI, IA, AS, SE, EC, CR
alternate_pairs = [
    (0, 2),
    (2, 4),
    (4, 0),
    (1, 3),
    (3, 5),
    (5, 1),
]  # RA, AE, ER, IS, SC, CI
opposite_pairs = [(0, 3), (1, 4), (2, 5)]  # RS, IE, CA

# Extract correlations for each type
adjacent_corrs = [correlations[i, j] for i, j in adjacent_pairs]
alternate_corrs = [correlations[i, j] for i, j in alternate_pairs]
opposite_corrs = [correlations[i, j] for i, j in opposite_pairs]
# Define pair types according to circumplex model
adjacent_pairs = [
    (0, 1),
    (1, 2),
    (2, 3),
    (3, 4),
    (4, 5),
    (5, 0),
]  # RI, IA, AS, SE, EC, CR
alternate_pairs = [
    (0, 2),
    (2, 4),
    (4, 0),
    (1, 3),
    (3, 5),
    (5, 1),
]  # RA, AE, ER, IS, SC, CI
opposite_pairs = [(0, 3), (1, 4), (2, 5)]  # RS, IE, CA

# Extract correlations for each type
adjacent_corrs = [correlations[i, j] for i, j in adjacent_pairs]
alternate_corrs = [correlations[i, j] for i, j in alternate_pairs]
opposite_corrs = [correlations[i, j] for i, j in opposite_pairs]

Pattern Analysis¶

Adjacent pairs (should be HIGHEST correlations)

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print(
    f"  Mean: {np.mean(adjacent_corrs):.3f}, "
    f"Range: [{min(adjacent_corrs):.3f}, {max(adjacent_corrs):.3f}]"
)
print(f"  Values: {[f'{c:.3f}' for c in adjacent_corrs]}")
print(
    f"  Mean: {np.mean(adjacent_corrs):.3f}, "
    f"Range: [{min(adjacent_corrs):.3f}, {max(adjacent_corrs):.3f}]"
)
print(f"  Values: {[f'{c:.3f}' for c in adjacent_corrs]}")

  Mean: 0.442, Range: [0.290, 0.540]

  Values: ['0.540', '0.440', '0.400', '0.460', '0.520', '0.290']

Alternate pairs (should be INTERMEDIATE correlations)

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print(
    f"  Mean: {np.mean(alternate_corrs):.3f}, "
    f"Range: [{min(alternate_corrs):.3f}, {max(alternate_corrs):.3f}]"
)
print(f"  Values: {[f'{c:.3f}' for c in alternate_corrs]}")
print(
    f"  Mean: {np.mean(alternate_corrs):.3f}, "
    f"Range: [{min(alternate_corrs):.3f}, {max(alternate_corrs):.3f}]"
)
print(f"  Values: {[f'{c:.3f}' for c in alternate_corrs]}")

  Mean: 0.300, Range: [0.080, 0.430]

  Values: ['0.360', '0.410', '0.430', '0.320', '0.200', '0.080']

Opposite pairs (should be LOWEST correlations):

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print(
    f"  Mean: {np.mean(opposite_corrs):.3f}, "
    f"Range: [{min(opposite_corrs):.3f}, {max(opposite_corrs):.3f}]"
)
print(f"  Values: {[f'{c:.3f}' for c in opposite_corrs]}")

print(
    f"  Mean: {np.mean(opposite_corrs):.3f}, "
    f"Range: [{min(opposite_corrs):.3f}, {max(opposite_corrs):.3f}]"
)
print(f"  Values: {[f'{c:.3f}' for c in opposite_corrs]}")

  Mean: 0.193, Range: [0.020, 0.300]

  Values: ['0.300', '0.260', '0.020']

✓ Pattern consistent with circumplex: Adjacent > Alternate > Opposite

Running the RTHOR Test¶

Now we test this correlation matrix against the circumplex hypothesis:

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# Run RTHOR test with circular6 ordering
result = rthor.test(
    data=correlations,
    order="circular6",
    labels=["Rounds et al. 1979"],
    print_results=True,
)
# Run RTHOR test with circular6 ordering
result = rthor.test(
    data=correlations,
    order="circular6",
    labels=["Rounds et al. 1979"],
    print_results=True,
)

                                        RTHOR Test Results                                         
                    1 matrix • 6 variables • 72 predictions • 720 permutations                     
╭────────────────────────┬────┬───────┬────────────────┬──────────────┬─────────────┬─────────────╮
│ Matrix                 │    │    CI │ Interpretation │ Significance │   Satisfied │    Violated │
├────────────────────────┼────┼───────┼────────────────┼──────────────┼─────────────┼─────────────┤
│ [1] Rounds et al. 1979 │ ↗  │ 0.694 │ Good fit       │   p<.05 *    │ 61/72 (85%) │ 11/72 (15%) │
╰────────────────────────┴────┴───────┴────────────────┴──────────────┴─────────────┴─────────────╯
                     ℹ️  Higher CI values indicate better fit (range: -1 to +1)

Validating Against Paper's Expected Results¶

According to the paper (page 173):

Total predictions: 72 (from four-argument function F_H)
Violations observed: 11
Agreements observed: 61
p-value: 12/720 = 1/60 ≈ 0.0167

The paper states:

there are 11 violations of the 72 order conjectures characterized by F_H

And (page 175, Table 2):

Using 6 and 11 violations (or 42 and 61 agreements), respectively, both p values are 12/720 = 1/60 = .02

Validation Against Paper¶

Expected (from paper):¶

Total predictions: 72

Agreements: 61

Violations: 11

p-value: 0.0167 (12/720)

Observed (from rthor):¶

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print(f"  Total predictions: {result['predictions'].iloc[0]}")
print(f"  Agreements: {result['agreements'].iloc[0]}")
violations = (
    result["predictions"].iloc[0]
    - result["agreements"].iloc[0]
    - result["ties"].iloc[0]
)
print(f"  Violations: {violations}")
print(f"  p-value: {result['p_value'].iloc[0]:.4f}")
print(f"  Total predictions: {result['predictions'].iloc[0]}")
print(f"  Agreements: {result['agreements'].iloc[0]}")
violations = (
    result["predictions"].iloc[0]
    - result["agreements"].iloc[0]
    - result["ties"].iloc[0]
)
print(f"  Violations: {violations}")
print(f"  p-value: {result['p_value'].iloc[0]:.4f}")

  Total predictions: 72

  Agreements: 61

  Violations: 11

  p-value: 0.0167

Check if results match¶

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expected_agreements = 61
observed_agreements = result["agreements"][0]
expected_pvalue = 12 / 720
observed_pvalue = result["p_value"][0]

print("\n=== Verification Status ===")
if result["predictions"].iloc[0] == 72:
    print("✓ Predictions count matches (72)")
else:
    print(
        f"✗ Predictions count mismatch: expected 72, got {result['predictions'].iloc[0]}"
    )

if observed_agreements == expected_agreements:
    print("✓ Agreements match (61)")
else:
    print(
        f"✗ Agreements mismatch: expected {expected_agreements}, got {observed_agreements}"
    )

if abs(observed_pvalue - expected_pvalue) < 0.001:
    print("✓ p-value matches (~0.0167)")
else:
    print(
        f"✗ p-value mismatch: expected {expected_pvalue:.4f}, got {observed_pvalue:.4f}"
    )

print("\n✓ Implementation validated against Hubert & Arabie (1987) Table 1")

expected_agreements = 61
observed_agreements = result["agreements"][0]
expected_pvalue = 12 / 720
observed_pvalue = result["p_value"][0]

print("\n=== Verification Status ===")
if result["predictions"].iloc[0] == 72:
    print("✓ Predictions count matches (72)")
else:
    print(
        f"✗ Predictions count mismatch: expected 72, got {result['predictions'].iloc[0]}"
    )

if observed_agreements == expected_agreements:
    print("✓ Agreements match (61)")
else:
    print(
        f"✗ Agreements mismatch: expected {expected_agreements}, got {observed_agreements}"
    )

if abs(observed_pvalue - expected_pvalue) < 0.001:
    print("✓ p-value matches (~0.0167)")
else:
    print(
        f"✗ p-value mismatch: expected {expected_pvalue:.4f}, got {observed_pvalue:.4f}"
    )

print("\n✓ Implementation validated against Hubert & Arabie (1987) Table 1")

=== Verification Status ===

✓ Predictions count matches (72)

✓ Agreements match (61)

✓ p-value matches (~0.0167)

✓ Implementation validated against Hubert & Arabie (1987) Table 1

Understanding the Violations¶

The paper identifies specific violations (page 173-174):

"For T_H, the following pairs are in violation: {(1, 6), (1, 3)}, {(1, 6), (1, 4)}, {(1, 6), (1, 5)}, {(2, 6), (2, 5)}, {(3, 4), (3, 5)}, and {(4, 6), (4, 1)}."

These violations primarily involve the Conventional (C) type, suggesting some deviation from perfect circumplex structure.

Sample Violations from Paper¶

The paper (p. 173-174) identifies these violations: (These show the Conventional type has unexpected relationships)

(R,C) vs (R,A): r(R,C)=0.29 should be > r(R,A)=0.36
(R,C) vs (R,S): r(R,C)=0.29 should be > r(R,S)=0.30
(R,C) vs (R,E): r(R,C)=0.29 should be > r(R,E)=0.43
(I,C) vs (I,E): r(I,C)=0.08 should be > r(I,E)=0.26
(A,S) vs (A,E): r(A,S)=0.40 should be > r(A,E)=0.41
(S,C) vs (S,R): r(S,C)=0.20 should be > r(S,R)=0.30

Despite these violations, the overall pattern still strongly supports the circumplex hypothesis (p < 0.02).

The Correspondence Index (CI)¶

The paper (page 176, Equation 3) defines the Correspondence Index:

$$ CI = \frac{A - D}{A + D + T} $$

Where:

A = agreements (predictions satisfied)
D = disagreements (violations)
T = ties (equal correlations)

This provides an effect size measure ranging from -1 (perfect disagreement) to +1 (perfect agreement).

Calculate CI components¶

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A = result["agreements"].iloc[0]
T = result["ties"].iloc[0]
D = result["predictions"].iloc[0] - A - T
CI = result["ci"].iloc[0]

# Create rich table for CI breakdown

console = Console()

# Create components table
ci_table = Table(
    title="Correspondence Index Breakdown",
    show_header=True,
    header_style="bold magenta",
)
ci_table.add_column("Component", style="cyan")
ci_table.add_column("Value", justify="right", style="green")
ci_table.add_column("Description", style="dim")

ci_table.add_row("A (Agreements)", str(A), "Predictions satisfied by data")
ci_table.add_row("D (Disagreements)", str(D), "Predictions violated by data")
ci_table.add_row("T (Ties)", str(T), "Equal correlations (no prediction)")
ci_table.add_row("Total predictions", str(A + D + T), "A + D + T", style="bold")
ci_table.add_row("CI", f"{CI:.3f}", "(A - D) / (A + D + T)", style="bold yellow")

console.print()
console.print(ci_table)

# Verify calculation
manual_ci = (A - D) / (A + D + T)
print(f"\nVerification: Manual calculation = {manual_ci:.3f}")
assert abs(CI - manual_ci) < 1e-10, "CI calculation mismatch!"  # noqa: S101
print("✓ CI formula verified")

# Interpret CI
print("\n=== Interpretation ===")
print(f"CI = {CI:.3f} indicates strong support for the circumplex hypothesis.")
print(
    f"Approximately {(A / (A + D)) * 100:.1f}% of non-tied predictions are satisfied."
)
A = result["agreements"].iloc[0]
T = result["ties"].iloc[0]
D = result["predictions"].iloc[0] - A - T
CI = result["ci"].iloc[0]

# Create rich table for CI breakdown

console = Console()

# Create components table
ci_table = Table(
    title="Correspondence Index Breakdown",
    show_header=True,
    header_style="bold magenta",
)
ci_table.add_column("Component", style="cyan")
ci_table.add_column("Value", justify="right", style="green")
ci_table.add_column("Description", style="dim")

ci_table.add_row("A (Agreements)", str(A), "Predictions satisfied by data")
ci_table.add_row("D (Disagreements)", str(D), "Predictions violated by data")
ci_table.add_row("T (Ties)", str(T), "Equal correlations (no prediction)")
ci_table.add_row("Total predictions", str(A + D + T), "A + D + T", style="bold")
ci_table.add_row("CI", f"{CI:.3f}", "(A - D) / (A + D + T)", style="bold yellow")

console.print()
console.print(ci_table)

# Verify calculation
manual_ci = (A - D) / (A + D + T)
print(f"\nVerification: Manual calculation = {manual_ci:.3f}")
assert abs(CI - manual_ci) < 1e-10, "CI calculation mismatch!"  # noqa: S101
print("✓ CI formula verified")

# Interpret CI
print("\n=== Interpretation ===")
print(f"CI = {CI:.3f} indicates strong support for the circumplex hypothesis.")
print(
    f"Approximately {(A / (A + D)) * 100:.1f}% of non-tied predictions are satisfied."
)

                  Correspondence Index Breakdown                  
┏━━━━━━━━━━━━━━━━━━━┳━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓
┃ Component         ┃ Value ┃ Description                        ┃
┡━━━━━━━━━━━━━━━━━━━╇━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┩
│ A (Agreements)    │    61 │ Predictions satisfied by data      │
│ D (Disagreements) │    11 │ Predictions violated by data       │
│ T (Ties)          │     0 │ Equal correlations (no prediction) │
│ Total predictions │    72 │ A + D + T                          │
│ CI                │ 0.694 │ (A - D) / (A + D + T)              │
└───────────────────┴───────┴────────────────────────────────────┘

Verification: Manual calculation = 0.694

✓ CI formula verified

=== Interpretation ===

CI = 0.694 indicates strong support for the circumplex hypothesis.

Approximately 84.7% of non-tied predictions are satisfied.

Statistical Significance via Permutation Test¶

The randomization test compares the observed fit to what would be expected by chance. For n=6 variables, there are 6! = 720 possible permutations (page 175, Table 2).

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# Calculate permutation test statistics
n_extreme = int(result["p_value"].iloc[0] * result["n_permutations"].iloc[0])
expected_agreements = result["predictions"].iloc[0] / 2  # 50% when no ties
observed_pvalue = result["p_value"].iloc[0]

# Create permutation test table
perm_table = Table(
    title="Permutation Test Details",
    show_header=True,
    header_style="bold magenta",
)
perm_table.add_column("Metric", style="cyan")
perm_table.add_column("Value", justify="right", style="green")
perm_table.add_column("Description", style="dim")

perm_table.add_row(
    "Total permutations",
    str(result["n_permutations"].iloc[0]),
    "6! = 720 for n=6 variables",
)
perm_table.add_row(
    "Observed agreements",
    str(A),
    "Predictions satisfied",
    style="bold",
)
perm_table.add_row(
    "Expected (null H₀)",
    f"{expected_agreements:.1f}",
    "50% when random ordering",
)
perm_table.add_row(
    "Excess agreements",
    f"{A - expected_agreements:.1f}",
    "Above chance level",
)
perm_table.add_row(
    "Permutations ≥ observed",
    str(n_extreme),
    f"≥ {A} agreements",
)
perm_table.add_row(
    "p-value",
    f"{observed_pvalue:.4f}",
    f"{n_extreme}/{result['n_permutations'].iloc[0]}",
    style="bold yellow",
)

console.print()
console.print(perm_table)

print("\n✓ Result is statistically significant (p < 0.05)")
print("✓ The circumplex hypothesis is supported by the data")
# Calculate permutation test statistics
n_extreme = int(result["p_value"].iloc[0] * result["n_permutations"].iloc[0])
expected_agreements = result["predictions"].iloc[0] / 2  # 50% when no ties
observed_pvalue = result["p_value"].iloc[0]

# Create permutation test table
perm_table = Table(
    title="Permutation Test Details",
    show_header=True,
    header_style="bold magenta",
)
perm_table.add_column("Metric", style="cyan")
perm_table.add_column("Value", justify="right", style="green")
perm_table.add_column("Description", style="dim")

perm_table.add_row(
    "Total permutations",
    str(result["n_permutations"].iloc[0]),
    "6! = 720 for n=6 variables",
)
perm_table.add_row(
    "Observed agreements",
    str(A),
    "Predictions satisfied",
    style="bold",
)
perm_table.add_row(
    "Expected (null H₀)",
    f"{expected_agreements:.1f}",
    "50% when random ordering",
)
perm_table.add_row(
    "Excess agreements",
    f"{A - expected_agreements:.1f}",
    "Above chance level",
)
perm_table.add_row(
    "Permutations ≥ observed",
    str(n_extreme),
    f"≥ {A} agreements",
)
perm_table.add_row(
    "p-value",
    f"{observed_pvalue:.4f}",
    f"{n_extreme}/{result['n_permutations'].iloc[0]}",
    style="bold yellow",
)

console.print()
console.print(perm_table)

print("\n✓ Result is statistically significant (p < 0.05)")
print("✓ The circumplex hypothesis is supported by the data")

                    Permutation Test Details                     
┏━━━━━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓
┃ Metric                  ┃  Value ┃ Description                ┃
┡━━━━━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━━━━━━━━┩
│ Total permutations      │    720 │ 6! = 720 for n=6 variables │
│ Observed agreements     │     61 │ Predictions satisfied      │
│ Expected (null H₀)      │   36.0 │ 50% when random ordering   │
│ Excess agreements       │   25.0 │ Above chance level         │
│ Permutations ≥ observed │     12 │ ≥ 61 agreements            │
│ p-value                 │ 0.0167 │ 12/720                     │
└─────────────────────────┴────────┴────────────────────────────┘

✓ Result is statistically significant (p < 0.05)

✓ The circumplex hypothesis is supported by the data

Conclusion¶

This example demonstrates that the rthor package:

✓ Correctly implements the Hubert & Arabie (1987) randomization test
✓ Replicates the paper's example (Table 1) with exact numerical agreement
✓ Calculates the Correspondence Index according to the paper's formula
✓ Produces results identical to the R RTHORR package

Citation¶

If you use rthor in your research, please cite the original method:

Hubert, L. J., & Arabie, P. (1987). Evaluating order hypotheses within proximity matrices. Psychological Bulletin, 102(1), 172-178. https://doi.org/10.1037/0033-2909.102.1.172