CORRELATION MATRIX ANALYSIS

Dataset Overview:

• Observations: 150
• Variables: 4
• Variable names: impressions, clicks, conversions, ctr
• Missing values: 0

Potential outliers detected:
conversions 2
dtype: int64

============================================================
⚠️ DERIVED METRIC DETECTED

ctr = clicks / impressions

Correlations involving CTR and its components (clicks, impressions)
are MATHEMATICAL ARTIFACTS, not empirical findings.

✓ Heatmap saved: correlation_heatmap.png

✓ Results saved: correlation_results.csv

============================================================
SEPARATING MATHEMATICAL VS EMPIRICAL CORRELATIONS

🚨 MATHEMATICAL ARTIFACTS (do not report as insights):
These are guaranteed by the formula: ctr = clicks / impressions

• clicks ↔ ctr: r = 0.644 [clicks in numerator] ***
• impressions ↔ ctr: r = 0.110 [impressions in denominator]

✓ EMPIRICAL FINDINGS (real relationships):
These reflect actual behavioral patterns in the data

1. clicks ↔ conversions
   r = 0.849 (p = 0.0000) *** STRONG POSITIVE

2. impressions ↔ clicks
   r = 0.752 (p = 0.0000) *** STRONG POSITIVE

3. impressions ↔ conversions
   r = 0.610 (p = 0.0000) *** MODERATE POSITIVE

4. conversions ↔ ctr
   r = 0.569 (p = 0.0000) *** MODERATE POSITIVE

============================================================
MULTICOLLINEARITY ALERTS (|r| > 0.8):

🚨 clicks ↔ conversions: r = 0.849
→ These variables are highly redundant
→ Do NOT include both in predictive models

============================================================
KEY INSIGHTS

1. STRONG EMPIRICAL RELATIONSHIPS:
   • clicks ↔ conversions (r = 0.849)
   → Campaigns that generate more clicks drive more conversions
   → This is the fundamental marketing funnel relationship

   • impressions ↔ clicks (r = 0.752)
   → Higher reach leads to more engagement
   → Validates the importance of broad campaign reach

2. MATHEMATICAL ARTIFACTS TO IGNORE:
   • ctr ↔ clicks (r = 0.644)
   → DO NOT REPORT as ‘higher clicks correlate with higher CTR’
   → This is mathematically guaranteed by the formula ctr = clicks/impressions

   • ctr ↔ impressions (r = 0.110)
   → Weak inverse relationship is also artifact of ratio formula

3. MULTICOLLINEARITY WARNING:
   • clicks and conversions are highly correlated (r = 0.849)
   → For predictive modeling, include ONLY ONE of these variables
   → Including both causes redundancy and unstable coefficient estimates
   → Recommendation: Keep clicks (it’s the earlier funnel stage)

4. CTR AS OUTCOME METRIC:
   • conversions ↔ ctr (r = 0.569) is empirical and meaningful
   → Higher conversion campaigns tend to have better CTR
   → Both reflect campaign quality/relevance
   → This relationship is NOT forced by formula

============================================================
RECOMMENDATIONS

✓ FOR EXPLORATORY ANALYSIS:

• Focus on empirical relationships: impressions → clicks → conversions
• Treat CTR as dependent outcome, not independent driver
• Investigate what campaign characteristics drive both CTR and conversions

✓ FOR PREDICTIVE MODELING:

• Include: impressions
• Choose ONE of: clicks OR conversions (not both)
• Exclude: ctr (it’s derived from included variables)
• Add campaign features if available (ad copy, targeting, creative, etc.)

✓ WHAT NOT TO REPORT:
✗ ‘Higher CTR correlates with more clicks’ (mathematical artifact)
✗ ‘CTR drives conversions’ (correlation ≠ causation)
✗ Include both clicks and conversions in same model (multicollinearity)

============================================================
FILES GENERATED

• marketing_campaigns.csv (150 campaigns)
• correlation_heatmap.png (visualization)
• correlation_results.csv (detailed results)

============================================================
⚠️ CRITICAL REMINDER

Correlation ≠ Causation

These correlations show associations but cannot establish
causal relationships. To make causal claims:
• Run A/B tests (vary impressions, measure clicks)
• Use time-series analysis (track campaigns over time)
• Control for confounders (campaign type, audience, timing)

Without experimental or quasi-experimental design,
these are descriptive patterns only.