S-Shape vs Concave
S-Shape (ICP) vs Concave (ADBUG)
Choosing the Right Curve Type for Your Marketing Channels
MixModeler offers two fundamental curve types for modeling saturation: S-Shape (ICP) for threshold-based effects and Concave (ADBUG) for immediate diminishing returns. Understanding when to use each is critical for accurate marketing attribution.
S-Shape Curves (ICP)
Visual Pattern
Effect
| ╱──────── (Saturation)
| ╱
| ╱ (Rapid growth)
| ╱
| ╱ (Slow start)
| ╱
|╱____________________
Spend →Three Distinct Phases:
Slow Start: Minimal effect at low spend (below threshold)
Rapid Acceleration: Steep effectiveness increase (threshold crossed)
Saturation: Diminishing returns, approaching maximum
When Marketing Shows S-Shape Behavior
Threshold Effects: Marketing needs critical mass before becoming effective
Network Effects: Value increases as more people are reached (social phenomena)
Viral Potential: Slow build, then rapid spread, then saturation
Brand Awareness: Need sufficient reach before brand recall kicks in
Real-World S-Shape Examples
Social Media Campaigns:
Posts need engagement threshold to trigger algorithm amplification
Slow initial shares → viral tipping point → saturation
Use ICP curves
New Product Launches:
Initial adopters slow (awareness building)
Word-of-mouth acceleration phase
Market saturation eventually
Use ICP curves
Brand TV Campaigns:
Need frequency/reach threshold before brand lift
Middle campaign period shows peak effectiveness
Late campaign shows saturation
Use ICP curves
PR and Sponsorships:
Slow credibility building phase
Recognition acceleration once established
Saturation at high awareness levels
Use ICP curves
S-Shape Parameters (ICP)
Available in MixModeler:
CDR Formula (3 parameters):
Alpha: Controls overall saturation level
Beta: Controls steepness of middle acceleration
Gamma: Controls switch point location (threshold)
ATAN Formula (2 parameters):
Alpha: Controls saturation level
Power: Controls curve shape (power > 1.5 creates S-shape)
Concave Curves (ADBUG)
Visual Pattern
Single Phase:
High initial impact
Immediate and continuous diminishing returns
Gradual approach to maximum (no inflection point)
When Marketing Shows Concave Behavior
Audience Sorting: Best prospects captured first, each additional reach less valuable
Frequency Fatigue: First impression most impactful, additional exposures less effective
Search Intent: High-intent keywords exhausted first, expanding reach lowers conversion
Price Sensitivity: Most price-sensitive customers respond first to promotions
Real-World Concave Examples
Paid Search:
Best keywords (high intent) captured immediately
Expanding to broader keywords lowers conversion rate
Each additional keyword less efficient
Use ADBUG curves
Performance Display:
High-intent audiences targeted first (retargeting, lookalikes)
Audience expansion reaches less-qualified users
Frequency caps cause diminishing impact
Use ADBUG curves
Email Marketing:
Best customers open/convert immediately
Additional sends show declining response rates
List fatigue sets in
Use ADBUG curves
Price Promotions:
Most price-sensitive shoppers respond first
Deep discounts capture progressively less-sensitive segments
Immediate saturation at high discount levels
Use ADBUG curves
Direct Mail:
Best prospects (highest propensity) mailed first
Additional targeting reaches lower-propensity segments
Immediate diminishing returns per piece
Use ADBUG curves
Concave Parameters (ADBUG)
Available in MixModeler:
CDR Formula (3 parameters):
Alpha: Controls saturation ceiling (typically 0.8-1.0)
Beta: Controls steepness of diminishing returns
Gamma: Controls how quickly saturation is reached
ATAN Formula (2 parameters):
Alpha: Controls saturation level
Power: Set to 1.0 for pure concave (no S-shape)
Direct Comparison
Key Differences
Visual Side-by-Side
S-Shape (ICP):
Concave (ADBUG):
Decision Framework
Choose S-Shape (ICP) When:
✅ Threshold Exists Need minimum investment before effectiveness begins
✅ Brand Building Awareness campaigns, sponsorships, PR
✅ Network/Viral Effects Social media, word-of-mouth dependent
✅ Long Sales Cycles B2B campaigns with awareness → consideration → conversion funnel
✅ New Market Entry Building presence from scratch
Typical Channels:
Social media (Facebook, Instagram, TikTok)
Brand TV campaigns
Sponsorships and events
Influencer marketing
PR campaigns
Choose Concave (ADBUG) When:
✅ Immediate Response First exposure drives immediate action
✅ Performance Marketing Direct response, conversion-focused
✅ Audience Targeting Best prospects captured first
✅ Short Sales Cycles Transactional, low consideration purchases
✅ Frequency-Based Multiple exposures show declining effectiveness
Typical Channels:
Paid search (Google Ads, Bing)
Performance display/programmatic
Email marketing
Price promotions
Retargeting campaigns
Direct mail
Test Both If:
⚠️ Unclear Pattern Can't determine from business logic alone
⚠️ Mixed Behavior Channel shows both threshold and immediate response characteristics
⚠️ New Channel No historical precedent to guide decision
⚠️ Experimental Testing new creative or targeting approach
Solution: Use Curve Testing interface to test both, choose based on statistical fit (t-statistics, R²)
Testing in MixModeler
Curve Testing Workflow
Step 1: Select Variable Choose marketing channel to test
Step 2: Test S-Shape (ICP)
Run curve tests with ICP type
Test multiple parameter combinations
Note best t-statistic and R²
Step 3: Test Concave (ADBUG)
Run curve tests with ADBUG type
Test multiple parameter combinations
Note best t-statistic and R²
Step 4: Compare Results
Which curve type has higher t-statistic?
Which shows better R² improvement?
Which makes more business sense?
Step 5: Select Winner
Use curve type with best statistical fit AND business logic alignment
Common Patterns by Channel
TV Advertising
Brand TV: S-Shape (ICP)
Need frequency for brand lift
Switch point around 500-1000 GRPs
Direct Response TV: Concave (ADBUG)
Immediate phone/web response
Diminishing returns with frequency
Digital Display
Programmatic: Concave (ADBUG)
Audience exhaustion
Frequency caps
Immediate diminishing returns
High-Impact (Takeovers): S-Shape (ICP)
Need threshold impressions for impact
Awareness building
Social Media
Organic/Viral: S-Shape (ICP)
Engagement threshold for algorithm boost
Network effects
Paid Social (Performance): Concave (ADBUG)
Best audiences first
Auction saturation
Radio
Brand Radio: S-Shape (ICP)
Need frequency for recall
Threshold effects
Direct Response Radio: Concave (ADBUG)
Immediate call-to-action response
Frequency fatigue
Parameter Implications
S-Shape Parameters
Alpha (higher values = more saturation):
2.0-3.0: Moderate saturation
3.0-4.0: Strong saturation
Power (ATAN formula):
1.5-1.8: Gentle S-shape
1.8-2.0: Moderate S-shape
2.0+: Strong S-shape (steep middle)
Switch Point: Location where acceleration begins (automatically calculated)
Should align with business threshold expectations
Concave Parameters
Alpha (saturation ceiling):
0.8-0.9: Lower ceiling (strong diminishing returns)
0.9-1.0: Higher ceiling (moderate diminishing returns)
Power (ATAN formula):
Always 1.0 for pure concave shape
Steepness: How quickly diminishing returns kick in
Controlled by beta (CDR) or implicit in ATAN
Interpretation Differences
S-Shape Model
If β₁ = 0.5:
Below threshold: TV has minimal impact
At switch point: TV shows rapid effectiveness increase
Above saturation: TV impact plateaus
Business Insight: "We need $X minimum spend for TV to work, then it's highly effective until $Y where it saturates"
Concave Model
If β₁ = 0.6:
First dollars: Very high ROI
Additional dollars: Continuously declining ROI
High spend: Approaches ceiling
Business Insight: "Search is immediately effective but quickly shows diminishing returns. Optimal spend is around $X before saturation."
Common Mistakes
Mistake 1: Using S-Shape for Performance Channels
Problem: Applying ICP to paid search or performance display
Why Wrong: These channels don't have threshold effects - they work immediately
Fix: Use Concave (ADBUG) for performance marketing
Mistake 2: Using Concave for Brand Campaigns
Problem: Applying ADBUG to brand TV or sponsorships
Why Wrong: Miss the threshold effect and acceleration phase
Fix: Use S-Shape (ICP) for brand/awareness campaigns
Mistake 3: Not Testing Both
Problem: Assuming curve type without validation
Why Wrong: May miss better-fitting alternative
Fix: Always test both types in Curve Testing, let data guide
Mistake 4: Ignoring Business Logic
Problem: Choosing curve based solely on statistics
Why Wrong: May select mathematically better but illogical curve
Fix: Statistics + business judgment = best choice
Summary
Key Takeaways:
📈 S-Shape (ICP) = Threshold effects - slow start, rapid middle, saturation
📉 Concave (ADBUG) = Immediate diminishing returns - high start, continuous decline
🎯 Brand → S-Shape - awareness, viral, network effects
💰 Performance → Concave - direct response, audience sorting
🧪 Test both when uncertain - let statistics + logic guide
📊 Different parameters - Alpha, Power/Beta/Gamma behave differently per type
✅ Match to marketing behavior - curve should reflect reality
Choose the curve type that matches your channel's actual behavior, not the one that just fits the data best. The goal is accurate representation of marketing dynamics, not just mathematical fit!
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