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:

  1. Slow Start: Minimal effect at low spend (below threshold)

  2. Rapid Acceleration: Steep effectiveness increase (threshold crossed)

  3. 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

Effect
  |╱──────────────────  (Saturation)
  |
  |
  |                      (Immediate diminishing returns)
  |
  |
  |____________________
       Spend →

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):

      ╱────
    ╱
  ╱
╱___________
  ↑
  Switch point (threshold)

Concave (ADBUG):

╱───────────
  ↓
  Immediate diminishing returns

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

Sales = β₀ + β₁ × ICP(TV_Spend)

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

Sales = β₀ + β₁ × ADBUG(Search_Spend)

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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