Curve Theory & When to Use
Understanding Saturation and Diminishing Returns in Marketing
Real-world marketing rarely shows linear relationships with outcomes. The first dollar spent typically has the highest impact, with each subsequent dollar yielding progressively less return. Saturation curves mathematically model this fundamental marketing reality.
The Saturation Problem
Why Linear Models Fail
Linear Assumption:
Sales = β₀ + β₁ × TV_SpendWhat this assumes:
Every dollar has the same impact
$10K → $X sales lift
$100K → 10× $X sales lift (proportional)
No diminishing returns
Real Marketing Reality:
First $10K → High ROI (fresh audience)
Next $10K → Lower ROI (frequency fatigue)
Next $10K → Even lower ROI (saturation)
Problem: Linear models overestimate returns at high spend and underestimate efficiency at low spend
What Are Saturation Curves?
Non-Linear Transformations
Saturation curves apply mathematical transformations that capture diminishing returns:
Input: Raw marketing spend (linear) Output: Transformed values (curved) that represent actual effectiveness
Visual:
Result: Model coefficients now accurately represent ROI at different spend levels
Two Fundamental Patterns
Pattern 1: S-Shape (Sigmoid)
Visual Pattern:
Characteristics:
Slow start (threshold effect)
Rapid middle acceleration
Eventual plateau (saturation)
When Marketing Behaves This Way:
Need critical mass before effectiveness kicks in
Tipping point phenomenon
Network effects or viral spread
Pattern 2: Concave (Immediate Diminishing Returns)
Visual Pattern:
Characteristics:
High initial impact
Immediate diminishing returns
Gradual approach to maximum
When Marketing Behaves This Way:
Captures high-intent audience first
Best prospects respond immediately
Each additional exposure has less value
When to Use Saturation Curves
Apply Curves When:
✅ Wide Spend Range Variable has 3× or more variation (min to max)
Example: TV spend ranges from $0 to $100K
✅ Expected Diminishing Returns Channel logically should saturate
Media channels (TV, Radio, Digital)
Price promotions
Broad reach advertising
✅ Large Absolute Spend High enough spend to experience saturation
Not needed if spend is consistently low
✅ Non-Linear Patterns Visible Scatter plot shows curved relationship
High spend weeks don't show proportional lift
Don't Apply Curves When:
❌ Limited Spend Variation Variable ranges narrowly (all values similar)
Example: Price constant at $49.99 all year
❌ No Saturation Expected Variable shouldn't experience diminishing returns
Seasonality dummies
Weather variables
Time trends
❌ Insufficient Data Too few observations to estimate curve parameters
Need 30+ observations minimum
❌ Linear Relationship Sufficient Scatter plot shows linear pattern
Curve adds unnecessary complexity
Saturation in Different Channel Types
Broadcast Media (TV, Radio)
Saturation Behavior: Moderate S-shape or concave
Why: Audience reach limits, frequency fatigue
Typical Pattern: Moderate threshold, then gradual saturation
Curve Type: S-shape for brand building, Concave for direct response
Digital Display
Saturation Behavior: Strong concave
Why: Frequency caps, banner blindness, auction saturation
Typical Pattern: High initial efficiency, rapid diminishing returns
Curve Type: Concave (ADBUG)
Paid Search
Saturation Behavior: Strong concave
Why: Best keywords first, expanding to lower-intent searches
Typical Pattern: Capture high-intent immediately, declining ROI
Curve Type: Concave (ADBUG)
Social Media
Saturation Behavior: S-shape
Why: Viral threshold, network effects, algorithm amplification
Typical Pattern: Slow start, rapid middle growth (viral), then saturation
Curve Type: S-shape (ICP)
Brand vs. Performance
Brand Marketing (Awareness):
Often shows S-shape behavior
Needs threshold reach before impact
Long-term cumulative effects
Use: S-shape curves (ICP)
Performance Marketing (Conversion):
Usually concave behavior
Immediate response from high-intent users
Direct attribution visible
Use: Concave curves (ADBUG)
Mathematical Intuition
Why Curves Work
S-Shape Captures:
Threshold: Need minimum investment before effectiveness starts
Acceleration: Once threshold passed, rapid growth phase
Saturation: Eventually hits ceiling regardless of spend
Concave Captures:
High Initial Returns: First dollars most effective
Gradual Decay: Each additional dollar slightly less effective
Asymptotic Limit: Approaches maximum value
Both Capture: The reality that marketing doesn't scale linearly
Impact on Model Interpretation
Without Curves (Linear)
Model:
Interpretation: Each $1 in TV → $5 in sales (always)
Problem:
Overestimates at high spend
Suggests infinite returns with infinite budget
Unrealistic optimization recommendations
With Curves (Saturated)
Model:
Interpretation: Impact varies by spend level
At low spend: $1 TV → ~$4 sales (high ROI)
At medium spend: $1 TV → ~$3 sales (moderate ROI)
At high spend: $1 TV → ~$1 sales (low ROI, saturated)
Benefit: Realistic ROI estimates enable optimal budget allocation
Curve Selection Decision Tree
Start Here → Is this a media/marketing variable?
Yes → Does it have wide spend variation (3× or more)?
Yes → What type of marketing?
Brand/Awareness:
Social media
TV brand campaigns
PR/Sponsorships
→ Use S-Shape (ICP)
Direct Response/Performance:
Paid search
Performance display
Email marketing
Price promotions
→ Use Concave (ADBUG)
Unclear/Test Both:
Test both S-shape and Concave
Compare model fit (R², t-stats)
Choose best performer
→ Use Curve Testing interface
No (narrow range or non-marketing):
→ Don't apply curves, use raw values
Testing Before Applying
Always Preview Curves
Before creating curve variables:
Use Curve Testing interface
Visualize curve shape with your actual data
Test multiple parameter combinations
Check statistical fit (t-statistics)
Verify business logic makes sense
Red Flags:
Curve looks flat (not capturing saturation)
Curve looks too extreme (unrealistic)
No improvement in model fit
Negative coefficients when positive expected
Common Curve Misconceptions
Misconception 1: "More complex curves are better"
Reality: Simpler curves often perform just as well
Start with 2-parameter ATAN curves
Only use 3-parameter CDR if needed
Complexity doesn't guarantee better fit
Misconception 2: "Apply curves to everything"
Reality: Curves only for variables with saturation behavior
Don't apply to seasonality
Don't apply to external factors
Don't apply to variables with narrow range
Misconception 3: "Curve choice is arbitrary"
Reality: Curve type should match marketing behavior
S-shape for brand/awareness (threshold effects)
Concave for performance (immediate returns)
Test both if uncertain
Misconception 4: "Set once and forget"
Reality: Curves may need adjustment over time
Market saturation changes
Competition evolves
Creative quality varies
Revisit quarterly
Integration with Other Transformations
Typical Transformation Sequence
Step 1: Adstock (if media channel)
Models carryover effects
Step 2: Saturation Curve
Models diminishing returns
Step 3: Use in Model
Order Matters: Adstock first (accumulation), then saturation (diminishing returns on accumulated effect)
Business Value of Curves
Accurate ROI at All Spend Levels
Without Curves: "TV ROI is $3 per dollar" (misleading average)
With Curves:
At $20K/week: TV ROI is $5 per dollar
At $50K/week: TV ROI is $3 per dollar
At $80K/week: TV ROI is $1.50 per dollar
Actionable: Know exactly where to invest and where you're saturated
Optimal Budget Allocation
Linear Model Recommendation: "Spend $100K on TV" (overinvests in saturated channel)
Saturated Model Recommendation: "Spend $50K on TV (near saturation point), reallocate $50K to unsaturated channels"
Result: 15-30% improvement in total marketing efficiency
Scenario Planning
Question: "What if we double TV spend?"
Linear Answer: "Sales will increase proportionally" (wrong)
Saturated Answer: "Sales will increase 40% (not 100%) due to saturation"
Better Decision: "Instead of doubling TV, increase 30% and invest rest in Digital"
Summary
Key Takeaways:
📉 Real marketing saturates - linear models are systematically wrong
📊 Two fundamental patterns - S-shape (threshold) and Concave (immediate diminishing returns)
🎯 Match curve to behavior - brand → S-shape, performance → concave
✅ Test before applying - use Curve Testing interface to validate
📈 Transforms ROI estimates - from misleading averages to accurate marginal returns
🔧 Enables optimization - identify saturation points, reallocate budgets
⚖️ Combine with adstock - apply adstock first, then saturation
💡 Start simple - ATAN 2-parameter curves often sufficient
Saturation curves are essential for realistic MMM. They transform models from descriptive to prescriptive, enabling confident budget optimization based on true marginal returns.
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