# 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_Spend
```

**What 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:**

```
Raw Spend:        ─────────────────> (linear)
Saturated Effect: ∿∿∿∿∿∿∿∿──────> (curved, plateaus)
```

**Result:** Model coefficients now accurately represent ROI at different spend levels

***

### Two Fundamental Patterns

#### Pattern 1: S-Shape (Sigmoid)

**Visual Pattern:**

```
|           ╱────
|         ╱
|       ╱
|     ╱
|   ╱
| ╱
|╱_______________
  Spend →
```

**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:**

```
|╱────────────────
|
|
|
|
|
|________________
  Spend →
```

**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:**

1. **Threshold:** Need minimum investment before effectiveness starts
2. **Acceleration:** Once threshold passed, rapid growth phase
3. **Saturation:** Eventually hits ceiling regardless of spend

**Concave Captures:**

1. **High Initial Returns:** First dollars most effective
2. **Gradual Decay:** Each additional dollar slightly less effective
3. **Asymptotic Limit:** Approaches maximum value

**Both Capture:** The reality that marketing doesn't scale linearly

***

### Impact on Model Interpretation

#### Without Curves (Linear)

**Model:**

```
Sales = 1000 + 5 × TV_Spend
```

**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:**

```
Sales = 1000 + 3 × f(TV_Spend)
Where f() is saturation curve
```

**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:**

1. Use Curve Testing interface
2. Visualize curve shape with your actual data
3. Test multiple parameter combinations
4. Check statistical fit (t-statistics)
5. 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)**

```
TV_Spend → TV_Spend_ads60
```

Models carryover effects

**Step 2: Saturation Curve**

```
TV_Spend_ads60 → TV_Spend_ads60_ATAN_curve
```

Models diminishing returns

**Step 3: Use in Model**

```
Sales ~ TV_Spend_ads60_ATAN_curve + ...
```

**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.
