Parameter Interpretation

Understanding ATAN Curve Parameters and Their Business Meaning

The Two Parameters

ATAN curves use just two parameters:

1. Alpha (α) - Inflection Point Parameter

2. Power - Shape Parameter

Alpha (α) - Inflection Point Parameter

What Alpha Controls

Alpha determines where the curve transitions from linear to saturated behavior. Higher alpha → later saturation.

Visual Impact

Low Alpha (0.1-0.3):

Effect│╱──────────── (saturates early)
      │╱
      │
      │_______________
         Spend →

High Alpha (0.7-1.0):

Effect│        ╱──────── (saturates late)
      │       ╱
      │      ╱
      │_____╱
         Spend →

Typical Alpha by Channel

Channel Type	Alpha Range	Reasoning
Paid Search	0.2 - 0.4	Limited keywords, quick saturation
Display Remarketing	0.3 - 0.5	Audience pools exhaust moderately fast
Social Performance	0.3 - 0.5	Audience saturation, frequency caps
TV Brand	0.6 - 0.9	Large reach potential, slow saturation
Email Marketing	0.2 - 0.4	Limited list size, quick fatigue
Sponsorships	0.7 - 1.0	Long-term brand building

Business Interpretation

Digital Performance (Alpha 0.2-0.4):

• Quick audience exhaustion

• Early saturation expected

• Best prospects captured immediately

Brand TV (Alpha 0.6-0.9):

• Needs sustained investment

• Saturation at higher spend levels

• Awareness builds gradually

Power - Shape Parameter

What Power Controls

Power determines the shape of the curve - concave (immediate diminishing returns) or S-shaped (threshold effects).

Critical Power Values

Power = 1.0 (Concave/ADBUG):

Effect│╱──────────────
      │  (immediate diminishing returns)
      │
      │_______________
         Spend →

Best customers respond first, continuous declining ROI

Power = 1.5-1.8 (Gentle S-Shape):

Effect│      ╱────────
      │    ╱
      │   ╱
      │  ╱
      │_╱____________
         Spend →

Mild threshold, moderate acceleration, then saturation

Power = 1.8-2.0 (Strong S-Shape):

Effect│       ╱──────
      │      │
      │     ╱
      │    │
      │___╱_________
         Spend →

Clear threshold, strong acceleration, then saturation

Selecting Power

Channel Characteristic	Recommended Power
Audience sorting (best first)	1.0
Frequency fatigue	1.0
Search intent (high intent first)	1.0
Needs frequency (recall requires repetition)	1.6-2.0
Threshold effects	1.6-2.0
Viral potential	1.8-2.0
Brand building	1.6-2.0

Business Interpretation

Power = 1.0 (Pure Concave) - Use for:

• Paid search (best keywords first)

• Email (best customers open first)

• Price promotions (price-sensitive shoppers first)

• Retargeting (warmest audiences first)

Power = 1.6-2.0 (S-Shaped) - Use for:

• Brand TV (need frequency for recall)

• New product launches (awareness building)

• Social viral campaigns (engagement threshold)

• Sponsorships (credibility building)

Parameter Interaction Examples

Low Alpha + Low Power (α=0.3, power=1.0)

Early, immediate diminishing returns Example: Small email list, quick fatigue

Low Alpha + High Power (α=0.3, power=1.8)

Early threshold with rapid acceleration Example: Local market TV, small but responsive audience

High Alpha + Low Power (α=0.8, power=1.0)

Late but immediate diminishing returns Example: Mature paid search with large keyword inventory

High Alpha + High Power (α=0.8, power=1.8)

Late threshold, sustained growth Example: National brand TV, large potential reach

Practical Examples

Example 1: Digital Display Campaign

Curve: ADBUG_ATAN with α=0.4, power=1.0

What this means:

• Alpha = 0.4: Saturation begins around $10K weekly

• Power = 1.0: Immediate diminishing returns, best audiences first

Business insight: "Display shows immediate effectiveness but declining ROI. Best audiences exhaust around $10K/week. Cap spend around $15K/week."

Example 2: National TV Brand Campaign

Curve: ICP_ATAN with α=0.8, power=1.9

What this means:

• Alpha = 0.8: Saturation at very high spend levels

• Power = 1.9: Needs threshold investment, then rapid acceleration

Business insight: "TV requires ~$50K/week minimum for impact. Sweet spot $50K-$150K/week. Above $200K/week, severe diminishing returns."

How Model Coefficients Change

Without Curve: Sales = β₀ + β₁ × TV_Spend Interpretation: Constant ROI per dollar

With ATAN Curve: Sales = β₀ + β₁ × TV_Spend|ICP_ATAN_a0.5_power1.8 Interpretation: β₁ on transformed value, ROI varies with spend level

This is why curve-based models enable optimization - they capture how ROI changes with spend.

Parameter Selection Process

1. Select variable to transform

2. Choose curve type (ICP or ADBUG)

3. Select formula (ATAN recommended)

4. Run tests (MixModeler tests all combinations)

5. Review results (sorted by R² increase)

6. Select winner based on:

   • Statistical fit (R² increase, p-value)

   • Business logic (does shape make sense?)

   • Coefficient sign (positive as expected?)

Common Mistakes

❌ Using S-shape (power > 1.5) for performance channels ❌ Using concave (power=1.0) for brand TV ❌ Alpha too low for high-reach channels ❌ Alpha too high for small audiences

✅ Best practice: Let Curve Testing guide you, validate with business intuition

Key Takeaways

• Alpha controls WHERE saturation occurs

• Power controls SHAPE (1.0 = concave, >1.5 = S-shape)

• Both must align with channel behavior

• Test systematically, validate business logic

• ROI varies with spend level in curve-based models