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Southwest Actuarial Forum. Interactions and Saddles. Serhat Guven, FCAS, MAAA June 10, 2011. Agenda. Background Interactions Saddles Validation. Simple Model: Age + Gender. Male. Female. Youthful. β 0 + β Y. β 0 + β Y + β F. Adult. β 0. β 0 + β F. Mature. β 0 + β M.

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southwest actuarial forum

Southwest Actuarial Forum

Interactions and Saddles

Serhat Guven, FCAS, MAAA

June 10, 2011

agenda
Agenda
  • Background
  • Interactions
  • Saddles
  • Validation
background

Simple Model: Age + Gender

Male

Female

Youthful

β0+ βY

β0+ βY + βF

Adult

β0

β0+ βF

Mature

β0+ βM

β0+ βM + βF

Seniors

β0+ βS

β0+ βS + βF

5 Parameters

Background

SIMPLE MODEL

  • Relationship between rating levels of one factor is constant for all levels of other rating variables
  • Assume two rating variables
    • Age: Youthful, Adult (Base), Mature, Senior
    • Gender: Male (Base), Female
background1

Simple Model: Age + Gender

Male

Female

Youthful

β0+ βY

β0+ βY + βF

Adult

β0

β0+ βF

Mature

β0+ βM

β0+ βM + βF

Seniors

β0+ βS

β0+ βS + βF

5 Parameters

Background

SIMPLE MODEL

Difference between male and female is the same regardless of age

interactions

Simple Model: Age + Gender

Male

Female

Youthful

β0+ βY

β0+ βY + βF

Adult

β0

β0+ βF

Mature

β0+ βM

β0+ βM + βF

Seniors

β0+ βS

β0+ βS + βF

5 Parameters

Interactions

INTERACTION MODEL

  • Relationship between rating levels of one rating factor is different for all levels of another rating variable
  • Assume two rating variables
    • Age: Youthful, Adult (Base), Mature, Senior
    • Gender: Male (Base), Female

Interaction: Age + Gender+Age.Gender

8 Parameters

background2
Background

INTERACTION MODEL

Interaction: Age + Gender+Age.Gender

Youthful males are significantly higher than youthful females – resulting structure is still volatile

8 Parameters

background3
Background
  • Why are interactions present
    • Because that's how the factors behave
    • Because the multiplicative model can go wrong at the edges
      • 1.5 * 1.4 * 1.7 * 1.5 * 1.8 * 1.5 * 1.8 = 26!
  • Interaction challenges
    • Identification
      • Given n factors there are n choose 2 possible interactions to study
    • Simplification
      • Once an interaction has been identified the structure needs to be simplified to avoid overfitting
interactions1

Gender

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Interactions
  • Age x Gender
    • Main effects construction
interactions2
Interactions
  • Imbalances within the cells exists – suggesting some form of interaction
interactions3

Gender

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Age

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Interactions
  • Age x Gender
    • Full Interaction
interactions4
Interactions
  • Rating Area x Vehicle Value
    • Interaction needs to be simplified

Vehicle Value

Vehicle Value

Vehicle Value

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Rating Area

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Rating Area

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Rating Area

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interactions5
Interactions
  • Rating Area x Vehicle Value
    • Simplification emphasizes interaction strength
saddles
Saddles
  • Quadrant saddle: revisiting an simple main effect model
saddles1
Saddles
  • Quadrant saddle: interaction terms twist the paper
saddles6
Saddles
  • Transforming predictors into single parameter variates

Response

Factor Levels

saddles7

b

Saddles
  • Transforming predictors into single parameter variates

Response

Factor Levels

saddles8

Simple Model: Age + Gender

Variate Model: a1*bAge+ a2*bGender

Male

Male

Female

Female

Youthful

Youthful

β0+ a1*βY

β0+ βY

β0+ βY + βF

β0+ a1*βY + a2*βF

Adult

Adult

β0

β0

β0+ a2* βF

β0+ βF

Mature

Mature

β0+ a1*βM

β0+ βM

β0+ βM + βF

β0+ a1*βM + a2*βF

Seniors

Seniors

β0+ βS

β0+ a1*βS

β0+ βS + βF

β0+ a1*βS + a2* βF

3 Parameters

5 Parameters

Saddles
  • Parameterization
  • Fitted values from both structures are the same
  • Variates
    • X-Values are the beta parameters from the simple model
    • Coefficient should be 1.000
saddles9
Saddles
  • Parameterization
    • Nonlinear terms introduced via interaction among variates

Full Saddle: a1*bAge + a2*bGender + a3*bAge * bGender

saddles10
Saddles
  • Advantages
    • Non linear variate parameter
      • Easier to validate
      • Simpler shape
    • Useful in identifying interactions in more volatile structures
      • High dimensional factors
      • N-way interaction structures
      • Severity models
  • Disadvantages
    • Difficult to detect interactions when a factor is not a main effect
saddles11
Saddles
  • Rating Area x Vehicle ValueRevisited
saddles12
Saddles
  • Rating Area x Vehicle ValueRevisited
saddles13
Saddles
  • Rating Area x Vehicle ValueRevisited
saddles14
Saddles
  • Driver Age x Vehicle Age
validation
Validation
  • Model Comparison
    • Auto frequency: Out of sample
validation1
Validation
  • Model Comparison
    • Auto frequency: Out of time
validation2
Validation
  • Model Comparison
    • Renewals: Out of sample
conclusion
Conclusion
  • Interactions are an important part of the model creation process
  • Interactions have their own challenges
    • Identification
    • Simplification
    • Dimensionality
  • Effort needed in simplifying and validating identified interaction
    • Saddles use the framework of variate vectors in the design matrix to quickly simplify and validate new interaction terms