Using Neural Networks to Predict Claim Duration in the Presence of Right Censoring

and Covariates

David SpeightsSenior Research Statistician

HNC Insurance Solutions

Irvine, California

Session CPP-53

Presentation Outline

• Introduction to Neural Networks

• Introduction to Survival Analysis

• Neural Networks with Right Censoring

• Simulated Example

• Predicting Claim Duration

Introduction to Neural NetworksMotivation

• Complex Classification– Character Recognition

– Voice Recognition

• Humans have no trouble with these concepts– We can read even distorted documents

– We can recognize voices over poor telephone lines.

• Attempt to model human brain

Introduction to Neural NetworksConnection to Brain Functionality

• Brain – made up of millions of neurons sending signals to the

body and each other

• Neural Networks – collection of “neurons” which send “signals” to

produce an output

Introduction to Neural NetworksCommon Representation

. . .

. . .

X1 X2 XP

Y

P predictors (inputs)

1 Hidden Layer with M Neurons

1 output

1 2 M

Introduction to Neural Networks Architecture of the ith Neuron

Represents a neuron in the brain

X1

X2XP

...

O=bi0 + bi1X1 + … + bipXp

s(O)

S is a function on the interval (0,1) representing the strength of the output

0

1

s

O

Activation Function

Introduction to Neural Networks Connection to Multiple Regressions

• Similarities– Both describe relationships between variables

– Both can create predictions

• Differences– Function describing the relationships is more complex

– Response variables are typically called outputs

– Predictor variables are typically called inputs

– Estimating the parameters is usually called training

Introduction to Neural NetworksFunctional Representation

Y = f(X1, …, Xp) + error

• Multiple Linear Regression – f() = linear combination of regressors– Forced to model only specified relationships

• Neural Network– f() = nonlinear combination of regressors– Can deal with nonlinearities and interactions without special

designation

Introduction to Neural NetworksFunctional Specification

• For a neural network f() is written

• Here g and s are transformation functions specified in advance

))((),...,(

Equation RegressionLinear Multiple

10

101

p

kkjk

M

jjp

XsgXXf

Introduction to Survival AnalysisWhat is Survival Analysis

• Used to model time to event data (example: time until a claim ends)

• Usually represented by (1) right skewed data (2) multiplicative error structure (3) right censoring

• Common in cancer clinical trials, component failure analysis, and AIDS data analysis among other examples

Introduction to Survival AnalysisNotation

• T1, ..., Tn - independent failure times with distribution F

and density function f

• C1, ..., Cn - independent censoring times with distribution

G and density function g

• Yi = min(Ti,Ci) - observed time

• i = I(Yi = Ti) - Censoring indicator

• Xi = (Xi1, ..., Xip) - vector of known covariates

associated with the ith individual

Introduction to Survival Analysis Likelihood Analysis (Parametric Models)

• (Yi, i, Xi) i=1, …, n , independent observations

• Likelihood written

n

iiiiXYfL

1

)|,()(

• f(Y,|X)=[f (Y|X)(1-G(Y|X))][g(Y|X)(1-F (Y|X))]

n

i

i

ii

i

iiXYFXYfLL

1

1

2))|(1()|()(

• Here L2 does not depend on

Neural Networks with Right CensoringModel Specification

• Neural Network Model

• Here has distribution function F and density f• = {0, …, p, 1, …, p}

• The likelihood isi

iin

i

i

iixmlpT

FxmlpT

fLL

1),()log(

1),()log(1

),(1

2

),(

)'()log(1

0

xmlp

xsTM

jjj

Neural Networks with Right Censoring Fitting Neural Networks without Censoring

• estimated by minimizing squared error

n

iii

n

iii

xCxmlpYC11

2),(),()log()(

n

i

ii

n

i

ixmlpiY

xmlpYnL

eL

1

2

2

1

2),()log(

21

2

),()log(21

)2log(2

)),(log(

21

),(

• Ifis normal minimizing squared error same as maximizing the likelihood.

Neural Networks with Right CensoringFitting Neural Networks without Censoring

• Gradient decent algorithm for estimating ),(

1:1:: iijiijijxC

• Algorithm updated at each observation• is known as the learning rate

• j:0=j-1:n

• Known as back-propagation algorithm• To generalize to right censored data, replace C() with

the likelihood for censored neural networks.

Neural Networks with Right CensoringFitting Neural Networks with Censoring

• Step 1 - Estimating – Fix and pass through data once using

• Step 2 - Estimating – fix at end of pass through data

– iterate until |j-j-1|<using Newton-Raphson algorithm

),( 1:1:: jijiijij

L

),(

),(

1

2

1

1

j

j

jj L

L

Neural Networks with Right CensoringFitting Neural Networks with Censoring

• With highly parameterized neural networks we risk over fitting

• We need to design the fitting procedure to find a good fit to the data

Neural Networks with Right CensoringFitting Neural Networks with Censoring

• The negative of the likelihood is calculated on both sets of data at the same time.

Negative Likelihood

75% Training Data 25% Testing Data

Parameter Estimates

Training Cycles Training Cycles

Neural Networks with Right CensoringFitting Neural Networks with Censoring

• Potential drawbacks to neural networks– Hard to tell the individual effects of each predictor

variable on the response.

– Can have poor extrapolation properties

• Potential Gains from neural networks– Can reduce preliminary analysis in modeling

• discovery of interactions and nonlinear relationships becomes automatic

– Increases predictive power of models

Neural Networks with Right CensoringFitting Neural Networks with Censoring

• True Time Model : log(t) = x2 + 0.5• Censoring Model: log(c) = 0.25 + x2 + 0.5• x ~ U(-3,3)

• ~ N(0,1)

• Censored if c < t

• ~ 35% censoring

• 3 node neural network fit

Simulated Example

log P

redic

tion/

Actua

l

-2

-1

0

1

2

3

4

5

6

7

8

9

10

X-3 -2 -1 0 1 2 3

• Scatter are true times versus x

• Solid line represents NN fit to data

Simulated Example

Predicting Claim Duration

• Predictor Variables– NCCI Codes

• Body Part Code

• Injury Type

• Nature of Injury

• Industry Class Code

– Demographic Information• Age

• Gender

• Weekly Wage

• Zip Code

• Response Variable– Time from report until the

claim is closed

Predicting Claim Duration

• Ratio of prediction to actual duration on log10 scale

• Difficult to represent open claim results

Open Claim Closed Claim

Conclusions

• Provides an intuitive method to address right censored data with a neural network

• Allows for more flexible mean function

• Can be used with many time to event data situations