1. Using Neural Networks to Predict Claim Duration in the Presence of Right Censoring and Covariates David Speights Senior Research Statistician HNC Insurance Solutions Irvine, California Session CPP-53

2. Presentation Outline Introduction to Neural Networks Introduction to Survival Analysis Neural Networks with Right Censoring Simulated Example Predicting Claim Duration

3. Introduction to Neural Networks Motivation 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

4. Introduction to Neural Networks Connection 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

5. Introduction to Neural Networks Common Representation... X1X1 X2X2 XPXP Y P predictors (inputs) 1 Hidden Layer with M Neurons 1 output 12 M

6. Introduction to Neural Networks Architecture of the i th Neuron Represents a neuron in the brain X1X1 X2X2 XPXP... O=b i0 + b i1 X 1 + … + b ip X p s(O) S is a function on the interval (0,1) representing the strength of the output 0 1 s O Activation Function

7. 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

8. Introduction to Neural Networks Functional Representation Y = f(X 1, …, X p ) + 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

9. Introduction to Neural Networks Functional Specification For a neural network f() is written Here g and s are transformation functions specified in advance

10. Introduction to Survival Analysis What 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

11. Introduction to Survival Analysis Notation T 1,..., T n - independent failure times with distribution F and density function f C 1,..., C n - independent censoring times with distribution G and density function g Y i = min(T i,C i ) - observed time  i = I(Y i = T i ) - Censoring indicator X i = (X i1,..., X ip ) - vector of known covariates associated with the i th individual

12. Introduction to Survival Analysis Likelihood Analysis (Parametric Models) (Y i,  i, X i ) i=1, …, n, independent observations Likelihood written f  (Y,  |X)=[f  (Y|X)(1-G(Y|X))]  [g(Y|X)(1-F  (Y|X))]  Here L 2 does not depend on 

13. Neural Networks with Right Censoring Model Specification Neural Network Model Here  has distribution function F  and density f   = {  0, …,  p,  1, …,  p } The likelihood is

14. Neural Networks with Right Censoring Fitting Neural Networks without Censoring  estimated by minimizing squared error If  is normal minimizing squared error same as maximizing the likelihood.

15. Neural Networks with Right Censoring Fitting Neural Networks without Censoring Gradient decent algorithm for estimating  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.

16. Neural Networks with Right Censoring Fitting 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

17. Neural Networks with Right Censoring Fitting Neural Networks with Censoring With highly parameterized neural networks we risk over fitting

18. We need to design the fitting procedure to find a good fit to the data Neural Networks with Right Censoring Fitting Neural Networks with Censoring

19. The negative of the likelihood is calculated on both sets of data at the same time. Negative Likelihood 75% Training Data25% Testing Data Parameter Estimates Training Cycles Neural Networks with Right Censoring Fitting Neural Networks with Censoring

20. 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 Censoring Fitting Neural Networks with Censoring

21. True Time Model : log(t) = x 2 + 0.5   Censoring Model: log(c) = 0.25 + x 2 + 0.5   x ~ U(-3,3)      ~ N(0,1) Censored if c < t ~ 35% censoring 3 node neural network fit Simulated Example

22. Scatter are true times versus x Solid line represents NN fit to data Simulated Example

23. 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

24. Predicting Claim Duration Ratio of prediction to actual duration on log 10 scale Difficult to represent open claim results

25. 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