Threshold Regression Models

1. Threshold Regression Models

2. 2 Outline An example to demonstrate the usefulness of the first-hitting time based threshold regression (TR) model. Introduction of the TR model Connections with the PH model Semi-parametric TR model Simulations

3. 3 A non-proportional hazard example: Time to infection of kidney dialysis patients with different catheterization procedures(Nahman et al 1992, Klein & Moesberger 2003) Surgical group: 43 patients utilized a surgically placed catheter Percutaneous group: 76 patients utilized a percutaneous placement of their catheter The survival time is defined by the time to cutaneous exit-site infection.

4. 4 Kaplan-Meier Estimate versus PH Cox Model

5. 5 Weibull versus Lognormal

6. 6 Loglogistic versus Gamma

7. 7 Kaplan-Meier Estimate versus First-hitting-time based Threshold Regression Model

8. 8 Outline Introduction of First Hitting Time Threshold Regression Parametric and Semi-parametric Models Comparison with PH Models

9. 9 First-hitting Time Based Threshold Regression:Modeling Event Times by a Stochastic Process Reaching a Boundary (Lee & Whitmore 2006, Statistical Sciences) Example: Equipment Failure: Equipment fails when its cumulative wear first reaches a failure threshold. Question: What is the influence of ambient temperature on failure?

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15. 15 First Hitting Time (FHT) Models

16. 16 Parameters for the FHT Model Model parameters for the latent process Y(t) : Process parameters: ? = (m, s2), where m is the mean drift and s2 is the variance Baseline level of process: Y(0) = y0 Because Y(t) is latent, we set s2 = 1.

17. 17 Likelihood Inference for the FHT Model The likelihood contribution of each sample subject is as follows. If the subject fails at S=s: f (s | y0, m) = Pr [ first-hitting-time in (s, s+ds) ] If the subject survives beyond time L: 1- F (L | y0 ,m) = Pr [ no first-hitting-time before L ]

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21. 21 References

22. 22 Threshold Regression Interpretation of PH functions Most survival distributions can be related to hitting time distributions for some stochastic processes. Families of PH functions can be generated by varying time scales or boundaries of a TR model The same family of PH functions can be produced by different TR models. Simulation studies: both TR model and PH hold simultaneously, based on standard Brownian motion with variation of time scale. (Julia Batishev�s presentation in sec 2 of Tract 2 on July 3rd)

23. 23 Threshold Regression If Y(t) has a Wiener Process, then the first hitting time S has an inverse Gaussian distribution.

24. 24 Semi-parametric Threshold Regression(Joint work with Z. Yu and W. Tu)

25. 25 Semi-parametric TR using Regression Spline Use linear link for covariates X1, �, Xp For covariate Z, consider the nonparametric function q(z) as a linear combination of a set of basis functions Bj(z). q(z) = Sj bj Bj(z). Select the smoothing parameter and the number of knots.

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27. 27 Semi-parametric TR using Penalized Spline In addition to regression spline, we also consider a cubic spline approach with penalty on the second derivative of the nonparametric function

28. 28 Cross Validation

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30. 30 Simulation Results

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35. 35 symbols A b c d e f g h I j k l m n o p q r s t u v w x y z S X W # Q

36. 36 Analyzing Longitudinal Survival Data Using Threshold Regression: Comparison with Cox Regression

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