Proportional Hazard Models Bo Gallant Introduction Presentation Topics

Proportional Hazard Models Bo Gallant

Introduction Presentation Topics Introduction • Background information • Main assumptions Model Structure • PH function • Survival function Partial Likelihoods • • Page 2 Without and with tied observations explanation Without tied observations equations With tied observations equations Newton-Raphson equation Model Structure Partial Likelihood

Introduction Model Structure Partial Likelihood Introduction Background Information • Non-parametric: don’t need to know distribution • Constant hazard function over time: similar to Exponential • Hazard function of different individuals are proportional and independent of time Main Assumptions • Hazard function of different individuals are proportional and independent of time PH Model Background Equation Fix Negative Proportional Hazard Function Constant and independent of time Page 3

Introduction Model Structure Proportional Hazard Function Survival Function General Survival Equation: PH Survival Equation: Page 4 Model Structure Partial Likelihood

Introduction Model Structure Partial Likelihoods Types of PL • 2 Types of Partial Likelihoods Equations: • Equation for data sets Without Tied Observations • Equations for data sets With Tied Observations Reasons? • PL equation depends on order of failures • For Tied observations, order is unknown • Example Without Tied Observations • Page 5 5 Individuals Without Tied Failures: With Tied Observations • 5 Individuals with 3 Tied Failures: Individual (i) Time (t) 1 5 2 10 2 5 3 15 3 5 4 20 4 10 5 25 5 15

Introduction Model Structure Partial Likelihoods – Without Tied Observations Probability Failure of an Individual at time t(i) R(t(i)): Risk set which is everyone at Risk at time t(i) Partial Likelihood Page 6

Introduction Model Structure Partial Likelihoods – Without Tied Observations • 5 Individuals Without Tied Failures: Individual (i) Time (t) 1 5 2 10 3 15 4 20 5 25 Partial Likelihood Page 7 Hazard functions:

Introduction Partial Likelihood - Tied Observations • 5 Individuals with 3 Tied Failures: Individual (i) Time (t) 1 5 2 5 3 5 4 10 5 15 Hazard functions: 1) Breslow Estimation (1974) R(t(i)): Risk set: everyone at Risk at time t(i) Page 8 m(i) : # individual fail at t(i) Model Structure Partial Likelihood

Introduction Partial Likelihood - Tied Observations 2) Efron Estimation (1977) R(t(i)): Risk set: everyone at Risk at time t(i) Page 9 m(i) : # individual fail at t(i) Model Structure Partial Likelihood

Introduction Model Structure Partial Likelihood Newton-Raphson Algorithm Follow these Steps to find Estimates: Only 1 Time Check Every Time Page 10