Other Survival Models. (1) Non-PH models. We briefly discussed the non-proportional hazards (non-ph) model

Size: px
Start display at page:

Download "Other Survival Models. (1) Non-PH models. We briefly discussed the non-proportional hazards (non-ph) model"

Transcription

1 Other Survival Models (1) Non-PH models We briefly discussed the non-proportional hazards (non-ph) model λ(t Z) = λ 0 (t) exp{β(t) Z}, where β(t) can be estimated by: piecewise constants (recall how); splines, which give a more smooth ˆβ(t) (Gray, 1992); other methods. More broadly, non-ph can mean any survival model that is not the Cox model. 1

2 (2) Parametric regression models First we will discuss some parametric regression models, some are in fact PH models. We have (X i, δ i, Z i ), i = 1,..., n, i.i.d., where X i = min(t i, C i ), Z i = (Z i1, Z i2,..., Z ip ). If we assume that T i follows an exponential distribution with a parameter λ i = exp(β 0 + β Z i ), then we have a exponential regression model: T i Z i Exp(e β 0+β Z i ). Since the hazard functions are constants, these are PH models. Q: what is the baseline hazard? How do we estimate the unknown parameters? (hint: see separate notes on parametric regressions) 2

3 We may also assume that T i Z i follows a Weibull distribution. Recall that the hazard function for a Weibull distribution is: λ(t) = κ λ t (κ 1). The Weibull regression model assumes that given covariates Z i, λ i = exp(β 0 +β Z i ) as in the exponential regression. The conditional hazard function is then λ i (t) = κ exp(β 0 + β Z i ) t (κ 1). Note that the shape parameter κ is the same for all subjects; Is this a PH model? (what is the baseline hazard?) How do we estimate the unknown parameters κ, β 0 and β? (hint: see separate notes on parametric regressions) 3

4 Comparison of Exponential with Kaplan-Meier We can see how well the Exponential model fits by comparing the survival estimates for males and females under the exponential model, i.e., P (T t) = e ( ˆλ z t), to the Kaplan- Meier survival estimates: S 0. 7 u r 0. 6 v i 0. 5 v 0. 4 a l L e n g t h o f S t a y ( d a y s ) 4

5 Comparison of Weibull with Kaplan-Meier We can see how well the Weibull model fits by comparing the survival estimates, P (T t) = e ( ˆλ z tˆκ), to the Kaplan- Meier survival estimates S 0. 7 u r 0. 6 v i 0. 5 v 0. 4 a l L e n g t h o f S t a y ( d a y s ) Which do you think fits better? 5

6 Comparison of Models Exponential Regression: λ(t Z) = exp(β 0 + β 1 Z β p Z p ) Weibull Regression: RR = exp(β 1 Z β p Z p ) λ(t Z) = κ t κ 1 exp(β 0 + β 1 Z β p Z p ) RR = exp(β 1 Z β p Z p ) Proportional Hazards Model: λ(t Z) = λ 0 (t) exp(β 1 Z β p Z p ) RR = exp(β 1 Z β p Z p ) 6

7 Remarks Exponential model is a special case of the Weibull model with κ = 1 Exponential and Weibull models are both special cases of the Cox PH model. If either the exponential model or the Weibull model is valid, then these models should be more efficient (somewhat smaller variances of the parameter estimates) than the semiparametric PH model. This is because they assume a particular form for λ 0 (t), with only one or two unknown parameters, rather than leaving it as an infinite dimensional parameter and estimating it at each distinct failure time. Note however, that the Cox partial likelihood estimator is semiparametrically efficient, meaning: as more and more parameters are used to model λ 0 (t), the limit of the (asymptotic) variance of the estimated β equals the variance of the (asymptotic) variance of the partial likelihood estimator. 7

8 (3) Accelerated Failure Time Model The accelerated failure time (AFT) model is a linear regression model with log(t ) as the response: where log(t i ) = β Z i + ɛ i log(t i ) is the log of the survival time; (why take log?) β is the vector of regression parameters including intercept (and Z includes a 1 with a slight change of notation here); ɛ i is a random error term. Note that the AFT model can be also written T i = T 0 exp(β Z i ), where T 0 has the same distribution as e ɛ. 8

9 Write φ = exp( β Z). It can be shown: S(t Z) = S 0 (φ t) That is, the effect of covariates is to accelerate (or decelerate) the time-scale. If S i (M i ) = 0.5, then S 0 (φ i M i ) = 0.5. This means M 0 = φ i M i, or: M i = M 0 /φ = M 0 exp(β Z i ) 9

10 We will first discuss the parametric AFT model, which is often written as: where log(t i ) = β Z i + σɛ i ɛ i is a random error term with known distribution; σ is a scale factor. By choosing different distributions for ɛ, we can obtain different parametric regression models: Exponential Weibull Logistic Log-logistic Normal Log-normal These can be fitted using the R function survreg(). 10

11 Both the Exponential and the Weibull regression models discussed earlier can be written as an AFT model, if we choose the proper distribution for ɛ. For the Exponential Model: log(t i ) = β ez i + ɛ i, where ɛ follows an extreme value distribution, i.e. e ɛ follows a unit exponential distribution. So β = β e (including intercept), σ = 1. For the Weibull Model: log(t i ) = σβ wz i + σɛ i, where ɛ again follows an extreme value distribution, and σ = 1/κ. So β = σβ w (including intercept). 11

12 AFT model with normal error (Log-normal regression) Here we let ɛ N(0, 1). log(t i ) = β Z i + σɛ i This is a very appealing model, because it is the same as the linear regression model with normal error, where the response is the log of the survival time. Therefore the interpretation of the model is straightforward and familiar. Recall that the distribution of T i is called log-normal. This family of distributions have non-monotone hazards. How would you fit the model? If there are no censored observations, how would you fit the model? 12

13 log-logistic regression model Here we assume that ɛ has a logistic distribution with density f(ɛ) = e ɛ (1 + e ɛ ) 2. If ɛ has a logistic distribution, so does log(t i ) (with non-zero mean). Then T i has a log-logistic distribution. Log-logistic can also have non-monotone hazards. In addition, as t, the hazard goes to zero. The log-logistic model has a simple survival function (Ex.) S(t Z) = (λt) γ where γ = 1/σ and λ = exp( β Z). 13

14 After some algebra, it can be shown that log where β = β/σ. S(t Z) 1 S(t Z) = β Z γ log(t) If t is fixed, the above is a logistic regression model (why?). Since S(t) is the probability of surviving to time t, S(t)/{1 S(t)} is the odds of surviving to time t. Furthermore, for individuals i and j, S i (t) 1 S i (t) = c S j (t) ij 1 S j (t) for all t, where c ij = exp{β (Z i Z j )}. Therefore the log-logistic model is also called a proportional odds (PO) model, since c ij does not depend on t. 14

15 It can also be shown that, as t, the hazard ratio λ i (t)/λ j (t) 1. Therefore the log-logistic regression (PO) model can be used to model attenuating hazard ratios, which provides a useful alternative to the PH model. When does Proportional hazards = AFT? We have seen before that the Weibull regression model (which includes the Exponential regression model as a special case), is both a PH model and an AFT model. It turns out that the Weibull (and Exponential) regression model is the only one for which the accelerated failure time and proportional hazards models coincide. See Chan et al. (2018) for examples with R 2 values. 15

16 (4) Semiparametric AFT model log(t ) = β Z + ɛ If we leave the distribution of ɛ unspecified, then it leads to the semiparametric AFT model. Inference under this model is much more difficult, since there is not immediately a likelihood for this model. Rank-based estimating equations were proposed in the literature, but numerical solutions to these equations are challenging. Semiparametric AFT model has rarely (if ever) been used in practice. 16

17 (5) Semiparametric transformation model Replace the log transformation on the survival times to be any unspecified monotone transformation g( ): g(t ) = β Z + ɛ and this leads to the semiparametric transformation model. Here ɛ still comes from a parametric distribution. In fact, when ɛ follows the extreme value distribution, the above is equivalent to the PH model. A useful family of distributions for ɛ is the G ρ family of Harrington and Fleming. Inference under this model is like under the PH model, using the nonparametric MLE (NPMLE). When ɛ is logistically distributed, this is the semiparametric proportional odds model. 17

18 (6) Additive hazards model This is a class of models that is gaining popularity (Aalen 1980, 1989): λ i (t Z) = λ 0 (t) + β Z(t), where λ 0 (t) is an unspecified baseline hazard. What do you think of the model? A: Indeed one needs to make sure that the hazard is not negative. Inference for β is based on the estimating equation (Lin and Ying, 1994) 0 = = n i=1 n i=1 0 0 Z i (t)dm i (t) Z i (t){dn i (t) Y i (t)λ 0 (t) Y i (t)β Z i (t)dt}. And similar to the Breslow estimate, if β were known, Λ 0 (t) = t 0 λ 0(s)ds can be estimated by t n i=1 ˆΛ 0 (t) = {dn i(u) Y i (u)β Z i (u)du} n i=1 Y. i(u) 0 Plugging this estimate back into the above estimating equation (this is known as profiling out Λ 0 ), we have after some algebra: 18

19 U(β) = where Z(t) = This gives [ n ˆβ = i=1 0 n i=1 0 n Z l Y l (t)/ l=1 l=1 {Z i Z(t)}{dN i (t) Y i (t)β Z i (t)dt}, n Y l (t). Y i (t){z i Z(t)} ] 1 [ n 2 dt i=1 Notice that the above is no longer just rank based. 0 {Z i Z(t)}dN ] i (t). The cumulative baseline hazard function Λ 0 (t) is then estimated by ˆΛ 0 (t) = t 0 n i=1 dn i(u) n i=1 Y i(u) ˆβ t 0 Z(u)du. Note that ˆΛ 0 (t) can be negative. Lin and Ying (1994) suggested to use a modified ˆΛ 0(t) = max 0 s t ˆΛ0 (s). Martingale theory applies so that ˆβ is asymptotically normal with variance estimated by a sandwich of the form A 1 BA 1. R package timereg fits this model. 19

20 Part of the reason the additive hazards model became popular (over the PH model) is the following: The PH models, unlike the normal linear regression models, are not nested. This is sometimes called non-collapsible (in causal inference). This is also true for other non-linear models like logistic regression. That is, when adjusting or not adjusting for covariate(s), at most one of the two models might be valid (Lancaster and Nickell 1980, Gail et al. 1984, Struthers and Kalbfleisch 1986, Bretagnolle and Huber-Carol 1988, Anderson and Fleming 1995, Ford et al. 1995). Suppose that we have λ(t z 1, z 2 ) = λ 0 (t) exp(β 1z 1 + β 2z 2 ), where z 1, z 2 are vectors of covariates. This implies that S(t z 1, z 2 ) = exp{ Λ(t z 1, z 2 )} = exp{ Λ 0 (t)e β 1z 1 +β 2z 2 }. Then S(t z 1 ) = S(t z 1, z 2 )dg 2 (z 2 z 1 ), where G 2 is the conditional distribution function of Z 2 given Z 1. This gives λ(t z 1 ) = S(t z 1) /S(t z 1 ) t S(t z1, z 2 ) = dg 2 (z 2 z 1 )/ t S(t z 1, z 2 )dg 2 (z 2 z 1 ) = λ 0 (t)e β 1z 1 e β 2z 2 exp{ Λ 0 (t)e β 1z 1 +β 2z 2 }dg 2 (z 2 z 1 ). exp{ Λ0 (t)e β 1 z 1+β 2 z 2 }dg2 (z 2 z 1 ) 20

21 It is clear that unless the ratio of the two integrals in the last line above can be written as a function of t multiplied by a function of z 1, the PH assumption will be violated. One such example is when Z 2 is the logarithm of a positive stable random variable (see e.g. Feller 1966, Hougaard 1986), then with or without z 2, the model will always be PH, though the estimated coefficients of Z 1 will have changed. [Read] Example 1 Let Λ 0 (t) = t, β 2 = 1. Denote ξ = exp(β 1Z 1 ). Let Z 1 and Z 2 be independent, and Z 2 = log α, where α has a positive stable distribution. A distribution is called stable if for each n and X 1, X 2,..., X n i.i.d. from this distribution, there exists a constant c n, with D(X 1 + X X n ) = D(c n X 1 ), where D(X) means the distribution of X. It turns out that the only constants possible for c n are n 1/γ, γ (0, 2]. The stable distributions with finite variance are the normal, γ = 2, and the degenerate distributions, γ = 1. The stable distributions on the positive numbers have γ (0, 1] and apart from scale factors have Laplace transform E(e sx ) = exp( s γ ), s 0. Then from the calculation above, S(t z 1, z 2 ) = exp( αξt), S(t z 1 ) = 0 e αξt dg γ (α) = exp( ξ γ t γ ), where G γ ( ) is the distribution function of α. Therefore λ(t z 1 ) = γt γ 1 ξ γ = γt γ 1 e γβ 1z 1, which follows a PH model, but the coefficient of z 1 is now γβ 1 instead of β 1. 21

22 In general, one is almost certain to end up with non-ph rather than PH models after deleting covariates. Below is one example here from Ford, Norrie and Ahmadi (1995). Ford, Norrie and Ahmadi (1995) also provided examples in which adding covariates makes a PH model into a non-ph one. Example 2 Let Λ 0 (t) = t, β 2 = 1. Denote ξ = exp(β 1Z 1 ). Let Z 1 and Z 2 be independent, and Z 2 be the logarithm of an Exp(1) random variable. Then S(t z 1 ) = = 0 0 exp( te β 1z 1 +z 2 ) exp( e z 2 )e z 2 dz 2 exp{ (tξ + 1)e z 2 }e z 2 dz 2 = e (tξ+1) /(tξ + 1), and this is a non-ph model. The fact that the Cox model is non-collapsible has important implications: 1. If an important covariate is missed, the estimated effects of (other) covariates including treatment is biased towards zero, which can lead to less efficient (i.e. powerful) tests (Lagakos and Schoenfeld, 1984). This is also known for other non-linear models like logistic regression. 2. Another side of the coin of the above is: even in randomized trials, adjusting for important covariates can lead to more efficient tests of the treatment effect. 3. Interpretation becomes difficult when there are unobserved heterogeneity or confounders. 22

23 On the other hand, the additive hazards model is collapsible. [Ex.] 23

UNIVERSITY OF CALIFORNIA, SAN DIEGO

UNIVERSITY OF CALIFORNIA, SAN DIEGO UNIVERSITY OF CALIFORNIA, SAN DIEGO Estimation of the primary hazard ratio in the presence of a secondary covariate with non-proportional hazards An undergraduate honors thesis submitted to the Department

More information

Survival Analysis Math 434 Fall 2011

Survival Analysis Math 434 Fall 2011 Survival Analysis Math 434 Fall 2011 Part IV: Chap. 8,9.2,9.3,11: Semiparametric Proportional Hazards Regression Jimin Ding Math Dept. www.math.wustl.edu/ jmding/math434/fall09/index.html Basic Model Setup

More information

Statistics 262: Intermediate Biostatistics Non-parametric Survival Analysis

Statistics 262: Intermediate Biostatistics Non-parametric Survival Analysis Statistics 262: Intermediate Biostatistics Non-parametric Survival Analysis Jonathan Taylor & Kristin Cobb Statistics 262: Intermediate Biostatistics p.1/?? Overview of today s class Kaplan-Meier Curve

More information

Analysis of Time-to-Event Data: Chapter 4 - Parametric regression models

Analysis of Time-to-Event Data: Chapter 4 - Parametric regression models Analysis of Time-to-Event Data: Chapter 4 - Parametric regression models Steffen Unkel Department of Medical Statistics University Medical Center Göttingen, Germany Winter term 2018/19 1/25 Right censored

More information

Cox s proportional hazards model and Cox s partial likelihood

Cox s proportional hazards model and Cox s partial likelihood Cox s proportional hazards model and Cox s partial likelihood Rasmus Waagepetersen October 12, 2018 1 / 27 Non-parametric vs. parametric Suppose we want to estimate unknown function, e.g. survival function.

More information

University of California, Berkeley

University of California, Berkeley University of California, Berkeley U.C. Berkeley Division of Biostatistics Working Paper Series Year 24 Paper 153 A Note on Empirical Likelihood Inference of Residual Life Regression Ying Qing Chen Yichuan

More information

Efficient Semiparametric Estimators via Modified Profile Likelihood in Frailty & Accelerated-Failure Models

Efficient Semiparametric Estimators via Modified Profile Likelihood in Frailty & Accelerated-Failure Models NIH Talk, September 03 Efficient Semiparametric Estimators via Modified Profile Likelihood in Frailty & Accelerated-Failure Models Eric Slud, Math Dept, Univ of Maryland Ongoing joint project with Ilia

More information

Lecture 6 PREDICTING SURVIVAL UNDER THE PH MODEL

Lecture 6 PREDICTING SURVIVAL UNDER THE PH MODEL Lecture 6 PREDICTING SURVIVAL UNDER THE PH MODEL The Cox PH model: λ(t Z) = λ 0 (t) exp(β Z). How do we estimate the survival probability, S z (t) = S(t Z) = P (T > t Z), for an individual with covariates

More information

Definitions and examples Simple estimation and testing Regression models Goodness of fit for the Cox model. Recap of Part 1. Per Kragh Andersen

Definitions and examples Simple estimation and testing Regression models Goodness of fit for the Cox model. Recap of Part 1. Per Kragh Andersen Recap of Part 1 Per Kragh Andersen Section of Biostatistics, University of Copenhagen DSBS Course Survival Analysis in Clinical Trials January 2018 1 / 65 Overview Definitions and examples Simple estimation

More information

STAT331. Cox s Proportional Hazards Model

STAT331. Cox s Proportional Hazards Model STAT331 Cox s Proportional Hazards Model In this unit we introduce Cox s proportional hazards (Cox s PH) model, give a heuristic development of the partial likelihood function, and discuss adaptations

More information

Lecture 22 Survival Analysis: An Introduction

Lecture 22 Survival Analysis: An Introduction University of Illinois Department of Economics Spring 2017 Econ 574 Roger Koenker Lecture 22 Survival Analysis: An Introduction There is considerable interest among economists in models of durations, which

More information

MAS3301 / MAS8311 Biostatistics Part II: Survival

MAS3301 / MAS8311 Biostatistics Part II: Survival MAS3301 / MAS8311 Biostatistics Part II: Survival M. Farrow School of Mathematics and Statistics Newcastle University Semester 2, 2009-10 1 13 The Cox proportional hazards model 13.1 Introduction In the

More information

PENALIZED LIKELIHOOD PARAMETER ESTIMATION FOR ADDITIVE HAZARD MODELS WITH INTERVAL CENSORED DATA

PENALIZED LIKELIHOOD PARAMETER ESTIMATION FOR ADDITIVE HAZARD MODELS WITH INTERVAL CENSORED DATA PENALIZED LIKELIHOOD PARAMETER ESTIMATION FOR ADDITIVE HAZARD MODELS WITH INTERVAL CENSORED DATA Kasun Rathnayake ; A/Prof Jun Ma Department of Statistics Faculty of Science and Engineering Macquarie University

More information

1 Introduction. 2 Residuals in PH model

1 Introduction. 2 Residuals in PH model Supplementary Material for Diagnostic Plotting Methods for Proportional Hazards Models With Time-dependent Covariates or Time-varying Regression Coefficients BY QIQING YU, JUNYI DONG Department of Mathematical

More information

Goodness-of-fit test for the Cox Proportional Hazard Model

Goodness-of-fit test for the Cox Proportional Hazard Model Goodness-of-fit test for the Cox Proportional Hazard Model Rui Cui rcui@eco.uc3m.es Department of Economics, UC3M Abstract In this paper, we develop new goodness-of-fit tests for the Cox proportional hazard

More information

PhD course in Advanced survival analysis. One-sample tests. Properties. Idea: (ABGK, sect. V.1.1) Counting process N(t)

PhD course in Advanced survival analysis. One-sample tests. Properties. Idea: (ABGK, sect. V.1.1) Counting process N(t) PhD course in Advanced survival analysis. (ABGK, sect. V.1.1) One-sample tests. Counting process N(t) Non-parametric hypothesis tests. Parametric models. Intensity process λ(t) = α(t)y (t) satisfying Aalen

More information

Semiparametric Regression

Semiparametric Regression Semiparametric Regression Patrick Breheny October 22 Patrick Breheny Survival Data Analysis (BIOS 7210) 1/23 Introduction Over the past few weeks, we ve introduced a variety of regression models under

More information

Hypothesis Testing Based on the Maximum of Two Statistics from Weighted and Unweighted Estimating Equations

Hypothesis Testing Based on the Maximum of Two Statistics from Weighted and Unweighted Estimating Equations Hypothesis Testing Based on the Maximum of Two Statistics from Weighted and Unweighted Estimating Equations Takeshi Emura and Hisayuki Tsukuma Abstract For testing the regression parameter in multivariate

More information

STAT 331. Accelerated Failure Time Models. Previously, we have focused on multiplicative intensity models, where

STAT 331. Accelerated Failure Time Models. Previously, we have focused on multiplicative intensity models, where STAT 331 Accelerated Failure Time Models Previously, we have focused on multiplicative intensity models, where h t z) = h 0 t) g z). These can also be expressed as H t z) = H 0 t) g z) or S t z) = e Ht

More information

STAT 6350 Analysis of Lifetime Data. Failure-time Regression Analysis

STAT 6350 Analysis of Lifetime Data. Failure-time Regression Analysis STAT 6350 Analysis of Lifetime Data Failure-time Regression Analysis Explanatory Variables for Failure Times Usually explanatory variables explain/predict why some units fail quickly and some units survive

More information

Survival Analysis for Case-Cohort Studies

Survival Analysis for Case-Cohort Studies Survival Analysis for ase-ohort Studies Petr Klášterecký Dept. of Probability and Mathematical Statistics, Faculty of Mathematics and Physics, harles University, Prague, zech Republic e-mail: petr.klasterecky@matfyz.cz

More information

On the Breslow estimator

On the Breslow estimator Lifetime Data Anal (27) 13:471 48 DOI 1.17/s1985-7-948-y On the Breslow estimator D. Y. Lin Received: 5 April 27 / Accepted: 16 July 27 / Published online: 2 September 27 Springer Science+Business Media,

More information

Lecture 5 Models and methods for recurrent event data

Lecture 5 Models and methods for recurrent event data Lecture 5 Models and methods for recurrent event data Recurrent and multiple events are commonly encountered in longitudinal studies. In this chapter we consider ordered recurrent and multiple events.

More information

Power and Sample Size Calculations with the Additive Hazards Model

Power and Sample Size Calculations with the Additive Hazards Model Journal of Data Science 10(2012), 143-155 Power and Sample Size Calculations with the Additive Hazards Model Ling Chen, Chengjie Xiong, J. Philip Miller and Feng Gao Washington University School of Medicine

More information

log T = β T Z + ɛ Zi Z(u; β) } dn i (ue βzi ) = 0,

log T = β T Z + ɛ Zi Z(u; β) } dn i (ue βzi ) = 0, Accelerated failure time model: log T = β T Z + ɛ β estimation: solve where S n ( β) = n i=1 { Zi Z(u; β) } dn i (ue βzi ) = 0, Z(u; β) = j Z j Y j (ue βz j) j Y j (ue βz j) How do we show the asymptotics

More information

11 Survival Analysis and Empirical Likelihood

11 Survival Analysis and Empirical Likelihood 11 Survival Analysis and Empirical Likelihood The first paper of empirical likelihood is actually about confidence intervals with the Kaplan-Meier estimator (Thomas and Grunkmeier 1979), i.e. deals with

More information

Cox s proportional hazards/regression model - model assessment

Cox s proportional hazards/regression model - model assessment Cox s proportional hazards/regression model - model assessment Rasmus Waagepetersen September 27, 2017 Topics: Plots based on estimated cumulative hazards Cox-Snell residuals: overall check of fit Martingale

More information

Application of Time-to-Event Methods in the Assessment of Safety in Clinical Trials

Application of Time-to-Event Methods in the Assessment of Safety in Clinical Trials Application of Time-to-Event Methods in the Assessment of Safety in Clinical Trials Progress, Updates, Problems William Jen Hoe Koh May 9, 2013 Overview Marginal vs Conditional What is TMLE? Key Estimation

More information

Frailty Models and Copulas: Similarities and Differences

Frailty Models and Copulas: Similarities and Differences Frailty Models and Copulas: Similarities and Differences KLARA GOETHALS, PAUL JANSSEN & LUC DUCHATEAU Department of Physiology and Biometrics, Ghent University, Belgium; Center for Statistics, Hasselt

More information

Survival Analysis. Stat 526. April 13, 2018

Survival Analysis. Stat 526. April 13, 2018 Survival Analysis Stat 526 April 13, 2018 1 Functions of Survival Time Let T be the survival time for a subject Then P [T < 0] = 0 and T is a continuous random variable The Survival function is defined

More information

Introduction to Empirical Processes and Semiparametric Inference Lecture 25: Semiparametric Models

Introduction to Empirical Processes and Semiparametric Inference Lecture 25: Semiparametric Models Introduction to Empirical Processes and Semiparametric Inference Lecture 25: Semiparametric Models Michael R. Kosorok, Ph.D. Professor and Chair of Biostatistics Professor of Statistics and Operations

More information

Part III. Hypothesis Testing. III.1. Log-rank Test for Right-censored Failure Time Data

Part III. Hypothesis Testing. III.1. Log-rank Test for Right-censored Failure Time Data 1 Part III. Hypothesis Testing III.1. Log-rank Test for Right-censored Failure Time Data Consider a survival study consisting of n independent subjects from p different populations with survival functions

More information

STAT Sample Problem: General Asymptotic Results

STAT Sample Problem: General Asymptotic Results STAT331 1-Sample Problem: General Asymptotic Results In this unit we will consider the 1-sample problem and prove the consistency and asymptotic normality of the Nelson-Aalen estimator of the cumulative

More information

You know I m not goin diss you on the internet Cause my mama taught me better than that I m a survivor (What?) I m not goin give up (What?

You know I m not goin diss you on the internet Cause my mama taught me better than that I m a survivor (What?) I m not goin give up (What? You know I m not goin diss you on the internet Cause my mama taught me better than that I m a survivor (What?) I m not goin give up (What?) I m not goin stop (What?) I m goin work harder (What?) Sir David

More information

CIMAT Taller de Modelos de Capture y Recaptura Known Fate Survival Analysis

CIMAT Taller de Modelos de Capture y Recaptura Known Fate Survival Analysis CIMAT Taller de Modelos de Capture y Recaptura 2010 Known Fate urvival Analysis B D BALANCE MODEL implest population model N = λ t+ 1 N t Deeper understanding of dynamics can be gained by identifying variation

More information

Competing risks data analysis under the accelerated failure time model with missing cause of failure

Competing risks data analysis under the accelerated failure time model with missing cause of failure Ann Inst Stat Math 2016 68:855 876 DOI 10.1007/s10463-015-0516-y Competing risks data analysis under the accelerated failure time model with missing cause of failure Ming Zheng Renxin Lin Wen Yu Received:

More information

Dynamic Prediction of Disease Progression Using Longitudinal Biomarker Data

Dynamic Prediction of Disease Progression Using Longitudinal Biomarker Data Dynamic Prediction of Disease Progression Using Longitudinal Biomarker Data Xuelin Huang Department of Biostatistics M. D. Anderson Cancer Center The University of Texas Joint Work with Jing Ning, Sangbum

More information

POWER AND SAMPLE SIZE DETERMINATIONS IN DYNAMIC RISK PREDICTION. by Zhaowen Sun M.S., University of Pittsburgh, 2012

POWER AND SAMPLE SIZE DETERMINATIONS IN DYNAMIC RISK PREDICTION. by Zhaowen Sun M.S., University of Pittsburgh, 2012 POWER AND SAMPLE SIZE DETERMINATIONS IN DYNAMIC RISK PREDICTION by Zhaowen Sun M.S., University of Pittsburgh, 2012 B.S.N., Wuhan University, China, 2010 Submitted to the Graduate Faculty of the Graduate

More information

From semi- to non-parametric inference in general time scale models

From semi- to non-parametric inference in general time scale models From semi- to non-parametric inference in general time scale models Thierry DUCHESNE duchesne@matulavalca Département de mathématiques et de statistique Université Laval Québec, Québec, Canada Research

More information

MAS3301 / MAS8311 Biostatistics Part II: Survival

MAS3301 / MAS8311 Biostatistics Part II: Survival MAS330 / MAS83 Biostatistics Part II: Survival M. Farrow School of Mathematics and Statistics Newcastle University Semester 2, 2009-0 8 Parametric models 8. Introduction In the last few sections (the KM

More information

Tests of independence for censored bivariate failure time data

Tests of independence for censored bivariate failure time data Tests of independence for censored bivariate failure time data Abstract Bivariate failure time data is widely used in survival analysis, for example, in twins study. This article presents a class of χ

More information

1 Glivenko-Cantelli type theorems

1 Glivenko-Cantelli type theorems STA79 Lecture Spring Semester Glivenko-Cantelli type theorems Given i.i.d. observations X,..., X n with unknown distribution function F (t, consider the empirical (sample CDF ˆF n (t = I [Xi t]. n Then

More information

Multivariate Survival Analysis

Multivariate Survival Analysis Multivariate Survival Analysis Previously we have assumed that either (X i, δ i ) or (X i, δ i, Z i ), i = 1,..., n, are i.i.d.. This may not always be the case. Multivariate survival data can arise in

More information

Key Words: survival analysis; bathtub hazard; accelerated failure time (AFT) regression; power-law distribution.

Key Words: survival analysis; bathtub hazard; accelerated failure time (AFT) regression; power-law distribution. POWER-LAW ADJUSTED SURVIVAL MODELS William J. Reed Department of Mathematics & Statistics University of Victoria PO Box 3060 STN CSC Victoria, B.C. Canada V8W 3R4 reed@math.uvic.ca Key Words: survival

More information

Survival Analysis. Lu Tian and Richard Olshen Stanford University

Survival Analysis. Lu Tian and Richard Olshen Stanford University 1 Survival Analysis Lu Tian and Richard Olshen Stanford University 2 Survival Time/ Failure Time/Event Time We will introduce various statistical methods for analyzing survival outcomes What is the survival

More information

Attributable Risk Function in the Proportional Hazards Model

Attributable Risk Function in the Proportional Hazards Model UW Biostatistics Working Paper Series 5-31-2005 Attributable Risk Function in the Proportional Hazards Model Ying Qing Chen Fred Hutchinson Cancer Research Center, yqchen@u.washington.edu Chengcheng Hu

More information

Survival Analysis: Weeks 2-3. Lu Tian and Richard Olshen Stanford University

Survival Analysis: Weeks 2-3. Lu Tian and Richard Olshen Stanford University Survival Analysis: Weeks 2-3 Lu Tian and Richard Olshen Stanford University 2 Kaplan-Meier(KM) Estimator Nonparametric estimation of the survival function S(t) = pr(t > t) The nonparametric estimation

More information

5. Parametric Regression Model

5. Parametric Regression Model 5. Parametric Regression Model The Accelerated Failure Time (AFT) Model Denote by S (t) and S 2 (t) the survival functions of two populations. The AFT model says that there is a constant c > 0 such that

More information

Outline. Frailty modelling of Multivariate Survival Data. Clustered survival data. Clustered survival data

Outline. Frailty modelling of Multivariate Survival Data. Clustered survival data. Clustered survival data Outline Frailty modelling of Multivariate Survival Data Thomas Scheike ts@biostat.ku.dk Department of Biostatistics University of Copenhagen Marginal versus Frailty models. Two-stage frailty models: copula

More information

Chapter 2 Inference on Mean Residual Life-Overview

Chapter 2 Inference on Mean Residual Life-Overview Chapter 2 Inference on Mean Residual Life-Overview Statistical inference based on the remaining lifetimes would be intuitively more appealing than the popular hazard function defined as the risk of immediate

More information

8. Parametric models in survival analysis General accelerated failure time models for parametric regression

8. Parametric models in survival analysis General accelerated failure time models for parametric regression 8. Parametric models in survival analysis 8.1. General accelerated failure time models for parametric regression The accelerated failure time model Let T be the time to event and x be a vector of covariates.

More information

Lecture 2: Martingale theory for univariate survival analysis

Lecture 2: Martingale theory for univariate survival analysis Lecture 2: Martingale theory for univariate survival analysis In this lecture T is assumed to be a continuous failure time. A core question in this lecture is how to develop asymptotic properties when

More information

Approximation of Survival Function by Taylor Series for General Partly Interval Censored Data

Approximation of Survival Function by Taylor Series for General Partly Interval Censored Data Malaysian Journal of Mathematical Sciences 11(3): 33 315 (217) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Journal homepage: http://einspem.upm.edu.my/journal Approximation of Survival Function by Taylor

More information

Goodness-Of-Fit for Cox s Regression Model. Extensions of Cox s Regression Model. Survival Analysis Fall 2004, Copenhagen

Goodness-Of-Fit for Cox s Regression Model. Extensions of Cox s Regression Model. Survival Analysis Fall 2004, Copenhagen Outline Cox s proportional hazards model. Goodness-of-fit tools More flexible models R-package timereg Forthcoming book, Martinussen and Scheike. 2/38 University of Copenhagen http://www.biostat.ku.dk

More information

Unobserved Heterogeneity

Unobserved Heterogeneity Unobserved Heterogeneity Germán Rodríguez grodri@princeton.edu Spring, 21. Revised Spring 25 This unit considers survival models with a random effect representing unobserved heterogeneity of frailty, a

More information

Accelerated Failure Time Models

Accelerated Failure Time Models Accelerated Failure Time Models Patrick Breheny October 12 Patrick Breheny University of Iowa Survival Data Analysis (BIOS 7210) 1 / 29 The AFT model framework Last time, we introduced the Weibull distribution

More information

Statistical Inference and Methods

Statistical Inference and Methods Department of Mathematics Imperial College London d.stephens@imperial.ac.uk http://stats.ma.ic.ac.uk/ das01/ 31st January 2006 Part VI Session 6: Filtering and Time to Event Data Session 6: Filtering and

More information

Empirical Likelihood in Survival Analysis

Empirical Likelihood in Survival Analysis Empirical Likelihood in Survival Analysis Gang Li 1, Runze Li 2, and Mai Zhou 3 1 Department of Biostatistics, University of California, Los Angeles, CA 90095 vli@ucla.edu 2 Department of Statistics, The

More information

ANALYSIS OF COMPETING RISKS DATA WITH MISSING CAUSE OF FAILURE UNDER ADDITIVE HAZARDS MODEL

ANALYSIS OF COMPETING RISKS DATA WITH MISSING CAUSE OF FAILURE UNDER ADDITIVE HAZARDS MODEL Statistica Sinica 18(28, 219-234 ANALYSIS OF COMPETING RISKS DATA WITH MISSING CAUSE OF FAILURE UNDER ADDITIVE HAZARDS MODEL Wenbin Lu and Yu Liang North Carolina State University and SAS Institute Inc.

More information

Efficient Estimation of Censored Linear Regression Model

Efficient Estimation of Censored Linear Regression Model 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 2 22 23 24 25 26 27 28 29 3 3 32 33 34 35 36 37 38 39 4 4 42 43 44 45 46 47 48 Biometrika (2), xx, x, pp. 4 C 28 Biometrika Trust Printed in Great Britain Efficient Estimation

More information

Typical Survival Data Arising From a Clinical Trial. Censoring. The Survivor Function. Mathematical Definitions Introduction

Typical Survival Data Arising From a Clinical Trial. Censoring. The Survivor Function. Mathematical Definitions Introduction Outline CHL 5225H Advanced Statistical Methods for Clinical Trials: Survival Analysis Prof. Kevin E. Thorpe Defining Survival Data Mathematical Definitions Non-parametric Estimates of Survival Comparing

More information

Survival Regression Models

Survival Regression Models Survival Regression Models David M. Rocke May 18, 2017 David M. Rocke Survival Regression Models May 18, 2017 1 / 32 Background on the Proportional Hazards Model The exponential distribution has constant

More information

Modelling geoadditive survival data

Modelling geoadditive survival data Modelling geoadditive survival data Thomas Kneib & Ludwig Fahrmeir Department of Statistics, Ludwig-Maximilians-University Munich 1. Leukemia survival data 2. Structured hazard regression 3. Mixed model

More information

Statistical Inference of Interval-censored Failure Time Data

Statistical Inference of Interval-censored Failure Time Data Statistical Inference of Interval-censored Failure Time Data Jinheum Kim 1 1 Department of Applied Statistics, University of Suwon May 28, 2011 J. Kim (Univ Suwon) Interval-censored data Sprring KSS 2011

More information

FULL LIKELIHOOD INFERENCES IN THE COX MODEL

FULL LIKELIHOOD INFERENCES IN THE COX MODEL October 20, 2007 FULL LIKELIHOOD INFERENCES IN THE COX MODEL BY JIAN-JIAN REN 1 AND MAI ZHOU 2 University of Central Florida and University of Kentucky Abstract We use the empirical likelihood approach

More information

Efficiency Comparison Between Mean and Log-rank Tests for. Recurrent Event Time Data

Efficiency Comparison Between Mean and Log-rank Tests for. Recurrent Event Time Data Efficiency Comparison Between Mean and Log-rank Tests for Recurrent Event Time Data Wenbin Lu Department of Statistics, North Carolina State University, Raleigh, NC 27695 Email: lu@stat.ncsu.edu Summary.

More information

Multistate Modeling and Applications

Multistate Modeling and Applications Multistate Modeling and Applications Yang Yang Department of Statistics University of Michigan, Ann Arbor IBM Research Graduate Student Workshop: Statistics for a Smarter Planet Yang Yang (UM, Ann Arbor)

More information

e 4β e 4β + e β ˆβ =0.765

e 4β e 4β + e β ˆβ =0.765 SIMPLE EXAMPLE COX-REGRESSION i Y i x i δ i 1 5 12 0 2 10 10 1 3 40 3 0 4 80 5 0 5 120 3 1 6 400 4 1 7 600 1 0 Model: z(t x) =z 0 (t) exp{βx} Partial likelihood: L(β) = e 10β e 10β + e 3β + e 5β + e 3β

More information

Frailty Modeling for clustered survival data: a simulation study

Frailty Modeling for clustered survival data: a simulation study Frailty Modeling for clustered survival data: a simulation study IAA Oslo 2015 Souad ROMDHANE LaREMFiQ - IHEC University of Sousse (Tunisia) souad_romdhane@yahoo.fr Lotfi BELKACEM LaREMFiQ - IHEC University

More information

Analysing geoadditive regression data: a mixed model approach

Analysing geoadditive regression data: a mixed model approach Analysing geoadditive regression data: a mixed model approach Institut für Statistik, Ludwig-Maximilians-Universität München Joint work with Ludwig Fahrmeir & Stefan Lang 25.11.2005 Spatio-temporal regression

More information

Chapter 4 Regression Models

Chapter 4 Regression Models 23.August 2010 Chapter 4 Regression Models The target variable T denotes failure time We let x = (x (1),..., x (m) ) represent a vector of available covariates. Also called regression variables, regressors,

More information

Improving Efficiency of Inferences in Randomized Clinical Trials Using Auxiliary Covariates

Improving Efficiency of Inferences in Randomized Clinical Trials Using Auxiliary Covariates Improving Efficiency of Inferences in Randomized Clinical Trials Using Auxiliary Covariates Anastasios (Butch) Tsiatis Department of Statistics North Carolina State University http://www.stat.ncsu.edu/

More information

Survival Analysis I (CHL5209H)

Survival Analysis I (CHL5209H) Survival Analysis Dalla Lana School of Public Health University of Toronto olli.saarela@utoronto.ca January 7, 2015 31-1 Literature Clayton D & Hills M (1993): Statistical Models in Epidemiology. Not really

More information

Does Cox analysis of a randomized survival study yield a causal treatment effect?

Does Cox analysis of a randomized survival study yield a causal treatment effect? Published in final edited form as: Lifetime Data Analysis (2015), 21(4): 579 593 DOI: 10.1007/s10985-015-9335-y Does Cox analysis of a randomized survival study yield a causal treatment effect? Odd O.

More information

Goodness-of-Fit Tests With Right-Censored Data by Edsel A. Pe~na Department of Statistics University of South Carolina Colloquium Talk August 31, 2 Research supported by an NIH Grant 1 1. Practical Problem

More information

Validation. Terry M Therneau. Dec 2015

Validation. Terry M Therneau. Dec 2015 Validation Terry M Therneau Dec 205 Introduction When I use a word, Humpty Dumpty said, in rather a scornful tone, it means just what I choose it to mean - neither more nor less. The question is, said

More information

General Regression Model

General Regression Model Scott S. Emerson, M.D., Ph.D. Department of Biostatistics, University of Washington, Seattle, WA 98195, USA January 5, 2015 Abstract Regression analysis can be viewed as an extension of two sample statistical

More information

9 Estimating the Underlying Survival Distribution for a

9 Estimating the Underlying Survival Distribution for a 9 Estimating the Underlying Survival Distribution for a Proportional Hazards Model So far the focus has been on the regression parameters in the proportional hazards model. These parameters describe the

More information

Survival Analysis. STAT 526 Professor Olga Vitek

Survival Analysis. STAT 526 Professor Olga Vitek Survival Analysis STAT 526 Professor Olga Vitek May 4, 2011 9 Survival Data and Survival Functions Statistical analysis of time-to-event data Lifetime of machines and/or parts (called failure time analysis

More information

Survival Distributions, Hazard Functions, Cumulative Hazards

Survival Distributions, Hazard Functions, Cumulative Hazards BIO 244: Unit 1 Survival Distributions, Hazard Functions, Cumulative Hazards 1.1 Definitions: The goals of this unit are to introduce notation, discuss ways of probabilistically describing the distribution

More information

Lecture 3. Truncation, length-bias and prevalence sampling

Lecture 3. Truncation, length-bias and prevalence sampling Lecture 3. Truncation, length-bias and prevalence sampling 3.1 Prevalent sampling Statistical techniques for truncated data have been integrated into survival analysis in last two decades. Truncation in

More information

Faculty of Health Sciences. Cox regression. Torben Martinussen. Department of Biostatistics University of Copenhagen. 20. september 2012 Slide 1/51

Faculty of Health Sciences. Cox regression. Torben Martinussen. Department of Biostatistics University of Copenhagen. 20. september 2012 Slide 1/51 Faculty of Health Sciences Cox regression Torben Martinussen Department of Biostatistics University of Copenhagen 2. september 212 Slide 1/51 Survival analysis Standard setup for right-censored survival

More information

Lecture 7 Time-dependent Covariates in Cox Regression

Lecture 7 Time-dependent Covariates in Cox Regression Lecture 7 Time-dependent Covariates in Cox Regression So far, we ve been considering the following Cox PH model: λ(t Z) = λ 0 (t) exp(β Z) = λ 0 (t) exp( β j Z j ) where β j is the parameter for the the

More information

DAGStat Event History Analysis.

DAGStat Event History Analysis. DAGStat 2016 Event History Analysis Robin.Henderson@ncl.ac.uk 1 / 75 Schedule 9.00 Introduction 10.30 Break 11.00 Regression Models, Frailty and Multivariate Survival 12.30 Lunch 13.30 Time-Variation and

More information

Semiparametric Models for Joint Analysis of Longitudinal Data and Counting Processes

Semiparametric Models for Joint Analysis of Longitudinal Data and Counting Processes Semiparametric Models for Joint Analysis of Longitudinal Data and Counting Processes by Se Hee Kim A dissertation submitted to the faculty of the University of North Carolina at Chapel Hill in partial

More information

( t) Cox regression part 2. Outline: Recapitulation. Estimation of cumulative hazards and survival probabilites. Ørnulf Borgan

( t) Cox regression part 2. Outline: Recapitulation. Estimation of cumulative hazards and survival probabilites. Ørnulf Borgan Outline: Cox regression part 2 Ørnulf Borgan Department of Mathematics University of Oslo Recapitulation Estimation of cumulative hazards and survival probabilites Assumptions for Cox regression and check

More information

Quantile Regression for Residual Life and Empirical Likelihood

Quantile Regression for Residual Life and Empirical Likelihood Quantile Regression for Residual Life and Empirical Likelihood Mai Zhou email: mai@ms.uky.edu Department of Statistics, University of Kentucky, Lexington, KY 40506-0027, USA Jong-Hyeon Jeong email: jeong@nsabp.pitt.edu

More information

Residuals and model diagnostics

Residuals and model diagnostics Residuals and model diagnostics Patrick Breheny November 10 Patrick Breheny Survival Data Analysis (BIOS 7210) 1/42 Introduction Residuals Many assumptions go into regression models, and the Cox proportional

More information

Step-Stress Models and Associated Inference

Step-Stress Models and Associated Inference Department of Mathematics & Statistics Indian Institute of Technology Kanpur August 19, 2014 Outline Accelerated Life Test 1 Accelerated Life Test 2 3 4 5 6 7 Outline Accelerated Life Test 1 Accelerated

More information

Statistical Methods for Alzheimer s Disease Studies

Statistical Methods for Alzheimer s Disease Studies Statistical Methods for Alzheimer s Disease Studies Rebecca A. Betensky, Ph.D. Department of Biostatistics, Harvard T.H. Chan School of Public Health July 19, 2016 1/37 OUTLINE 1 Statistical collaborations

More information

Models for Multivariate Panel Count Data

Models for Multivariate Panel Count Data Semiparametric Models for Multivariate Panel Count Data KyungMann Kim University of Wisconsin-Madison kmkim@biostat.wisc.edu 2 April 2015 Outline 1 Introduction 2 3 4 Panel Count Data Motivation Previous

More information

Linear life expectancy regression with censored data

Linear life expectancy regression with censored data Linear life expectancy regression with censored data By Y. Q. CHEN Program in Biostatistics, Division of Public Health Sciences, Fred Hutchinson Cancer Research Center, Seattle, Washington 98109, U.S.A.

More information

Session 9: Introduction to Sieve Analysis of Pathogen Sequences, for Assessing How VE Depends on Pathogen Genomics Part I

Session 9: Introduction to Sieve Analysis of Pathogen Sequences, for Assessing How VE Depends on Pathogen Genomics Part I Session 9: Introduction to Sieve Analysis of Pathogen Sequences, for Assessing How VE Depends on Pathogen Genomics Part I Peter B Gilbert Vaccine and Infectious Disease Division, Fred Hutchinson Cancer

More information

Likelihood Construction, Inference for Parametric Survival Distributions

Likelihood Construction, Inference for Parametric Survival Distributions Week 1 Likelihood Construction, Inference for Parametric Survival Distributions In this section we obtain the likelihood function for noninformatively rightcensored survival data and indicate how to make

More information

Exercises. (a) Prove that m(t) =

Exercises. (a) Prove that m(t) = Exercises 1. Lack of memory. Verify that the exponential distribution has the lack of memory property, that is, if T is exponentially distributed with parameter λ > then so is T t given that T > t for

More information

β j = coefficient of x j in the model; β = ( β1, β2,

β j = coefficient of x j in the model; β = ( β1, β2, Regression Modeling of Survival Time Data Why regression models? Groups similar except for the treatment under study use the nonparametric methods discussed earlier. Groups differ in variables (covariates)

More information

USING MARTINGALE RESIDUALS TO ASSESS GOODNESS-OF-FIT FOR SAMPLED RISK SET DATA

USING MARTINGALE RESIDUALS TO ASSESS GOODNESS-OF-FIT FOR SAMPLED RISK SET DATA USING MARTINGALE RESIDUALS TO ASSESS GOODNESS-OF-FIT FOR SAMPLED RISK SET DATA Ørnulf Borgan Bryan Langholz Abstract Standard use of Cox s regression model and other relative risk regression models for

More information

Longitudinal + Reliability = Joint Modeling

Longitudinal + Reliability = Joint Modeling Longitudinal + Reliability = Joint Modeling Carles Serrat Institute of Statistics and Mathematics Applied to Building CYTED-HAROSA International Workshop November 21-22, 2013 Barcelona Mainly from Rizopoulos,

More information

SEMIPARAMETRIC METHODS FOR ESTIMATING CUMULATIVE TREATMENT EFFECTS IN THE PRESENCE OF NON-PROPORTIONAL HAZARDS AND DEPENDENT CENSORING

SEMIPARAMETRIC METHODS FOR ESTIMATING CUMULATIVE TREATMENT EFFECTS IN THE PRESENCE OF NON-PROPORTIONAL HAZARDS AND DEPENDENT CENSORING SEMIPARAMETRIC METHODS FOR ESTIMATING CUMULATIVE TREATMENT EFFECTS IN THE PRESENCE OF NON-PROPORTIONAL HAZARDS AND DEPENDENT CENSORING by Guanghui Wei A dissertation submitted in partial fulfillment of

More information

THESIS for the degree of MASTER OF SCIENCE. Modelling and Data Analysis

THESIS for the degree of MASTER OF SCIENCE. Modelling and Data Analysis PROPERTIES OF ESTIMATORS FOR RELATIVE RISKS FROM NESTED CASE-CONTROL STUDIES WITH MULTIPLE OUTCOMES (COMPETING RISKS) by NATHALIE C. STØER THESIS for the degree of MASTER OF SCIENCE Modelling and Data

More information