Other Survival Models. (1) Non-PH models. We briefly discussed the non-proportional hazards (non-ph) model
|
|
- Arron Edwards
- 5 years ago
- Views:
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 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 informationSurvival 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 informationStatistics 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 informationAnalysis 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 informationCox 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 informationUniversity 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 informationEfficient 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 informationLecture 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 informationDefinitions 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 informationSTAT331. 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 informationLecture 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 informationMAS3301 / 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 informationPENALIZED 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 information1 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 informationGoodness-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 informationPhD 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 informationSemiparametric 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 informationHypothesis 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 informationSTAT 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 informationSTAT 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 informationSurvival 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 informationOn 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 informationLecture 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 informationPower 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 informationlog 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 information11 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 informationCox 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 informationApplication 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 informationFrailty 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 informationSurvival 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 informationIntroduction 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 informationPart 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 informationSTAT 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 informationYou 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 informationCIMAT 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 informationCompeting 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 informationDynamic 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 informationPOWER 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 informationFrom 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 informationMAS3301 / 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 informationTests 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 information1 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 informationMultivariate 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 informationKey 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 informationSurvival 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 informationAttributable 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 informationSurvival 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 information5. 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 informationOutline. 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 informationChapter 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 information8. 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 informationLecture 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 informationApproximation 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 informationGoodness-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 informationUnobserved 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 informationAccelerated 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 informationStatistical 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 informationEmpirical 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 informationANALYSIS 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 informationEfficient 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 informationTypical 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 informationSurvival 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 informationModelling 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 informationStatistical 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 informationFULL 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 informationEfficiency 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 informationMultistate 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 informatione 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 informationFrailty 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 informationAnalysing 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 informationChapter 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 informationImproving 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 informationSurvival 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 informationDoes 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 informationGoodness-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 informationValidation. 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 informationGeneral 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 information9 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 informationSurvival 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 informationSurvival 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 informationLecture 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 informationFaculty 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 informationLecture 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 informationDAGStat 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 informationSemiparametric 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
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 informationQuantile 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 informationResiduals 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 informationStep-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 informationStatistical 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 informationModels 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 informationLinear 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 informationSession 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 informationLikelihood 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 informationExercises. (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 informationβ 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 informationUSING 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 informationLongitudinal + 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 informationSEMIPARAMETRIC 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 informationTHESIS 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