GoodnessOfFit for Cox s Regression Model. Extensions of Cox s Regression Model. Survival Analysis Fall 2004, Copenhagen


 Zoe Burke
 9 months ago
 Views:
Transcription
1 Outline Cox s proportional hazards model. Goodnessoffit tools More flexible models Rpackage timereg Forthcoming book, Martinussen and Scheike. 2/38 University of Copenhagen GoodnessOfFit for Cox s Regression Model. Extensions of Cox s Regression Model. Survival Analysis Fall 24, Copenhagen Torben Martinussen and Thomas Scheike 1/38
2 Cox s proportional hazards model In practice one has covariates: X i (pdimensional). Hazard conditional on covariates: α i (t, X i ). The absolute dominant model is Cox s proportional hazards model: α i (t) = α (t) exp (β T X i ) where α (t) is unspecified baseline hazard (hazard for X i = ). Flexible model. Model is easily fitted using for example SAS or R (SPlus). Primary model for survival data because of its nice properties. Suppose X is 1dim. (or fix other covariates) then the relative risk α(t, X + 1) α(t, X) = exp (β) 4/38 is not depending on time (key assumption)! This assumption is often violated! Survival analysis Standard setup for rightcensored survival data. IID copies of (T, D) where T = T C D = I(T C) with T being the true survival time and C the (potential) censoring time and possibly covariates X i (t). Hazardfunction Counting process Martingale 1 α(t) = lim h h P (t T < t + h T t, F ). N i (t) = I(T i t, D i = 1) M i (t) = N i (t) Λ i (t) 3/38 where Λ i (t) = Y i(s)α(s) ds (compensator), Y i (t) = I(t T i ) (at risk process).
3 > library(survival) > data(pbc) > attach(pbc) > cbind(time,status,age,edema,bili,protime,alb)[1:5,] time status age edema bili protime alb [1,] [2,] [3,] [4,] [5,] > sum(status) [1] 161 > fit.pbc<coxph(surv(time/365, status) age+edema+log(bili)+log(protime)+log(alb)) > fit.pbc Call: coxph(formula = Surv(time/365, status) age + edema + log(bili) + log(protime) + log(alb)) coef exp(coef) se(coef) z p age e7 edema e3 log(bili) e+ log(protime) e3 log(alb) e4 Likelihood ratio test=234 on 5 df, p= n= 418 6/38 PBCdata PBC data (primary biliary cirrhosis): 418 patients are followed until death or censoring. PBC is a fatal chronic liver disease. Important explanatory variables: Age Albumin Bilirubin Edema Prothrombin time Fitting Cox s model in R. 5/38
4 Graphical GOF for Cox s regression model 8/38 Figure 1: Estimated logcumulative hazards difference along with 95% pointwise confidence intervals. The straight lines (dashed lines) are based on the Cox model. Cox s proportional hazards model Traditional goodnessoffit tools. Model: α i (t) = α (t) exp(β 1 X i β p X ip ) Investigate if each of the covariates are consistent with the proportional hazards assumption. Stratify based on a grouping (k=1,...,k) based on X i1 s values: α i (t) = α k (t) exp(β 2 X i β p X ip ); ifx i1 A k 7/38 Now, if the underlying full Coxmodel is true the baseline estimates α k (t) should be proportional, as α k (t) = α (t) exp( K β 1k I(X i1 A k )). k=1 Graphical modelcheck of proportionality by making graphs of estimates of log( α k(s)ds). Plotted against t they should be parallel.
5 Cumulative martingale residuals Alternative: Cumulative martingale residuals, (Lin et al., 1993). The martingales under the Cox regression model can be written as M i (t) = N i (t) Y i (s) exp(x T β)dλ (s); ˆMi (t) = N i (t) Y i (s) exp(x T ˆβ)dˆΛ (s) The score function, evaluated in the estimate ˆβ, and seen as a function of time, can for example be written as U( ˆβ, t) = n i=1 X i (s)d ˆM i (s) = n i=1 (X i (s) E(t, ˆβ))d ˆM i (s). and is asymptotically equivalent to a Gaussian process (not a martingale) that can easily be simulated (LWY,93). Can now proceed to suggest some appropriate test statistic like sup U( ˆβ, t) t [,τ] 1/38 This is essentially a test for timeconstant effects of all covariates!! Cox s proportional hazards model Traditional goodnessoffit tools. Make tests against specific deviations: Replace X 1 with (X 1, X 1 (log (t))), say (β 1 β 1 + β p+1 log (t)). Test the null β p+1 =. These methods are quite useful but also have some limitations : Graphical method: Not parallel. What is acceptable? What if a given covariate is continuous? Test: Ad hoc method. Which transformation to use? Both methods: They assume that model is ok for all the other covariates. 9/38
6 Resampling technique The observed score process is given as U( ˆβ, t) and its asymptotic distribution is equivalent to the asymptotic distribution of i (Ŵ1i (t) + I( ˆβ, t)i 1 ( ˆβ, τ)ŵ1i(τ)) G i where G 1,..., G n are independent standard normals, and independent of the observed data. 12/38 Score process The score process evaluated at β can be written as U(β, t) = i W 1i (t) (1) where with W 1i (t) = ( Zi Z T Y (β )(Y T (β )W Y (β )) 1 Y i ) dmi (s), Y (β, t) = (Y 1 exp(z T 1 β),..., Y n exp(z T n β)) W (t) = diag(y i exp( Z T i β)) Variance can then be estimated robustly by { } ˆΣ β = ni 1 ( ˆβ, τ) Ŵ 1i (t) 2 I 1 ( ˆβ, τ). i 11/38 where Ŵ1i is obtained by replacing M i by ˆM i.
7 PBCdata 14/38 PBCdata > library(timereg) > fit<cox.aalen(surv(time/365,status) prop(age)+prop(edema)+ + prop(log(bilirubin))+prop(log(albumin))+prop(log(protime)), + weighted.score=,pbc); Right censored survival times CoxAalen Survival Model Simulations starts N= 5 > summary(ourcox) CoxAalen Model Score Test for Proportionality sup hat U(t) pvalue H_: U(t) Proportional prop(age) prop(edema) prop(log(bilirubin)) prop(log(albumin)) prop(log(protime)) > plot(fit,score=t,xlab="time (years)") 13/38
8 Cumulative Residuals Now, ˆM(t) = G(β, s)dm(s) + { Y (β, s)y (β, s) Y ( ˆβ, s)y ( ˆβ, } s) dn(s). The second term can be Taylor series expanded [ G(β, s)diag { Y (β, s)y (β, s)dn(s) } ] Z(s) ( ˆβ β ) [ = G(s)diag { Y (β, s)y (β, s)dn(s) } ] Z(s) I 1 (β, τ)u(β, τ) where β and β are on line segment between ˆβ and β. Therefore ˆM(t) is asymptotically equivalent (see below) to 16/38 M(t) + B(β, t)m(τ) Cumulative Residuals Model can be written as the n 1 vector N(t) = Y (t)λ (t) + M(t) where M(t) is the meanzero martingale, and then the martingales are estimated by ˆM(t) = N(t) Y ( ˆβ, s)y ( ˆβ, s)dn(s) = G(s)dN(s). where and G( ˆβ, s) = I Y ( ˆβ, s)y ( ˆβ, s) Y (β, s) = (Y T W Y ) 1 Y T W. 15/38
9 Cumulative Residuals M U (t) is asymptotically equivalent to n i=1 [ U i (t) U T (s)y (β, s) { Y T (β, s)w (s)y (β, s) } 1 Yi (s)dm i (s) U T (s)g(s)diag { Y (β, s)y (β, s)dn(s) } ] n Z(s) = W i (t) + o p (n 1/2 ), where W i (t) are i.i.d. an W 1,i is an i.i.d. decomposition of ˆβ β. Resample construction Ŵi (t)g i, i=1 W 1,i + o p (n 1/2 ), and G 1,..., G n are standard normals have the same asymptotic distribution. 18/38 Where Ŵ i (t) is obtained by using ˆM i instead of M i. Cumulative Residuals A cumulative residual process is then defined by M U (t) = U T (t)d ˆM(s) and this process is asymptotically equivalent to [ U T (t)g(β, s)dm(s) U T (s)g(s)diag { Y (β, s)y (β, s)dn(s) } ] Z(s) ( ˆβ β ). Denote the second integral in the latter display by B U (β, t). The variance of M U (t) can be estimated by the optional variation process [M U ] (t) = U T (s)g( ˆβ, s)diag(dn(s))u(s)g( ˆβ, s) + B U ( ˆβ, t) [U] (τ)b T U ( ˆβ, t) B U ( ˆβ, t) [M U, U] (t) [U, M U ] (t)b T U ( ˆβ, t). 17/38
10 Cumulative residuals > # our PBC version with no ties!!!!!!!!!!!!!!!!!!!! > fit<cox.aalen(surv(time/365,status) prop(age)+ prop(edema)+prop(log(bilirubin))+prop(log(albumin))+ prop(log(protime)),pbc, + weighted.score=,resid.mg=1); CoxAalen Survival Model Simulations starts N= 5 > > X<model.matrix( 1+cut(Bilirubin,quantile(Bilirubin),include.lowest=T),pbc) > colnames(x)<c("1. quartile","2. quartile","3. quartile","4. quartile"); > > resids<mg.resids(fit,pbc,x,n.sim=1,cum.resid=1) > summary(resids) Test for cumulative MGresiduals Grouped Residuals consistent with model sup hat B(t) pvalue H_: B(t)= 1. quartile quartile quartile quartile int ( B(t) )ˆ2 dt pvalue H_: B(t)= 1. quartile quartile quartile quartile /38 Residual versus covariates consistent with model sup hat B(t) pvalue H_: B(t)= prop(age) prop(log(bilirubin)) prop(log(albumin)) prop(log(protime)) Cumulative Residuals (Lin et al., 1993) suggest to cumulate the residuals over the covariate space as well as over time, and thus considers the double cumulative processes M j (x, t) = = K T (j, x, s)d ˆM(s) K T (j, x, s)g(s)dm(s) for j = 1,.., p where K(j, x, t) is an n 1 vector with ith I(Z i,j (t) x) for i = 1,.., n. Integrating over time we get a process in x M j (x) = M j (x, τ) (2) 19/38
11 Cumulative martingale residuals 22/38 Cumulative martingale residuals 21/38
12 Cumulative martingale residuals 24/38 Cumulative residuals > nfit<cox.aalen(surv(time/365,status) prop(age)+prop(edema)+prop(bilirubin)+prop(log(albumin))+prop(log(protime)),pbc, weighted.score=,resid.mg=1); > nresids<mg.resids(nfit,pbc,x,n.sim=1,cum.resid=1) > summary(nresids) Test for cumulative MGresiduals Grouped Residuals consistent with model sup hat B(t) pvalue H_: B(t)= 1. quartile quartile quartile quartile int ( B(t) )ˆ2 dt pvalue H_: B(t)= 1. quartile quartile quartile quartile Residual versus covariates consistent with model sup hat B(t) pvalue H_: B(t)= prop(age) prop(bilirubin) prop(log(albumin)) prop(log(protime)) /38
13 Cox s model with timedependent effects A typical deviation from Cox s model is timedependent covariate effects. Treatment is effective for some time, but then effect levels off. Takes some time before treatment has an effect. Model α i (t) = exp (β(t) T X i ) where coefficients β(t) are now depending on time! Scoreequation: X(t) T (dn(t) λ(t)dt) = (3) Cannot solve (3). Taylor expansion and integration of (3) yields an algorithm. 26/38 Cumulative martingale residuals 25/38
14 Cox s model with timedependent effects Algorithm: g( B)(t) = β(s) ds + Ã(s) 1 X(s) T dn(s) Ã(s) 1 X(s) T λ(s) ds, (4) with Ã(t) = A β(t) = i Y i(t)e x i(t) T β(t) x i (t)x i (t) T and β(t) obtained from B(t) by smoothing. Theorem (4) has a solution g( ˆB) = ˆB n 1/2 ( ˆB B) D U ˆΣ(t) = n Â(s) 1 ds ˆB is efficient 28/38 Reference:(Martinussen et al., 22). Cox s model with timedependent effects If consistent estimate, β(t), is present for estimating β(t) NewtonRaphson suggests that λ(t)dt = Ã(s) 1 X(s) T dn(s) Ã(s) 1 X(s) T λ(s) ds, where Ã(t) = A β(t) = i Y i(t)e x i(t) T β(t) x i (t)x i (t) T 27/38
15 PBCdata > fit<timecox(surv(time/365,status) Age+Edema+log(Bilirubin)+log(Albumin)+log(Protime),pbc, + maxtime=3/365,band.width=.5); Right censored survival times Nonparametric Multiplicative Hazard Model Simulations starts N= 5 > plot(fit,ylab="cumulative coefficients",xlab="time (years)"); > summary(fit) Multiplicative Hazard Model Test for nonparametric terms 3/38 Test for nonsiginificant effects sup hat B(t)/SD(t) pvalue H_: B(t)= (Intercept) Age Edema log(bilirubin) log(albumin) log(protime) Test for time invariant effects sup B(t)  (t/tau)b(tau) pvalue H_: B(t)=b t (Intercept) Age Edema log(bilirubin) log(albumin) log(protime) Cox s model with timedependent effects Important: can also handle the semiparametric model λ(t) = Y (t)λ (t) exp(x T (t)β(t) + Z T (t)γ) (5) Can investigate the important H : β p (t) γ q+1 of nontimedependency; Notice that it may be done in a model allowing other covariates to have timedependent effects! 29/38
16 PBCdata > fit.semi<timecox(surv(time/365,status) semi(age)+edema+semi(log(bilirubin))+semi(log(albumin))+log(protime),pbc, maxtime=3/365,band.width=.5) Right censored survival times Semiparametric Multiplicative Risk Model Simulations starts N= 5 > summary(fit.semi) Multiplicative Hazard Model Test for nonparametric terms Test for nonsiginificant effects sup hat B(t)/SD(t) pvalue H_: B(t)= (Intercept) Edema log(protime) Test for time invariant effects sup B(t)  (t/tau)b(tau) pvalue H_: B(t)=b t (Intercept) Edema log(protime) /38 Parametric terms : Coef. Std. Error Robust Std. Error semi(age) semi(log(bilirubin)) semi(log(albumin)) PBCdata 31/38
17 Mix of Aalens and Cox s models CoxAalen model α i (t) = α(t) T X i (t) exp (β T Z i (t)), Gives a mix of Aalens and Cox s models Flexible modelling in additive part and multiplicative relative risk parameters for Z. Reference : (Scheike and Zhang, 22; Scheike and Zhang, 23) Model can also be fitted in timereg. Gives excess risk interpretation of additive part and relative risk interpretation for multiplicative part. Cox s regression model and stratified versions of it is a special case of the model. 34/38 Cox s model with timedependent effects Model with timedependent and constant effects is from a theoretically point of view much more satisfactory Model is available using the Rpackage timereg. Practical experience is needed. May need quite a bit of data to get reliable inference. Needs to choose a bandwidth. Optimal, crossvalidation. Additional reference: (Scheike and Martinussen, 24) Alternative models that allow for timedependent effects without the unpleasant bandwidth choice are for example : Aalens s additive hazards model CoxAalen model 33/38 Proportional Excess hazard models These can also be fitted using timereg!
18 CoxAalen model 36/38 PBCdata > logbili.m<log(pbc$bilirubin)mean(log(pbc$bilirubin)); > logalb.m<log(pbc$albumin)mean(log(pbc$albumin)); > Age.m<pbc$Agemean(pbc$Age); > fit<cox.aalen(surv(time/365,status) prop(age.m)+edema+ + prop(logbili.m)+prop(logalb.m)+log(protime),resid.mg=1, + max.time=3/365,pbc) CoxAalen Survival Model Simulations starts N= 5 > summary(fit) Test for nonsiginificant effects sup hat B(t)/SD(t) pvalue H_: B(t)= (Intercept) Edema log(protime) Test for time invariant effects sup B(t)  (t/tau)b(tau) pvalue H_: B(t)=b t (Intercept) Edema.269. log(protime) Proportional Cox terms : Coef. Std. Error Robust SE D2log(L)ˆ1 prop(age.m) prop(logbili.m) prop(logalb.m) /38 Score Tests for Proportionality sup hat U(t) pvalue H_ prop(age.m) prop(logbili.m) prop(logalb.m)
19 Summary Cox s proportional hazards model. Are the relative risks really not depending on time? Check model carefully. More flexible models Multiplicative model with timevarying covariate effects and also constant effects. Inference. Other flexible models: CoxAalen model and excess risk models. Aalens additive hazards model (and the semiparametric version). Can all be fitted in the Rpackage timereg: ts/timereg.html 38/38 Mix of Aalens and Cox s models Excessrisk type model α i (t) = α(t) T X i (t) + ρ i λ (t) exp (β T Z i (t)), ρ i = 1, all i, gives a mix of Aalens and Cox s models Model is perhaps most naturally seen as an excess risk model: ρ i is excess indicator eg I(d i > ) with d i dosis for ith subject. Has proven useful in cancer studies, see (Zahl, 23). Notice α i (t) = α(t) T X i (t) + ρ i λ (t) exp (β T Z i (t)) = ψ(t) T X i where ψ(t) = (α(t), λ (t)), XT i = (X T i, φ i (β)), φ i (β) = ρ i exp (β T Z i ). 37/38 May derive estimators of unknown parameters and also their large sample properties, see (Martinussen and Scheike, 22) Model can also be fitted in timereg.
20 References Lin, D.Y., Wei, L.J., and Ying, Z. (1993). Checking the Cox model with cumulative sums of martingalebased residuals. Biometrika 8, Martinussen, T. and Scheike, T.H. (22). A flexible additive multiplicative hazard model. Biometrika 89, Martinussen, T., Scheike, T.H., and Skovgaard, I.M. (22). Efficient estimation of fixed and timevarying covariate effects in multiplicative intensity models. Scandinavian Journal of Statistics 28, Scheike, T.H. and Martinussen, T. (24). On efficient estimation and tests of timevarying effects in the proportional hazards model. Scandinavian Journal of Statistics 31, Scheike, T.H. and Zhang, M.J. (22). An additivemultiplicative CoxAalen model. Scandinavian Journal of Statistics 28, Scheike, T.H. and Zhang, M.J. (23). Extensions and applications of the CoxAalen survival model. Biometrics 59, Zahl, 38/38 P. (23). Regresion analysis with multiplicative and timevarying additive regression coefficients with examples from
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 rightcensored survival
More informationSurvival analysis in R
Survival analysis in R Niels Richard Hansen This note describes a few elementary aspects of practical analysis of survival data in R. For further information we refer to the book Introductory Statistics
More informationSurvival analysis in R
Survival analysis in R Niels Richard Hansen This note describes a few elementary aspects of practical analysis of survival data in R. For further information we refer to the book Introductory Statistics
More informationJournal of Statistical Software
JSS Journal of Statistical Software January 2011, Volume 38, Issue 2. http://www.jstatsoft.org/ Analyzing Competing Risk Data Using the R timereg Package Thomas H. Scheike University of Copenhagen MeiJie
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 informationLoglinearity for Cox s regression model. Thesis for the Degree Master of Science
Loglinearity for Cox s regression model Thesis for the Degree Master of Science Zaki Amini Master s Thesis, Spring 2015 i Abstract Cox s regression model is one of the most applied methods in medical
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 informationThe coxvc_111 package
Appendix A The coxvc_111 package A.1 Introduction The coxvc_111 package is a set of functions for survival analysis that run under R2.1.1 [81]. This package contains a set of routines to fit Cox models
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 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 informationLecture 8 Stat D. Gillen
Statistics 255  Survival Analysis Presented February 23, 2016 Dan Gillen Department of Statistics University of California, Irvine 8.1 Example of two ways to stratify Suppose a confounder C has 3 levels
More informationEstimation and Inference of Quantile Regression. for Survival Data under Biased Sampling
Estimation and Inference of Quantile Regression for Survival Data under Biased Sampling Supplementary Materials: Proofs of the Main Results S1 Verification of the weight function v i (t) for the lengthbiased
More informationOn the Breslow estimator
Lifetime Data Anal (27) 13:471 48 DOI 1.17/s19857948y 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 informationFULL LIKELIHOOD INFERENCES IN THE COX MODEL
October 20, 2007 FULL LIKELIHOOD INFERENCES IN THE COX MODEL BY JIANJIAN REN 1 AND MAI ZHOU 2 University of Central Florida and University of Kentucky Abstract We use the empirical likelihood approach
More informationAttributable Risk Function in the Proportional Hazards Model
UW Biostatistics Working Paper Series 5312005 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. 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 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 informationTied survival times; estimation of survival probabilities
Tied survival times; estimation of survival probabilities Patrick Breheny November 5 Patrick Breheny Survival Data Analysis (BIOS 7210) 1/22 Introduction Tied survival times Introduction Breslow approximation
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 informationLogistic regression model for survival time analysis using timevarying coefficients
Logistic regression model for survival time analysis using timevarying coefficients Accepted in American Journal of Mathematical and Management Sciences, 2016 Kenichi SATOH ksatoh@hiroshimau.ac.jp Research
More informationLecture 7. Proportional Hazards Model  Handling Ties and Survival Estimation Statistics Survival Analysis. Presented February 4, 2016
Proportional Hazards Model  Handling Ties and Survival Estimation Statistics 255  Survival Analysis Presented February 4, 2016 likelihood  Discrete Dan Gillen Department of Statistics University of
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 informationFrom semi to nonparametric inference in general time scale models
From semi to nonparametric inference in general time scale models Thierry DUCHESNE duchesne@matulavalca Département de mathématiques et de statistique Université Laval Québec, Québec, Canada Research
More informationCox regression: Estimation
Cox regression: Estimation Patrick Breheny October 27 Patrick Breheny Survival Data Analysis (BIOS 7210) 1/19 Introduction The Cox Partial Likelihood In our last lecture, we introduced the Cox partial
More informationSTAT 331. Martingale Central Limit Theorem and Related Results
STAT 331 Martingale Central Limit Theorem and Related Results In this unit we discuss a version of the martingale central limit theorem, which states that under certain conditions, a sum of orthogonal
More informationStat 642, Lecture notes for 04/12/05 96
Stat 642, Lecture notes for 04/12/05 96 HosmerLemeshow Statistic The HosmerLemeshow Statistic is another measure of lack of fit. Hosmer and Lemeshow recommend partitioning the observations into 10 equal
More informationUsing Estimating Equations for Spatially Correlated A
Using Estimating Equations for Spatially Correlated Areal Data December 8, 2009 Introduction GEEs Spatial Estimating Equations Implementation Simulation Conclusion Typical Problem Assess the relationship
More informationMultivariate Survival Data With Censoring.
1 Multivariate Survival Data With Censoring. Shulamith Gross and Catherine HuberCarol Baruch College of the City University of New York, Dept of Statistics and CIS, Box 11220, 1 Baruch way, 10010 NY.
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 informationMODELING THE SUBDISTRIBUTION OF A COMPETING RISK
Statistica Sinica 16(26), 13671385 MODELING THE SUBDISTRIBUTION OF A COMPETING RISK Liuquan Sun 1, Jingxia Liu 2, Jianguo Sun 3 and MeiJie Zhang 2 1 Chinese Academy of Sciences, 2 Medical College of
More informationMüller: Goodnessoffit criteria for survival data
Müller: Goodnessoffit criteria for survival data Sonderforschungsbereich 386, Paper 382 (2004) Online unter: http://epub.ub.unimuenchen.de/ Projektpartner Goodness of fit criteria for survival data
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 models and health sequences
Survival models and health sequences Walter Dempsey University of Michigan July 27, 2015 Survival Data Problem Description Survival data is commonplace in medical studies, consisting of failure time information
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 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 informationSEMIPARAMETRIC ADDITIVE RISKS REGRESSION FOR TWOSTAGE DESIGN SURVIVAL STUDIES
Statistica Sinica 20 (2010, 15811607 SEMIPARAMETRIC ADDITIVE RISKS REGRESSIO FOR TWOSTAGE DESIG SURVIVAL STUDIES Gang Li and Tong Tong Wu University of California, Los Angeles and University of Maryland,
More informationExtensions of Cox Model for NonProportional Hazards Purpose
PhUSE 2013 Paper SP07 Extensions of Cox Model for NonProportional Hazards Purpose Jadwiga Borucka, PAREXEL, Warsaw, Poland ABSTRACT Cox proportional hazard model is one of the most common methods used
More informationThe covariate order method for nonparametric exponential regression and some applications in other lifetime models.
The covariate order method for nonparametric exponential regression and some applications in other lifetime models. Jan Terje Kvaløy and Bo Henry Lindqvist Department of Mathematics and Natural Science
More informationADVANCED STATISTICAL ANALYSIS OF EPIDEMIOLOGICAL STUDIES. Cox s regression analysis Time dependent explanatory variables
ADVANCED STATISTICAL ANALYSIS OF EPIDEMIOLOGICAL STUDIES Cox s regression analysis Time dependent explanatory variables Henrik Ravn Bandim Health Project, Statens Serum Institut 4 November 2011 1 / 53
More informationSmoothing the NelsonAalen Estimtor Biostat 277 presentation Chihong Tseng
Smoothing the NelsonAalen Estimtor Biostat 277 presentation Chihong seng Reference: 1. Andersen, Borgan, Gill, and Keiding (1993). Statistical Model Based on Counting Processes, SpringerVerlag, p.229255
More informationANALYSIS OF COMPETING RISKS DATA WITH MISSING CAUSE OF FAILURE UNDER ADDITIVE HAZARDS MODEL
Statistica Sinica 18(28, 219234 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 informationKernel density estimation in R
Kernel density estimation in R Kernel density estimation can be done in R using the density() function in R. The default is a Guassian kernel, but others are possible also. It uses it s own algorithm to
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 informationLecture 3. Truncation, lengthbias and prevalence sampling
Lecture 3. Truncation, lengthbias 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 informationOutline. Recall that Aalen additive hazards model and the semiparametric version
Outlne Clustered survval data (addtve models) AddtveMultplcatve hazards model Advanced Survval Analyss 21 Copenhagen Gulana Cortese gco@bostat.ku.dk Clustered survval data Margnal addtve models AddtveMultplcatve
More informationLecture 11. Interval Censored and. DiscreteTime Data. Statistics Survival Analysis. Presented March 3, 2016
Statistics 255  Survival Analysis Presented March 3, 2016 Motivating Dan Gillen Department of Statistics University of California, Irvine 11.1 First question: Are the data truly discrete? : Number of
More informationEvaluating Predictive Accuracy of Survival Models with PROC PHREG
Paper SAS4622017 Evaluating Predictive Accuracy of Survival Models with PROC PHREG Changbin Guo, Ying So, and Woosung Jang, SAS Institute Inc. Abstract Model validation is an important step in the model
More informationA class of mean residual life regression models with censored survival data
1 A class of mean residual life regression models with censored survival data LIUQUAN SUN Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing,
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 informationConstrained Maximum Likelihood Estimation for Model Calibration Using Summarylevel Information from External Big Data Sources
Constrained Maximum Likelihood Estimation for Model Calibration Using Summarylevel Information from External Big Data Sources YiHau Chen Institute of Statistical Science, Academia Sinica Joint with Nilanjan
More informationGeneral Linear Model: Statistical Inference
Chapter 6 General Linear Model: Statistical Inference 6.1 Introduction So far we have discussed formulation of linear models (Chapter 1), estimability of parameters in a linear model (Chapter 4), least
More informationUNIVERSITÄT POTSDAM Institut für Mathematik
UNIVERSITÄT POTSDAM Institut für Mathematik Testing the Acceleration Function in Life Time Models Hannelore Liero Matthias Liero Mathematische Statistik und Wahrscheinlichkeitstheorie Universität Potsdam
More informationCTDLPositive Stable Frailty Model
CTDLPositive Stable Frailty Model M. Blagojevic 1, G. MacKenzie 2 1 Department of Mathematics, Keele University, Staffordshire ST5 5BG,UK and 2 Centre of Biostatistics, University of Limerick, Ireland
More informationModeling and Analysis of Recurrent Event Data
Edsel A. Peña (pena@stat.sc.edu) Department of Statistics University of South Carolina Columbia, SC 29208 New Jersey Institute of Technology Conference May 20, 2012 Historical Perspective: Random Censorship
More informationFrailty Modeling for Spatially Correlated Survival Data, with Application to Infant Mortality in Minnesota By: Sudipto Banerjee, Mela. P.
Frailty Modeling for Spatially Correlated Survival Data, with Application to Infant Mortality in Minnesota By: Sudipto Banerjee, Melanie M. Wall, Bradley P. Carlin November 24, 2014 Outlines of the talk
More informationFull likelihood inferences in the Cox model: an empirical likelihood approach
Ann Inst Stat Math 2011) 63:1005 1018 DOI 10.1007/s104630100272y Full likelihood inferences in the Cox model: an empirical likelihood approach JianJian Ren Mai Zhou Received: 22 September 2008 / Revised:
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 informationTwo LikelihoodBased Semiparametric Estimation Methods for Panel Count Data with Covariates
Two LikelihoodBased Semiparametric Estimation Methods for Panel Count Data with Covariates Jon A. Wellner 1 and Ying Zhang 2 December 1, 2006 Abstract We consider estimation in a particular semiparametric
More informationAFT Models and Empirical Likelihood
AFT Models and Empirical Likelihood Mai Zhou Department of Statistics, University of Kentucky Collaborators: Gang Li (UCLA); A. Bathke; M. Kim (Kentucky) Accelerated Failure Time (AFT) models: Y = log(t
More informationA fast routine for fitting Cox models with time varying effects
Chapter 3 A fast routine for fitting Cox models with time varying effects Abstract The Splus and R statistical packages have implemented a counting process setup to estimate Cox models with time varying
More informationDepartment of Biostatistics University of Copenhagen
Cox regression with missing covariate data using a modified partial likelihood method Torben Martinussen Klaus K. Holst Thomas Scheike Research Report 14/7 Department of Biostatistics University of Copenhagen
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 Nonparametric Estimates of Survival Comparing
More informationST495: Survival Analysis: Maximum likelihood
ST495: Survival Analysis: Maximum likelihood Eric B. Laber Department of Statistics, North Carolina State University February 11, 2014 Everything is deception: seeking the minimum of illusion, keeping
More informationasymptotic normality of nonparametric Mestimators with applications to hypothesis testing for panel count data
asymptotic normality of nonparametric Mestimators with applications to hypothesis testing for panel count data Xingqiu Zhao and Ying Zhang The Hong Kong Polytechnic University and Indiana University Abstract:
More informationGrouped variable selection in high dimensional partially linear additive Cox model
University of Iowa Iowa Research Online Theses and Dissertations Fall 2010 Grouped variable selection in high dimensional partially linear additive Cox model Li Liu University of Iowa Copyright 2010 Li
More informationCURE MODEL WITH CURRENT STATUS DATA
Statistica Sinica 19 (2009), 233249 CURE MODEL WITH CURRENT STATUS DATA Shuangge Ma Yale University Abstract: Current status data arise when only random censoring time and event status at censoring are
More informationA COMPARISON OF POISSON AND BINOMIAL EMPIRICAL LIKELIHOOD Mai Zhou and Hui Fang University of Kentucky
A COMPARISON OF POISSON AND BINOMIAL EMPIRICAL LIKELIHOOD Mai Zhou and Hui Fang University of Kentucky Empirical likelihood with right censored data were studied by Thomas and Grunkmier (1975), Li (1995),
More informationBIAS OF MAXIMUMLIKELIHOOD ESTIMATES IN LOGISTIC AND COX REGRESSION MODELS: A COMPARATIVE SIMULATION STUDY
BIAS OF MAXIMUMLIKELIHOOD ESTIMATES IN LOGISTIC AND COX REGRESSION MODELS: A COMPARATIVE SIMULATION STUDY Ingo Langner 1, Ralf Bender 2, Rebecca LenzTönjes 1, Helmut Küchenhoff 2, Maria Blettner 2 1
More informationOutline of GLMs. Definitions
Outline of GLMs Definitions This is a short outline of GLM details, adapted from the book Nonparametric Regression and Generalized Linear Models, by Green and Silverman. The responses Y i have density
More informationAssessing the effect of a partly unobserved, exogenous, binary timedependent covariate on APPENDIX
Assessing the effect of a partly unobserved, exogenous, binary timedependent covariate on survival probabilities using generalised pseudovalues Ulrike Pötschger,2, Harald Heinzl 2, Maria Grazia Valsecchi
More informationIntegrated likelihoods in survival models for highlystratified
Working Paper Series, N. 1, January 2014 Integrated likelihoods in survival models for highlystratified censored data Giuliana Cortese Department of Statistical Sciences University of Padua Italy Nicola
More informationEstimation of Conditional Kendall s Tau for Bivariate Interval Censored Data
Communications for Statistical Applications and Methods 2015, Vol. 22, No. 6, 599 604 DOI: http://dx.doi.org/10.5351/csam.2015.22.6.599 Print ISSN 22877843 / Online ISSN 23834757 Estimation of Conditional
More informationDescription Syntax for predict Menu for predict Options for predict Remarks and examples Methods and formulas References Also see
Title stata.com stcrreg postestimation Postestimation tools for stcrreg Description Syntax for predict Menu for predict Options for predict Remarks and examples Methods and formulas References Also see
More informationContinuous Time Survival in Latent Variable Models
Continuous Time Survival in Latent Variable Models Tihomir Asparouhov 1, Katherine Masyn 2, Bengt Muthen 3 Muthen & Muthen 1 University of California, Davis 2 University of California, Los Angeles 3 Abstract
More informationBusiness Statistics. Lecture 9: Simple Regression
Business Statistics Lecture 9: Simple Regression 1 On to Model Building! Up to now, class was about descriptive and inferential statistics Numerical and graphical summaries of data Confidence intervals
More informationSTAT 512 MidTerm I (2/21/2013) Spring 2013 INSTRUCTIONS
STAT 512 MidTerm I (2/21/2013) Spring 2013 Name: Key INSTRUCTIONS 1. This exam is open book/open notes. All papers (but no electronic devices except for calculators) are allowed. 2. There are 5 pages in
More informationMultivariate spatial modeling
Multivariate spatial modeling Pointreferenced spatial data often come as multivariate measurements at each location Chapter 7: Multivariate Spatial Modeling p. 1/21 Multivariate spatial modeling Pointreferenced
More informationF9 F10: Autocorrelation
F9 F10: Autocorrelation Feng Li Department of Statistics, Stockholm University Introduction In the classic regression model we assume cov(u i, u j x i, x k ) = E(u i, u j ) = 0 What if we break the assumption?
More informationCASE STUDY: Bayesian Incidence Analyses from CrossSectional Data with Multiple Markers of Disease Severity. Outline:
CASE STUDY: Bayesian Incidence Analyses from CrossSectional Data with Multiple Markers of Disease Severity Outline: 1. NIEHS Uterine Fibroid Study Design of Study Scientific Questions Difficulties 2.
More informationTwolevel lognormal frailty model and competing risks model with missing cause of failure
University of Iowa Iowa Research Online Theses and Dissertations Spring 2012 Twolevel lognormal frailty model and competing risks model with missing cause of failure Xiongwen Tang University of Iowa Copyright
More informationA Course in Applied Econometrics Lecture 14: Control Functions and Related Methods. Jeff Wooldridge IRP Lectures, UW Madison, August 2008
A Course in Applied Econometrics Lecture 14: Control Functions and Related Methods Jeff Wooldridge IRP Lectures, UW Madison, August 2008 1. LinearinParameters Models: IV versus Control Functions 2. Correlated
More informationMultivariable Fractional Polynomials
Multivariable Fractional Polynomials Axel Benner September 7, 2015 Contents 1 Introduction 1 2 Inventory of functions 1 3 Usage in R 2 3.1 Model selection........................................ 3 4 Example
More informationGeneralized Linear Models with Functional Predictors
Generalized Linear Models with Functional Predictors GARETH M. JAMES Marshall School of Business, University of Southern California Abstract In this paper we present a technique for extending generalized
More informationUnderstanding the Cox Regression Models with TimeChange Covariates
Understanding the Cox Regression Models with TimeChange Covariates Mai Zhou University of Kentucky The Cox regression model is a cornerstone of modern survival analysis and is widely used in many other
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 informationNotes from Survival Analysis
Notes from Survival Analysis Cambridge Part III Mathematical Tripos 20122013 Lecturer: Peter Treasure Vivak Patel March 23, 2013 1 Contents 1 Introduction, Censoring and Hazard 4 1.1 Introduction..............................
More informationAsymptotic statistics using the Functional Delta Method
Quantiles, Order Statistics and LStatsitics TU Kaiserslautern 15. Februar 2015 Motivation Functional The delta method introduced in chapter 3 is an useful technique to turn the weak convergence of random
More informationA Blockwise Descent Algorithm for Grouppenalized Multiresponse and Multinomial Regression
A Blockwise Descent Algorithm for Grouppenalized Multiresponse and Multinomial Regression Noah Simon Jerome Friedman Trevor Hastie November 5, 013 Abstract In this paper we purpose a blockwise descent
More informationNonparametric Tests for the Comparison of Point Processes Based on Incomplete Data
Published by Blackwell Publishers Ltd, 108 Cowley Road, Oxford OX4 1JF, UK and 350 Main Street, Malden, MA 02148, USA Vol 28: 725±732, 2001 Nonparametric Tests for the Comparison of Point Processes Based
More informationAnalysis of competing risks data and simulation of data following predened subdistribution hazards
Analysis of competing risks data and simulation of data following predened subdistribution hazards Bernhard Haller Institut für Medizinische Statistik und Epidemiologie Technische Universität München 27.05.2013
More informationVersion of record first published: 01 Jan 2012.
This article was downloaded by: [North Carolina State University] On: 15 October 212, At: 8:45 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 172954 Registered
More informationA NOTE ON ROBUST ESTIMATION IN LOGISTIC REGRESSION MODEL
Discussiones Mathematicae Probability and Statistics 36 206 43 5 doi:0.75/dmps.80 A NOTE ON ROBUST ESTIMATION IN LOGISTIC REGRESSION MODEL Tadeusz Bednarski Wroclaw University email: t.bednarski@prawo.uni.wroc.pl
More informationLattice Data. Tonglin Zhang. Spatial Statistics for Point and Lattice Data (Part III)
Title: Spatial Statistics for Point Processes and Lattice Data (Part III) Lattice Data Tonglin Zhang Outline Description Research Problems Global Clustering and Local Clusters Permutation Test Spatial
More informationSemiparametric maximum likelihood estimation in normal transformation models for bivariate survival data
Biometrika (28), 95, 4,pp. 947 96 C 28 Biometrika Trust Printed in Great Britain doi: 1.193/biomet/asn49 Semiparametric maximum likelihood estimation in normal transformation models for bivariate survival
More informationStatistical Modeling and Analysis for Survival Data with a Cure Fraction
Statistical Modeling and Analysis for Survival Data with a Cure Fraction by Jianfeng Xu A thesis submitted to the Department of Mathematics and Statistics in conformity with the requirements for the degree
More informationVariable Selection and Model Choice in Survival Models with TimeVarying Effects
Variable Selection and Model Choice in Survival Models with TimeVarying Effects Boosting Survival Models Benjamin Hofner 1 Department of Medical Informatics, Biometry and Epidemiology (IMBE) FriedrichAlexanderUniversität
More informationLinear Methods for Prediction
This work is licensed under a Creative Commons AttributionNonCommercialShareAlike License. Your use of this material constitutes acceptance of that license and the conditions of use of materials on this
More informationGeneralized additive modelling of hydrological sample extremes
Generalized additive modelling of hydrological sample extremes Valérie ChavezDemoulin 1 Joint work with A.C. Davison (EPFL) and Marius Hofert (ETHZ) 1 Faculty of Business and Economics, University of
More informationAssessment of time varying long term effects of therapies and prognostic factors
Assessment of time varying long term effects of therapies and prognostic factors Dissertation by Anika Buchholz Submitted to Fakultät Statistik, Technische Universität Dortmund in Fulfillment of the Requirements
More informationChapter 1 Statistical Inference
Chapter 1 Statistical Inference causal inference To infer causality, you need a randomized experiment (or a huge observational study and lots of outside information). inference to populations Generalizations
More information