Threshold Regression for Time-to-event Data Mei-Ling Ting Lee University of Maryland, College Park MLTLEE@UMD.EDU
Outline The proportional hazards (PH) model is widely used in analyzing time-to-event data. It requires, however, a strong PH assumption. Threshold regression (TR) is an alternative model without the PH assumption. Brief introduction of the TR model. Extensions and Examples Connections between TR and PH models
Hazard Function: The instantaneous failure rate Pr( t T < t +Δ t T t ) ht () = lim Δ t 0 Δt t ht () = f () t St ()
The Proportional Hazards (PH) Model For any covariates x 1 and x 2, the PH model assumes that the hazard functions are proportional to a baseline hazard h 0 (t). h t x h t x ( 1) = 0( )exp( 1β ) h ( t x 2 ) = h 0 ( t ) exp( x 2 β ) h ( t x 1) e x p ( x 1β ) = = h ( t x 2 ) exp( x 2 β ) exp[( x 1 x 2) ] β The hazard ratio of the PH model has a simple e a a d a o o e ode as a s p e form that does not depend on time t.
Hazard rate is really an elusive concept, especially when one tries to interpret its shape considered as a function of time. Aalen, Gjessing (2001), Understanding the shape of the hazard rate: a process point of view, Statistical Science, 16, 1-22. S i l d Aalen, Borgan, and Gjessing (2008). Survival and Event History Analysis: a Process Point of view. Springer.
Consequences of the PH Assumption As a consequence of the proportional hazards assumption, the survivorship functions of the two groups must be parallel.
Example 1 Time to infection of kidney dialysis i patients t with different catheterization procedures (Nahman et al 1992, Klein & Moesberger 2003) Surgical group: 43 patients utilized a surgically placed catheter Percutaneous group: 76 patients utilized a percutaneous placement of their catheter Th t ti i d fi d b th ti t The event time is defined by the time to cutaneous exit-site infection.
Example 1 Kaplan-Meier Estimate t versus PH Cox Model
Example 1 Generalized Gamma Threshold Regression 9
Threshold Regression Model can estimate hazard ratios at different time points timevalue=2 timevalue=20 xi: threg,lny0(i.group) mu(i.group) failure(infection) /* */ time(time) hr(group) timevalue(2) graph(hz) xi: threg,lny0(i.group) mu(i.group) failure(infection) /* */ time(time) hr(group) timevalue(20) graph(hz) Hazard Rate Hazard Rate 0 0 h(t).02.04.0 06.08.1 h(t).02.04.0 06.08.1 0 2 10 20 30 Calendar Time 0 10 20 30 Calendar Time reference _Igroup_2==1 reference _Igroup_2==1 Proportional Hazard Cox Model can only estimate t a constant t hazard ratio stset time, failure(infection) xi: stcox i.group
Example 2 Time to progression of disease for patients with limited Stage II or IIIA ovarian carcinoma (Fleming et al., 1980; Hess, 1995) Low-grade tumor group: 15 patients with low-grade or well-differentiated tumors High-grade tumor group: 20 patients with high-grade grade or undifferentiated tumors 12
Example 2 Kaplan-Meier Estimates 13
Example 2 Generalized Gamma Threshold Regression Threshold 14
Example 3 Time to death for patients with bile duct cancer (Fleming et al., 1980; Lawless, 2003) Control group: 25 patients without treatment Treatment group: 22 patients with a combined treatment of radiation and the drug 5-fluorouracil (5-FU) 15
Example 3 Kaplan-Meier Estimates 16
Example 3 Generalized Gamma Threshold Regression 17
Threshold Regression is based on the First-hitting-time (FHT) Model Modeling event times by a stochastic process reaching a boundary
Threshold Regression is based on First-hitting-time time Methodology : Example: Engineering Equipment fails when its cumulative wear first reaches a failure threshold. Example: Health and Medicine A patient dies of heart disease when condition of the heart deteriorates to a critical state. Example: Social Sciences Divorce occurs when tensions in a marriage reach a breaking point. A review article: Modeling Event Times by Stochastic Process Reaching a Boundary (Lee & Whitmore 2006, Statistical Sciences)
y 0 Process Sample path Y(t) 0 0 time t S First hitting time S of a fixed boundary at level zero for a stochastic process of interest Y(t)
Process Y(t) y 0 Sample paths Y(t) 0 Time t S L Two sample paths of a stochastic process of interest: (1) One path experiences failure at first hitting time S (2) One path is surviving at end of follow up at time L
First Hitting Time (FHT) Models {Y(t)} : the stochastic process of interest Β : the threshold or boundary set First hitting time S defined by S = inf { t : Y(t) B }
Examples of First-hitting-time (FHT) Models: Wiener diffusion to a fixed boundary Progress of multiple l myeloma until death Renewal process to a fixed count of renewal events Time to the n th epileptic seizure Semi-Markov process to an absorbing state Multi-state model for disease with death as an absorbing state
First Hitting Time (FHT) Models The first hitting time (FHT) model describes many time-to-event applications The stochastic process of interest {Y(t)} may represent the latent (unobservable) health status of a subject. The threshold constitutes the critical level of the process that triggers the failure event (e.g., symptomatic cancer, death). The event occurs when health status {Y(t)} first decreases to the zero threshold.
Parameters for the FHT Model Model parameters for a latent process Y(t) : Process parameters: θ = (μ, σ 2 ), where μ is the mean drift and σ 2 is the variance Baseline level of process: Y(0) = y 0 Because Y(t) () is latent, we set σ 2 = 1.
Threshold Regression: FHT Model with Regression Structure Including covariates to FHT model parameters θ = (μ, σ 2 ) using link functions, where σ 2 = 1. Mean drift parameters: μ = g 1 ( x 1, x 2, ) Baseline level: y(0) = g 2 (x 1, x 2, ) A variety of link functions g have been considered: Linear combinations of covariates X1,, Xp Polynomial combinations of X1,, Xp Semi-parametric regression splines Penalized regression splines Random effects and Bayesian models 26
Likelihood Inference for the FHT Model The likelihood contribution of each sample subject is as follows. If the subject fails at S=s: f (s y 0, μ) = Pr [ first-hitting-time in (s, s+ds) ] If the subject survives beyond time L: 1- F (L y 0,μ) = Pr [ no first-hitting-time before L ]
n { } i i i i ln L, x = d ln f t, x + 1 d ln F t, x. where ( θ ) ( θ ) ( ) ( θ ) d i 0 0 0 i= 1 is the failure indicator for subject t is a censored survival time t = s if subject i fails i i i i ( ) f and F denote the FHT p.d.f and complementary c.d.f.
Extensions of Threshold Regression (TR) 1. Process Y(t): Wiener process, gamma process, etc 2. Boundary: straight lines or curves 3. Time scale: calendar time, running or analytical time
Example 4 Comparing TR and PH for two possible scenarios Scenario 1: A favorable surgical intervention might double the health level of a patient (relative to control) at the outset of the study but leaves the rate of decline of health unchanged on both study arms. Scenario 2: Drug treatment for chronic disease may slow the rate of decline in health (relative to control) but leaves the initial health level unchanged on both study arms. 30
Scenario 1 Scenario 2 2 2 Process Level y 1-1 Treatment t Process Level 1 y Treatment t -1 Control -1-0.5 Control Time Time
Example 4 Consider a simulated randomized clinical trial with two study arms and n=200 subjects on each arm. (control: arm=0; treatment: arm=1). The TR model is a Wiener diffusion model. Regression links for the TR model ln (y 0 )=α α 0 + α 1 arm μ = β 0 + β 1 arm In Scenario 1: α 1 =ln(2) = 0.6931, β 1 =0 In Scenario 2: α 1 = ln (1) = 0, β 1 = 0.5 32
Example 4 (simulation) The TR model: ln (y 0 ) = α 0 + α 1 arm μ = β 0 + β 1 arm The mean survival time on any arm is given by E(S)= y 0 / μ Both scenarios have the same mean of 1 year survival on the control arm and 2 years on the treatment arm. 33
A Comparison of TR and PH Regression Results
Example 4: Modeling Results for Scenario 1 In scenario 1, the PH assumption is untenable. The analyst would then need to proceed to some other model. If the TR model is considered, the table shows a significant positive treatment effect for ln y 0 but an insignificant effect for μ (as expected). The significant coefficient for α 1 translates into an estimated t exp(0.7752)=2.2 2 2 multiplier li for initial health level y 0 (note that the true value is 2, i.e. the treatment arm is twice as large at y 0 )
Example 4: Modeling Results for Scenario 2 In scenario 2, the PH assumption is tenable. The analyst using PH regression would discover a significant hazard ratio of 0.5581, indicating that the treatment is about 56% of the control risk. However, the question of how the treatment is acting on the subject is not known from the PH model. If the TR model is considered, as expected, the table shows a significant ifi positive treatment t t effect for μ but an insignificant effect for ln y 0 (). The significant ifi coefficient i for β 1 translates t into an estimated 0.4282 multiplier in the value of μ (the treatment is about half of the control). 36
2 2 Scenario 1 1.2 1.6 Control hazard Treatment hazard 1.2 1.6.4.8.4.8 Log-hazard ratio 0 0 0 5 10 Time (years) 0 2 4 6 8 10 Time (years) Scenario 2 1.2 1.6 1.2 1.6 2 2 Control hazard Log-hazard ratio Treatment hazard 0 0.4.8.4.8 0 5 10 Time (years) 0 5 10 Time (years)
Connection of the TR model with the PH model Most survival distributions are hitting time distributions for stochastic processes (the basic TR model) Families of PH functions can be generated by varying time scales or boundaries of a TR model Th f il f PH f ti b d d The same family of PH functions can be produced by different TR mode
Connection of the TR model with PH model Families of PH functions can be generated by 1. Varying process Y and time scales of a TR model 2. Varying process Y and boundaries of a TR model. Hence, the same family of PH functions can be produced by different TR models. For details, see M-LT Lee, Whitmore GA. Proportional hazards and threshold regression: their theoretical and practical connections. Lifetime Data Analysis (2010). 39
Benefits of the TR Model Probing the causal forces behind a hazard function is always a worthwhile endeavor. A TR model represents more fundamental knowledge about the underlying science than its corresponding PH model. These insights would deepen the investigator s understanding of the case application.
Whitmore, G. A. (1986). First-passage-time models for duration data: Regression structures and competing risks. The Statistician 35 207 219. Aalen O.O. and Gjessing H.K. (2001). Understanding the shape of the hazard rate: a process point of view. Statistical Science, 16: 1-22. Lee, M.-L. T. and G. A. Whitmore (2006). Threshold regression for survival analysis: modeling event times by a stochastic process reaching a boundary. Statistical Sciences. Aalen O.O., Borgon O, and Gjessing H.K (2008). Survival and Event History Analysis: A process Point of View. Springer. 41
Yu Z, Tu W, Lee M-LT (2009). A Semiparametric Threshold Regression Analysis of Sexually Transmitted Infections in Adolescent Women, Statistics in Medicine. Pennell M, GA Whitmore, Lee M-LT (2009). Bayesian random effects threshold regression with application to survival data with nonproportional hazards, Biostatistics. Lee M-LT, Whitmore GA (2010). Proportional hazards and threshold regression: their theoretical and practical connections. Lifetime Data Analysis. Lee M-LT Whitmore GA Rosner B (2010) Threshold Lee M LT, Whitmore GA, Rosner B. (2010) Threshold regression for survival data with time-varying covariates. Statistics in Medicine.
Nested boundaries Z=0 Z=1 Z=2 Process Level y Representative sample path for z=2 Time
.4 Z=0 Z=1 Z=2.2.3 Hazard h(t) 0.1 0 10 20 30 40 Time x t
Boundary b(t z) y(t z) Process Level y 0 0 s Time t
y(t z) Process Level y Overshoot y(s z)-b(s z) Boundary b(s z) Boundary b(t z) Time s t
Aalen O.O. and Gjessing H.K. (2001). Understanding the shape of the hazard rate: a process point of view. Statistical Science, 16: 1-22. Lee, M.-L. T. and G. A. Whitmore (2006). Threshold regression for survival analysis: modeling event times by a stochastic process reaching a boundary. Statistical Sciences. Aalen O.O., Borgon O, and Gjessing H.K (2008). Survival and Event History Analysis: A process Point of View. Springer. Yu Z, Tu W, Lee M-LT (2009). A Semiparametric Threshold h Regression Analysis of Sexually Transmitted Infections in Adolescent Women, Statistics in Medicine (in press). Pennell M, GA Whitmore, Lee M-LT (2009). Bayesian random effects threshold regression with application to survival data with nonproportional hazards, Biostatistics, (In Press).
The Connection of Threshold Regression Models With Cox Proportional Hazard Models Mei-Ling Ting Lee, University of Maryland Alex Whitmore, McGill University Lifetime Data Analysis 2010
Relation of Cox Proportional Hazards and Threshold Regression The class of survival functions that are firsthitting-time time (FHT) distributions includes most distributions of practical interest Almost any family of proportional hazard (PH) functions can be generated from an FHT context by an appropriate choice of process time or boundary for the stochastic process defining the FHT.
Cox regression can be considered d as a special case of threshold regression (TR) for the class of ffhtdi distributions tib ti that thave the proportional hazard property Suitable methods allow TR to replicate Cox PH regression with either fixed or time-varying i covariates
Generating PH Functions in an FHT Context Three building blocks for threshold regression 1. A stochastic process that describes the evolution of the subject's underlying health state 2. A critical level, condition or boundary (the threshold) that triggers the event of interest when it is reached by the process for the first time (the FHT) 3. A time scale on which the process unfolds.
Generating PH Functions in an FHT Context The three building blocks of TR give many combinations of conditions that can generate the same family of Cox proportional hazard functions: We now show two methods of construction of a PH family from an FHT context.
1. Generating PH by Varying Process Time Let denote a process time function, defined as a function of calendar time t, conditional on covariate z. Let the survival function for FHT context be denoted by: We have the following two correspondences: Varying process time:
Generating PH by Varying Process Time Equating the two right-hand hand sides and solving for gives process time functions that generate a family of PH functions.
Examples: Poisson process: A Poisson process with unit intensity has the following exponential survival time distribution: Solving for the process time function gives the following. Observe that the process time is proportional to the cumulative baseline hazard function H 0 (t).
Examples: Brownian motion process: A Brownian motion process with a unit variance, starting at x 0, has the following survival time distribution: where Φ denotes the standard normal c.d.f. Solving for the process time function gives:
2. Generating PH by Varying Boundaries A family of PH functions can be generated by varying the boundary function. Example: Brownian motion process The following identity links the boundary b(t z) to the c.d.f. of survival time F(t z) for a given covariate vector z in standard Brownian motion starting at zero:
Generating PH by Varying Boundaries Replacing F(t z) ( by the c.d.f. for each member of a PH family and solving for boundary function b(t z) ( )gives a set of nested boundaries that generate FHTs with proportional hazard functions.
Example: The next figure shows a nested set of three boundaries for standard Brownian motion that correspond to three levels of a fixed covariate z. Covariate z might represent, for example, level of occupational exposure to a carcinogen The figure shows an illustrative sample path for a case where exposure level z=2 generates a survival time S.
10 0 b0 b1 b2 Nested boundaries z=0 z=1 z=2-10 Simulated path x(t) for z=2 0 20 40 x S Three boundaries in Brownian motion that correspond to three levels of covariate z
Example: The next figure shows the hazard functions for survival time corresponding to the preceding three boundaries for Brownian motion. The boundaries and process have been chosen p so the hazard functions are, in fact, proportional for the three levels of fixed covariate z.
haz0 haz1 haz2 h(t) ().3.2 z=1 z=0.1 z=2 0 0 20 40 x Hazard functions generated by the FHT model. The functions are proportional p for the three levels of covariate z.
The threg package in R
Install threg package in R Open R in Windows From menu Packages > > Install package(s) from local zip files..., select the file "threg_1.0.1.zip 1 zip" and click "Open. Type the following command in R prompt to load this package: > library(threg) Now you can use the threg package in R.
An example The following example can be found in the help document of the threg package, by typing the following command in R prompt: >?threg
The myeloma dataset *The myeloma dataset is a simulated dataset with 49 observations. time fail age gender treat Variable Name survival time Description censoring indicator variable 0 indicates a right censored time and 1 indicates an observed event time. age of the corresponding subject. It is a continuous variable. gender of the corresponding subject. It is a categorical variable. treatment type of the corresponding subject. It is a categorical variable with 4 levels.
Load the myeloma dataset load the data "myeloma in the package: > data("myeloma") Transform the "gender" and "treat" variables into factor variables "f.gender gender and "ftreat" f.treat. > f.gender< factor(myeloma$gender) > f.treat< factor(myeloma$treat) f bind the two new factor variables "f.gender" and "f.treat" to the dataset "myeloma". > myeloma< cbind(myeloma,f.gender,f.treat) fgenderftreat)
Fit a threshold regression model by using threg function Fit a threshold h regression model, dlby using age and f.gender to predict parameter lny0 in the threshold regression model, and by using age, g, f.gender and f.treat to predict parameter mu in the threshold regression model: > threg(data=myeloma, t + lny0=age+f.gender, + mu=age+f.gender+f.treat, gender+f + time=time, + failure=fail + )
Output (regression coefficient estimates, log likelihood value and AIC value) $coef.estest Coef. Est. Std. Err. z P> z lny0: (Intercept) 3.2982293 1.2223463 2.698 0.0070 lny0: age 0.0141177 0.0169046 0.835 0.4037 lny0: f.gender(1) 0.0779063 0.3553153 0.219 0.8267 mu: (Intercept) 4.6616101 4.9709192 0.938 0.3482 mu: age 0.0172552 0.0707793 0.244 0.8072 mu: f.gender(1) 0.1134031 1.3306915 0.085 0.9323 mu: f.treat(1) 0.1710347 0.2071231 0.826 0.4088 mu: f.treat(2) 0.0941807 0.2110082 0.446 0.6556 mu: f.treat(4) 0.7977747 0.6781792 1.176 0.2396 $log.likelihood [1] 29.91943 $AIC [1] 77.83886
Hazard ratio calculation Fit the same threshold h regression model dlas previous, and calculate the hazard ratios for the factor variable "f.treat", at a fixed time point 3, for a specified scenario: age=50, gender=0: > threg(data=myeloma, + lny0=age+f.gender, + mu=age+f.gender+f.treat, + time=time, + failure=fail, + hr=f.treat, + timevalue=3, + scenario=age(50)+f.gender(0) + )
Output (hazard ratios) (some output tomitted) $scenario [1] "f.gender = 0 age=50 at time = 3" $hazard.ratio Haz. Ratio f.treat(1) 0.6416700 f.treat(2) 0.7891791 f.treat(4) 4.1874754
Plot of hazard functions Generate the overlaid lidestimated t hazard dfunctions corresponding to different levels of "treat", for a specified scenario: age=50,,gender=0: > threg(data=myeloma, + lny0=age+f.gender, + mu=age+f.gender+f.treat, + time=time, + failure=fail, + hr=f.treat, + timevalue=3, + scenario=age(50)+f.gender(0), + graph="hz" + )
Output (estimated hazard functions )
Plot of survival functions Generate the overlaid lidestimated t survival lfunctions corresponding to different levels of "treat", for a specified scenario: age=50,,gender=0: > threg(data=myeloma, + lny0=age+f.gender, + mu=age+f.gender+f.treat, + time=time, + failure=fail, + hr=f.treat, + timevalue=3, + scenario=age(50)+f.gender(0), + graph= sv" + )
Output (estimated survival functions )
Plot of probability density functions Generate the overlaid estimated t probability bilit density functions corresponding to different levels of "treat", for a specified scenario: age=50,,gender=0: > threg(data=myeloma, + lny0=age+f.gender, + mu=age+f.gender+f.treat, + time=time, + failure=fail, + hr=f.treat, + timevalue=3, + scenario=age(50)+f.gender(0), + graph= ds" + )
Output (estimated probability density functions )