Survival Analysis. René Böheim November 2013

Transcription

1 Survival Analysis Based on Cleves, Gould, and Gutierrez (2004), An Introduction to Survival Analysis using Stata, Revised Edition, Stata Press, Texas. René Böheim November 2013

2 Introduction The key question is: How long does it take before a certain incidence takes place? Examples: Medicine: How long does it take to die after a surgery? Technics: How long does it take before a machine breaks down? Economics: How long does it take before an unemployed person can find a job? Also: time to failure, survival time, duration data. Ökonometrie 1 / 66

3 Example Point in time Characteristic of the incidence (t) (x) Ökonometrie 2 / 66

4 OLS? time i = β 0 + β 1 x i + ε i ε i N(0, σ 2 ) time i N(β 0 + β 1 x i, σ 2 ) Not always suitable, e.g. if the risk is constant over time. Assumption about the distribution of the ε i lead to parametric models. Ökonometrie 3 / 66

5 Probit o.ä.? Probability of a incidence after exactly one time unit: Pr[incidence i = 1] = F(β x i ). After t time units? In general yes, but inefficient. Moreover, we want to hold e.g., β constant over time. These considerations lead to semiparametric models. (Semiparametric, because there is no specific assumption about the distribution of time, but the x are parametrised.) Ökonometrie 4 / 66

6 Sampling Stock sampling: random sample of those, who were at point of time t in the observed state (of interest), e.g. Observation of the unemployment duration of all, who were unemployed on Dec., 22th. Problem: long durations are systematically collected more often! Inflow sampling: random sample of those, who start a epsiode at a specific point in timen, e.g. Observation of unemployment durations of those, who were unemployed between Jan., 1st and March, 31st as random sample for those, who were unemployed once. (Problem: seasonal variations.) Outflow sampling: random sample of those, who end one period, e.g. how long have unemployed people had a job? Population sampling: Evaluation of episodes, which were e.g. obtained from a Population Survey. Ökonometrie 5 / 66

7 Problems with data Censoring : not observed starting or ending points Data are called left-censored if the starting point cannot be observed. One knows that someone was already 10 days in hospital, but not the point in time when she felt ill. right-censored : at the observed point in time the transition did not take place. E.g. a unemployed person is still unemployed. Truncation : systematic exclusion of specific episodes left-truncated data : if only observations which have a minimum duration experienced are included in the sample ( Delayed entry, stock sampling with follow-up.) short durations are now systematically excluded, e.g. unemployed who were at least 4 weeks without employment. right-truncated data : if only observations which make a transition at a specific point in time are included in the sample. Long durations are systematically excluded, e.g. unemployed people who find a job during the year. Ökonometrie 6 / 66

8 continuous or discrete Discrete: time is a discrete sequence of fixed intervals. Data is often organized in discrete units, e.g. weeks or quarters. Continuous: time is a continuum and the duration of an episode can be described by positive real numbers. Ökonometrie 7 / 66

9 A simple example In a simple search model, the search for a job of an unemployed person with respect to a reservation wage, r, the frequency of a job offer, ξ(t) and the magnitude of the wage offer are explained by θ(t) = ξ(t)[1 W(t)]. The duration of the job search (the hazard of re-employment) is a function of the reservation wage. Ökonometrie 8 / 66

10 Explanatory Variables Fixed in the episode: e.g. sex, location. Time-dependent variables : Varied with calendar: e.g., age. Varied with duration of the episode: e.g., unemployment compensation. Ökonometrie 9 / 66

11 Search model The duration could be modeled the following way: θ(t) = θ(x(t, s), t), where t is a vector of the characteristics which vary in the curse of unemployment (t) or over time (calendar time) s. Ökonometrie 10 / 66

12 Time-varying covariates This permits e.g., Unemployment compensation varies with the duration of unemployment and/or over time (e.g. changed laws), Employers select job seeker and do not employ long-term unemployed, θ/ t < 0, the reservation wage decreases with lasting unemployment, r/ t < 0 θ/ t > 0!. discouraged seekers search less intensively, the longer they are unemployed, θ/ t < 0 Benefit-exhaustion effect : the search intensity increases, the sooner the unemployment compensation ends, θ(t)/ t > 0... Ökonometrie 11 / 66

13 Overview of some models 1. Non-parametric Kaplan-Meier estimator 2. Parametric Proportional Hazard Models Exponential Weibull Gompertz Accelerated Failure Time Exponential (Weibull) 3. Semi-parametric Cox Regression Ökonometrie 12 / 66

14 Stata -st-: survival time commands -stset-: declare data (analogue to -tsset-) -stdes-: describe survival time data -stsum-: summarise survival time data -sts-: graphs, lists, and tests -ltable-: Kaplan-Meier estimate -streg-: Weibull and other models -stcox-: Cox s model Ökonometrie 13 / 66

15 Non-parametric methods No assumptions about: Distribution of the survival function (of the hazard) Impact of covariates Ökonometrie 14 / 66

16 The Kaplan-Meier estimator The KM-estimator is a estimator of the survival function S(t), the probability of surviving beyond t : ( ) nj d j Ŝ(t) =, j t j t where n j is the number of observations which is in the risk set, this are those who have survived until point in time t. d j denotes those observations which died in the interval, e.g. either the incidence happened or it denotes censored observations. (It is also named product limit estimate.) n j Ökonometrie 15 / 66

17 Beispiel The observations are ordered after the duration until the occurrence of the incidence (or the end of the observation period (censored)). Person t died? (1=yes) Ökonometrie 16 / 66

18 t n_j d_j number number at risk failed Ökonometrie 17 / 66

19 t n_j d_j number number at risk failed p / / /3 Ökonometrie 18 / 66

20 t n_j d_j number number. at risk failed p S(t) /6 5/ /5 1/ /3 1/3 Ökonometrie 19 / 66

21 Graphically Ökonometrie 20 / 66

22 Censored data Person t died? (1=yes, 0=not observed) Ökonometrie 21 / 66

23 n_j d_j t n at risk n failed censored Ökonometrie 22 / 66

24 n_j d_j t n at risk n failed censored p / / / Ökonometrie 23 / 66

25 n_j d_j. t n at risk n failed censored p S(t) /6 5/ /5 1/ / /2 1/ /4 Ökonometrie 24 / 66

26 Graphically per type Ökonometrie 25 / 66

27 Stata-Example Beg. Net Survivor Std. Time Total Fail Lost Function Error [95% Conf. Int.] output omitted Net lost...censored (1=yes) Ökonometrie 26 / 66

28 The survival function [T is a non-negative random variable and describes the time until an incidence, with density f(t) and distribution F(t).] The survival function is S(t) Pr[T > t] = 1 F(t), the probability to live longer than t. S(t = 0) = 1, monotonous declining function of time. Ökonometrie 27 / 66

29 The hazard rate (Also: conditional failure function.) is the (instantaneous rate) of the occurrence of an incidence in the next (small) interval, under the condition that the incidence has not occurred yet: Pr[t + t > T > t T > t] h(t) = lim = f(t) t 0 t S(t) Ökonometrie 28 / 66

30 The hazard rate ist a rate, no probabilites, with metric 1/t ; [0, ], d.h. no risk until certain risk, can increase with increasing exposition, decrease, or can stay constant. Ökonometrie 29 / 66

31 Simple Relationships Knowledge of one of the four functions (hazard rate, survival, density, distribution) leads to the other three: H(t) = t 0 t h(u)du = f(u) 0 S(u) du = S(t) = exp( H(t)), F(t) = 1 exp( H(t)), f(t) = h(t) exp( H(t)), h(t) = f(t) S(t). accumulated hazard rate t 0 [ ] 1 d S(u) du S(u) du = ln(s(t)), Ökonometrie 30 / 66

32 Parametric models Parametric models are divided in proportional hazard models (PH) and in accelerated failure-time models (AFT) In economics, PH models; in engineering, typically AFT models. Ökonometrie 31 / 66

33 Parametric Proportional Hazard Models h(t x i ) = h 0 (t) exp(β x i ), where h 0 (t), the baseline hazard, can have different functional forms, parametrization. Those are dependent on time t, but not on the x i. For example: h 0 (t) = exp(c), exponential model (constant hazard) h 0 (t) = αt α 1 exp(c), Weibull model h 0 (t) = exp(γt) exp(c), Gompertz model Ökonometrie 32 / 66

34 Exponential model (PH) Is the simple model because the hazard is constant: h(t x i ) = h 0 (t) exp(β x i ) = exp(c) exp(β x i ) = exp(c + β x i ), H(t x i ) = exp(c + β x i )t, S(t x i ) = exp( exp(c + β x i )t). Ökonometrie 33 / 66

35 Weibull Modell (PH) h 0 (t) = αt α 1 exp(c) h(t x i ) = h 0 (t) exp(β x i ) = αt α 1 exp(c) exp(β x i ) = αt α 1 exp(c + β x i ), H(t x i ) = exp(c + β x i )t α, S(t x i ) = exp( exp(c + β x i )t α ). Ökonometrie 34 / 66

36 Weibull hazard rates Ökonometrie 35 / 66

37 Weibull hazard rates Ökonometrie 36 / 66

38 Weibull hazard rates Ökonometrie 37 / 66

39 Weibull survivor functions Ökonometrie 38 / 66

40 Example Weibull regression t Coef. Std. Err. z P> z treatment age _cons /ln_p h 0 (t) = αt α 1 exp(β 0 ) 1.68t 0.68 exp( 11.67) = t 0.68 Ökonometrie 39 / 66

41 Example Ökonometrie 40 / 66

42 Semi-parametric model: Cox h(t x i ) = h 0 (t) exp(β x i ), h 0 (t) is the baseline hazard and exp(β x i ) the relative risk/hazard. (β x the log-relative Hazard or also risk score.) The baseline hazard h 0 (t) is not parameterized and not estimated. The baseline hazard can take every arbitrary course, decreasing, increasing or oscillating the only assumption is that it is the same for all observations. Ökonometrie 41 / 66

43 Cox-model Assumption: proportional hazard in the Cox model. h(t x i ) h(t x n ) = exp(β x i ) exp(β x n ), i.e., the hazard is related to the relative values f the x i. Ökonometrie 42 / 66

44 Example _t Coef. Std. Err. z P> z treatment age Ökonometrie 43 / 66

45 Interpretation estimated coefficient z.b. for age 0,105: an increase of age of 1 year leads to an increase of hazard of 11% (exp(0, 105) = 1, 11). z.b. für Treatment (1=Yes, 0=No) -2,256: 90% lower risk leads to treated (exp( 2, 256) = 0, 105). Ökonometrie 44 / 66

46 Changes i-th observation with k covariates: h(t x 1, x 2,..., x k ) = h 0 (t) exp(β 1 x 1 + β 2 x β k x k ), h(t x 1, (x 2 + 1),..., x k ) = h 0 (t) exp(β 1 x 1 + β 2 (x 2 + 1) + β k x k ). h(t x 1, (x 2 + 1),..., x k ) h(t x 1, x 2,..., x k ) = exp(β 2 ) Ökonometrie 45 / 66

47 cumulated baseline hazard (H 0 (t)) Ökonometrie 46 / 66

48 Estimated Survival Function (S 0 (t)) Ökonometrie 47 / 66

49 Baseline??? A Cox-Regression without covariables results in the Kaplan-Meier estimator. h 0 : is the first derivative of H 0 (t). This is not defined at the point in times, as this is a step function. The estimation of the baseline requires the estimation of the hazard contributions. These are the increases of the cumulated hazards at the incidences. Ökonometrie 48 / 66

50 Estimated baseline hazard (h 0 (t)) Ökonometrie 49 / 66

51 Stratified Analysis Instead of h 0 (t) for all, divided into groups: h(t x i ) = h 0 (t) exp(β x i ) h(t x i ) = h 01 (t) exp(β x i ), if i is in group 1, h(t x i ) = h 02 (t) exp(β x i ), if i is in group 2, etc. The h 0 are different, but the ˆβ are all the same. Ökonometrie 50 / 66

52 Generalization: Shared Frailty Model Problem: Random-effects for group j ( within-group correlation ): h i,j (t i x i, α j ) = α j h 0 (t) exp(β x i ), h i,j (t i x i, α j ) = h 0 (t) exp(β x i + ν j ) and ν j = log α j α j unobserved. For α often a Gamma-distribution with mean 1 and variance θ assumed. Cox Random-effects Model. Ökonometrie 51 / 66

53 Time-varying Covariates E.g., the search behavior of an unemployed is constant during the unemployment benefit, but changes after the end of the benefit. id beginning end unemployment benefit per week Ökonometrie 52 / 66

54 Example A drug is taken and with the exponential rate of exp(0.35t) (=half-life of 2 days) absorbed. Other variables are constant: log(h(t x, t)) = log(h 0 ) + β X = log(h 0 ) + β 1 x β tv [initial drug dose exp( 0.35t)]: Cox regression _t Haz. Ratio Std. Err. z P> z rh treatment t drug-level rh: hazard ratio which is constant over time. t: hr varies over time. Interpretation: higher drug concentration reduces the risk/hazard (about 20%=exp( 0.12)). Ökonometrie 53 / 66

55 Diagnostics Define the Schoenfeld residuals, r u,i : n R x u,n exp(ˆβ x n ) r u,i = x u,i, n R exp(ˆβ x n ) are the difference between the explanatory variable x u,i and the mean of the other persons in the risk set, weighted by their estimated relative hazard. Under the H 0 of the PH-assumption, the slope of the residuals are zero: r u,i = δ 0 + δ t t. Ökonometrie 54 / 66

56 Schoenfeld Residuen Ökonometrie 55 / 66

57 Diagnostics graphically ln[ ln{ŝ(t)}] and ln(t) (Ŝ(t) is the Kaplan-Meier estimator): h(t x) = h 0 (t) exp(β x) S(t x) = S 0 (t) exp(β x) ln[ ln{s(t x)}] = ln[ ln{s 0 (t)}] β x, under the H 0 the curves for different values of x i are parallel. Ökonometrie 56 / 66

58 Test for proportionality (1) Ökonometrie 57 / 66

59 Test for proportionality (2) Ökonometrie 58 / 66

60 Diagnosis residuals Cox-Snell residuals: CSr i = Ĥ0(t i ) exp(ˆβ x i ), and Ĥ0(t i ), where ˆβ from the Cox-model. Under the H 0, the CSr have an exponential distribution with a hazard rate=1 for all t. (The cumulated hazard is a 45 -line.) Ökonometrie 59 / 66

61 Goodness of fit Ökonometrie 60 / 66

62 Independent Competing Risks Until now: one single destination, e.g. unemployed people find jobs. Now: unemployed persons find jobs, they leave the labor market, they emigrate, etc. Necessary: mutually excluding events (i.e., the probabilities of leaving to a destination sum to 1). Definition: h a (t): latent hazard rate for the destination a, with respective density f a (t), and point of time of the incidence T a ; h b (t): b, f b (t), T b. Observed point of time of the incidence T = min{t a, T b }. Ökonometrie 61 / 66

63 Likelihood h(t) = h a (t) + h b (t) independent! S(t) = S a (t)s b (t). The likelihood function for the hazard rate model with independent destinations has following components: L = L a L b, L a : Contribution of all observations which are missing after a, L a = i {a} f a (t) L b : Contribution of all observations which are missing after b, L b = i {b} f b (t) Ökonometrie 62 / 66

64 Likelihood δ a i and δ b i are the following indicators: δ a i = δ b i = { 1 depart from a, 0 leaving to b. { 1 i depart from b, 0 leaving to a. Ökonometrie 63 / 66

65 log-likelihood L = L a L b, = [f a (t i )] δa i [f b (t i )] δb i ln L = alle i δ a i ln[f a (t i )] + δ b i ln[f b (t i )], alle i the log-likelihood function for independent competing risks can be separated into parts which are independent from each other. Every of these parts depends only on the parameters which affect the particular destination. Ökonometrie 64 / 66

66 Simplifies the estimation 1. Define indicator variables for every destination. 2. Observations which leave to another destination are censored. 3. Estimate the hazard rate for each destination. Ökonometrie 65 / 66

67 Further Issues Unobserved heterogeneity Dependent competing risks Initial conditions... Ökonometrie 66 / 66