Survival Analysis 存活率分析

Survival Analysis存活率分析 張新儀 7/12/2004

Introduction • What is Survival Analysis? • Outcome variable: time until an event occurs • Time origin: precisely defined, comparable, needs not to be the same calendar date time Event 0

Time × × × × × × × × × × 1980 1985 1987 0 28 Year since beginning of the study Year since diagnosis/entry Time: years, months, age, etc, positive Event: death, disease incidence, relapse (we only consider 1 event at this moment)

Our Interests • Distribution of failure times • Comparison of the failure times • Effects of explaintary variables on survival • If the exact failure time of each individual is known, we can apply all the statistical techniques

Censoring • Incomplete observation of failure time • Study ends before event happens • Lost to follow up • Withdraw from the study • Example Entered the study No event happened A Study ended Study started

Censoring (con’t) Ex. AIDS patients survival years Steroid: 1, 1, 1, 1+, 4+, 5 Placebo: 1+, 2+, 3, 3+ + indicates no event occurs (1) Ignore +: Mean survival time of steroid treatment: 13/6 Mean survival time of placebo: 9/4 (2) Delete +: 3 left in the steroid group, only 1 remained in the placebo Traditional techniques do not work here!

Type of Censoring • Notations Ti: potential failure time for ith individual Ci: potential censoring time… Xi: min (ti, Ci), observed time δi: 1, if Ti≦Ci(un censored) 0, if Ti > Ci (censored)

Right Censoring (most of the cases) • The person’s exact survival time becomes incomplete at the right side of the follow-up time A × B End of study C ○ withdraw D End of study ○ lost E × F 0 2 4 6 8 10 12 2 events: A, F, 4 censored: B,C,D,E,

Right Censoring (con’t) • Type I censoring: Study ends when a fixed time point is reached • Ex. 1 year study, Ci=C= 1 year, fixed in advance • Type II censoring: Study ends when a fixed number of failures occur • Ex. Study the life time of light buld, study ends when 10 light bulbs failed9

Left Censoring A person’s survival time becomes incomplete at the left side of the follow-up period Interval Censoring Left censoring Survival time × HIV exposure HIV tested + 0 6 3 Event occurred between 3 and 6, but no exact information

More Notation and Terminology • S(t)=survival function fundamental to survival analysis S(t)=P( T > t ) • The probability that a person survives longer than a specific time t • Obtaining survival probabilities for different values of t provides crucial summary information for survival pattern In practice: step function is observed 1 non-increasing S(0)=1 S(∞)=0 S(0)=1 S(t) 0

Hazard Function Conditional probability Instantaneous potential per unit time for the event to occur given that the individual survived up to time t Hazard function does not give a probability. It gives a rate ranging from 0 to ∞. Properties: λ(t) ≧0, λ(t) has no upper bound

Survival DistributionProbability distribution for(a)continuous (b) non-negative random variables Probability Density Function (pdf) f(t) Cumulative distribution Function P[t≦T<t+Δt] F(t) t t t+Δt 0 t F(t)=P[T≦t]=∫f(t)ds

Some Special Distributions • Exponential Distribution (with parameter ρ） • f(t)= ρexp(﹣ρt) • S(t)=exp (﹣ρt), F(t)=1-exp (﹣ρt) • λ(t)=ρ, Λ(t)= ρt • E(T)=1/ρ • Lack of memory • Constant hazard • Coefficient of variation=s.d/mean=1

Gamma Distribution (parameters: ρ,κ) • f(t)=ρ(ρt) (κ-1)exp(-ρt)/Γ(ρ) • S(t)=incomplete gamma function • E(T)=κ/ρ • λ(t) is monotone increasing from 0 ifκ>1, is monotone decreasing from∞, ifκ<1, and in either case approaches ρas t→∞ • If κ=1, the gamma distribution reduces to the exponential distribution

Weibull Distribution (parameters: ρ,κ) • f(t)=κρ(ρt) κ-1exp[-(ρt) κ] • S(t)=exp [-(ρt) κ], F(t)=1-exp[-(ρt) κ] • λ(t)=κρ(ρt) κ-1, Λ(t)=(ρt) κ • Important generalization of the exponential distribution allows for a power dependence of the hazard on time • λ(t) is monotone increasing from 0 ifκ>1, is monotone decreasing from∞, ifκ<1 • If κ=1, the it reduces to the exponential hazard

Other Distributions • Log-normal Distribution • Log logistic Distribution • Generalized Gamma Distribution • Gompertz-Makeham Distribution • Ways to select different distributions • the density function is not effective • plot (t) or log (t) vs. t or log t • plot (t) or log -S(t) or other transformation vs. t or log t

RELATIONSHIPS • S(t), (t), (t), F(t)

The Likelihood Function

Statistical Inference • Ho: = o • Likelihood Ratio Statistic W(o)=W=2[ln (, )-ln(o, o)], where(, ) is the joint MLE of (, ), and  o is the MLE under null hypothesis W(o)~2 with d.f.=dim(w) • Wald statistic • Score statistic

Example: Exponential Distribution • Exponential Distributed Failure Times • f(t)=exp(ρ), S(t)=exp(-ρt), λ(t)=ρ • l=㏑ (likelihood) =Σδi㏑ ρ－ ρΣxi=d ㏑ρ－ρΣxi,where d is the total number of failures,Σxi is the censored + uncensored times (total time at risk) • MLE: d/Σxi=total # of failure/total time at risk

Numerical Example • Gregory et al. NEJM 1978 (Severe viral hepatitis): 20 patients, follow-up 16 weeks, 14 steroid treatment, 15 control • Steroid: 1, 1, 1,1+, 4+, 5, 7, 8,10,10+, 12+,16+, 16+, 16+ • Control: 1+, 2+, 3,3,3+,5+, 5+, 16+(8) ρ μ d Σxi ln (ρ) Ratio of hazard: 0.0648/0.0133≒5, which group is at higher risk of dying?

Advantages of Parametric Methods • Convenience for statistical inferences • Existence of explicit, reasonably simple forms for S(t), (t), and (t) • Capability of representing both over- and under-dispersion relative to exponential distribution • Qualitative shape of the hazard function • Behavior of S(t) for small times and large times are reasonably easy to study

NONPARAMETRIC METHODS • Product-limit (Kaplan-Meier) estimator • Order the failure time. Then, the probability of surviving k ( 2) or more years from the beginning of the study is a product of k observed survival probability S(t) = P1 P2  P3 … Pk • Comparing two survival distributions: do not require the actual survival times, only need the order of the survival time.

Product Limit Estimator (example) For t in S(t) 1 [0,1) [1,3) [3,4) [4,5) [5, ∞) 1 1×5/6=0.833 1×5/6×3/4=0.625 1×5/6×3/4×2/3=0.417 1×5/6×3/4×2/3×1/2=0.208 0 1 2 3 4 5 6 7 t How would you estimate the median survival time?

Comparing Survival Times Same difference at 5 year, which is better? • Log rank (Mantel-Haenzel) test: order failure times in the pooled sample, e.g. samples from two treatments, then create a series of 22 tables according to ordered death times. The statistics compares observed value of deaths with what would be expected if the hazard at ti were the same in both groups.

t1=2 t2=4 t3=10 t4=15 t5=19 t6=23 d s • Example A B χ2statistic will be calculated for this series of tables. As long as the difference between hazards has a consistent sign, logrank test usually does well. Use PROC LIFETEST in SAS to do the test. Gehan/Breslow Generalized Wilcoxon test, let Wi=Ni, better than longrank test in detecting early difference; worse at detecting later differences. Peto/Prentice generalized Wilcoxon test, let Wi=S(ti), similar to previous, but it doesn’t jump as wildly due to censoring

Modeling • To compare two or more sets of data • To compare the effects of explanatory variables • What type of explanatory variables are usually of interests? • Treatments, characteristics of individuals (e.g. sex, age, medical history etc.) • Environmental variables (e.g. living conditions, working environment etc.) • interactions, etc. • Other way to classify: constant or time dependent covariates which may be influenced by the treatments under investigation.

Types of Models • Accelerated Life Model • These models assume that the effect of independent variables on an event-time distribution is multiplicative on the event time. • T = ψ ( z ) T0= exp(x’) T0 • log (T) = log (T0 ) + x’ • choices of ψ ( z ) • ψ ( z ) ﹥0 • simple interpretation • mathematically simple if possible • ψ ( z )= exp(x’) is most commonly used

Example (output from SAS PROC LIFEREG) • PROC LIFEREG; • MODEL time*event(0)=group/distribution=weibull; Variable Estimate Intercept 2.248 Group 1.267 Scale .732 In SAS T=Toψ ( z ) = To exp(x’) , E(T; control)=E(To)exp (β1); E(T; treat)=E(To)exp(β1+ β2) E(T; treat)/E(T; control)=exp(1.267)> 1, E(To) can be ignored.

However, some may want to know the actual expected survival time, not just a relative measure • κ=1/0.732=1.35, ρ=exp (-2.248) • For the control group, all the covariates are zero, we have E[T; control]=1/ρΓ(1+1/κ) ≒exp(2.248) Γ(1+.732) ≒8.7 E[T; treat]=8.7exp(β2)=8.7exp (1.267) ≒3.9

Note • κis not really of interest in the accelerated life model. It affects So(t) but not ψ ( z ) . Remember S(t; z)=So(t) ×ψ ( z ) =exp(-ρt) κ exp(x’) • Generalization  ψ ( z ) such thatS(t, z)=So(t ψ ( z ) )f(t;z)=fo(t ψ ( z ) ) ψ ( z )(t;z)= (t ψ ( z ) ) ψ ( z ) T=To/ ψ ( z ) o=E(lnTo)ln T=o - ln ψ ( z )+, where z and  are independent, E[  }=0.

Proportional Hazard Model • Similar to accelerated life model, but focus on hazard (t,z)=ψ( z ) o(t) • A common form forψ( z ) =exp[z’] • Example, Weibull distribution o(t)=(t)κ1(t)= ψo(t)=(ψ1/κt)κ= (*t)κ • 1(t) is another Weibull with = * • Is proportional hazard model related to accelerated life time? Could a model be proportional hazard and ALT? • Only true in Weibull (try by yourself, if interested)

Cox PH model • Response variable: observed (x, ), of interest: T or just (t), T’s distribution is some survival distribution with S(t), (t)[notice that no specific function is assumed)Observed covariates: z1, z2, … zk • (t| z1, z2, … zk)= o(t) exp(o+ 1 z1+ 2 z2 …+ k zk) unspecified

Noteo(t) = (t| z1= z2= … zk=0) exp(o+ 1 z1+ 2 z2 …+ k zk)=RR= (t| z1, z2, … zk) / (t| z1= z2= … zk=0) • This is so called Proportional Hazards (very popular) • First proposed by Cox (JRSS, 1972) • Proportional Hazard model only good for survival functions do not cross; Relative risk (hazards) of death comparing covariates z1,z2,…zk to z1= z2= … zk=0

Interpreting Coefficients for Cox’s Model • k is the log RR (hazard ratio) for a unit change in zk, given that all other covariates remain constant. • The RR comparing 2 set of values for the covariates (z1,z2,… zk) vs.(z1’,z2’,… z’k)RR= (t| z1, z2, … zk) / (t| z’1, z’2, … z’k)=exp[1(z1- z’1)+1(z2- z’2)… +1(zk -z’k)] • Fitting Cox model by maximizing the partial likelihood function • Linear assumption of the covariates • Observations are independent • SAS: PROC PHREG;

References • Cox and Oaks (1984) Analysis of Survival Data • Lee E. (1992?) Statistical Methods for Survival Data Analysis • SAS manual for Survival Analysis • Some websites http://www.ats.ucla.edu/stat/sas/seminars/sas_survival/default.htm

PROC lifetest data=uis plots=(s); time time*censor(0); strata treat; run; Other outputs omitted. Pr > Test Test of Equality over Strata Chi-Square DF Chi-Square Log-Rank 6.7979 1 0.0091 Wilcoxon 9.4608 1 0.00212Log(LR) 7.8267 1 0.0051

Survival Analysis 存活率分析