λ(t, Z) = λ 0 (t) exp(βz)

Transcription

1 Parametric Survival Analysis So far, we have focused primarily on nonparametric and semi-parametric approaches to survival analysis, with heavy emphasis on the Cox proportional hazards model: λ(t, Z) = λ 0 (t) exp(βz) We used the following estimating approach: We estimated λ 0 (t) nonparametrically, using the Kaplan-Meier estimator, or using the Kalbfleisch/Prentice estimator under the PH assumption We estimated β by assuming a linear model between the log HR and covariates, under the PH model Both estimates were based on maximum likelihood theory. Reading: for parametric models see Collett. 1

2 There are several reasons why we should consider some alternative approaches based on parametric models: The assumption of proportional hazards might not be appropriate (based on major departures) If a parametric model actually holds, then we would probably gain efficiency We may want to handle non-standard situations like interval censoring incorporating population mortality We may want to make some connections with other familiar approaches (e.g. use of the Poisson likelihood) We may want to obtain some estimates for use in designing a future survival study. 2

3 A simple start: Exponential Regression Observed data: (X i, δ i, Z i ) for individual i, Z i = (Z i1, Z i2,..., Z ip ) represents a set of p covariates. Right censoring: Assume that X i = min(t i, U i ) Survival distribution: Assume T i follows an exponential distribution with a parameter λ that depends on Z i, say λ i = Ψ(Z i ). Then we can write: T i exponential(ψ(z i )) 3

4 First, let s review some facts about the exponential distribution (from our first survival lecture): f(t) = λe λt for t 0 S(t) = P (T t) = t f(u)du = e λt F (t) = P (T < t) = 1 e λt λ(t) = f(t) S(t) = λ constant hazard! Λ(t) = t 0 λ(u) du = t 0 λ du = λt 4

5 Now, we say that λ is a constant over time t, but we want to let it depend on the covariate values, so we are setting λ i = Ψ(Z i ) The hazard rate would therefore be the same for any two individuals with the same covariate values. Although there are many possible choices for Ψ, one simple and natural choice is: WHY? Ψ(Z i ) = exp[β 0 + Z i1 β 1 + Z i2 β Z ip β p ] ensures a positive hazard for an individual with Z = 0, the hazard is e β 0. The model is called exponential regression because of the natural generalization from regular linear regression 5

6 Exponential regression for the 2-sample case: Assume we have only a single covariate Z = Z, i.e., (p = 1). Hazard Rate: Ψ(Z i ) = exp(β 0 + Z i β 1 ) Define: Z i = 0 if individual i is in group 0 Z i = 1 if individual i is in group 1 What is the hazard for group 0? What is the hazard for group 1? What is the hazard ratio of group 1 to group 0? What is the interpretation of β 1? 6

7 Likelihood for Exponential Model Under the assumption of right censored data, each person has one of two possible contributions to the likelihood: (a) they have an event at X i (δ i = 1) contribution is L i = S(X i ) }{{} λ(x i) }{{} = e λx i λ survive to X i fail at X i (b) they are censored at X i (δ i = 0) contribution is L i = S(X i ) }{{} = e λx i survive to X i 7

8 The likelihood is the product over all of the individuals: L = i L i = i ( λe λx i ) δi }{{} events ( e λx i ) (1 δi ) }{{} censorings = i λ δ i ( e λx i ) 8

9 Maximum Likelihood for Exponential How do we use the likelihood? first take the log then take the partial derivative with respect to β then set to zero and solve for β this gives us the maximum likelihood estimators 9

10 The log-likelihood is: [ log L = log i λ δ i ( e λx i ) ] = i [δ i log(λ) λx i ] = i [δ i log(λ)] i λx i For the case of exponential regression, we now substitute the hazard λ = Ψ(Z i ) in the above log-likelihood: log L = i [δ i log(ψ(z i ))] i Ψ(Z i )X i (1) 10

11 General Form of Log-likelihood for Right Censored Data In general, whenever we have right censored data, the likelihood and corresponding log likelihood will have the following forms: L = i [λ i (X i )] δ i S i (X i ) log L = i [δ i log (λ i (X i ))] i Λ i (X i ) where λ i (X i ) is the hazard for the individual i who fails at X i Λ i (X i ) is the cumulative hazard for an individual at their failure or censoring time For example, see the derivation of the likelihood for a Cox model on p of Lecture 4 notes. We started with the likelihood above, then substituted the specific forms for λ(x i ) under the PH assumption. 11

12 Consider our model for the hazard rate: λ = Ψ(Z i ) = exp[β 0 + Z i1 β 1 + Z i2 β Z ip β p ] We can write this using vector notation, as follows: Let Z i = (1, Z i1,...z ip ) T and β = (β 0, β 1,...β p ) (Since β 0 is the intercept (i.e., the log hazard rate for the baseline group), we put a 1 as the first term in the vector Z i.) Then, we can write the hazard as: Ψ(Z i ) = exp[βz i ] Now we can substitute Ψ(Z i ) = exp[βz i ] in the log-likelihood shown in (1): log L = n δ i (βz i ) i=1 n X i exp(βz i ) i=1 12

13 Score Equations Taking the derivative with respect to β 0, the score equation is: log L n = [δ i X i exp(βz i )] β 0 i=1 For β k, k = 1,...p, the equations are: log L n = [δ i Z ik X i Z ik exp(βz i )] β k = i=1 n i=1 Z ik [δ i X i exp(βz i )] To find the MLE s, we set the above equations to 0 and solve (simultaneously). The equations above imply that the MLE s are obtained by setting the weighted number of failures ( i Z ikδ i ) equal to the weighted cumulative hazard ( i Z ikλ(x i )). 13

14 To find the variance of the MLE s, we need to take the second derivatives: 2 log L β k β j = n Z ik Z ij X i exp(βz i ) i=1 Some algebra (see Cox and Oakes section 6.2) reveals that where V ar( β) = I(β) 1 = [ Z(I Π)Z T ] 1 Z = (Z 1,..., Z n ) is a (p + 1) n matrix (p covariates plus the 1 for the intercept β 0 ) Π = diag(π 1,..., π n ) (this means that Π is a diagonal matrix, with the terms π 1,..., π n on the diagonal) 14

15 π i is the probability that the i-th person is censored, so (1 π i ) is the probability that they failed. Note: The information I(β) (inverse of the variance) is proportional to the number of failures, not the sample size. This will be important when we talk about study design. 15

16 The Single Sample Problem (Z i = 1 for everyone): First, what is the MLE of β 0? We set log L β 0 = n i=1 [δ i X i exp(β 0 Z i )] equal to 0 and solve: n δ i = i=1 exp( β 0 ) = n [X i exp(β 0 )] i=1 d = exp(β 0 ) d n i=1 X i n i=1 X i ˆλ = d t where d is the total number of deaths (or events), and t = X i is the total person-time contributed by all individuals. 16

17 If d/t is the MLE for λ, what does this imply about the MLE of β 0? Using the previous formula V ar( ˆβ) = [ Z(I Π)Z T ] 1, what is the variance of β 0?: With some matrix algebra, you can show that it is: V ar( β 0 ) = 1 n i=1 (1 π i) = 1 d 17

18 What about ˆλ = e ˆβ 0? By the delta method, V ar(ˆλ) = ˆλ 2 V ar( β 0 ) =? 18

19 The Two-Sample Problem: Z i Subjects Events Follow-up Group 0: Z i = 0 n 0 d 0 t 0 = n 0 i=1 X i Group 1: Z i = 1 n 1 d 1 t 1 = n 1 i=1 X i 19

20 The log-likelihood: log L = n δ i (β 0 + β 1 Z i ) i=1 n X i exp(β 0 + β 1 Z i ) i=1 so log L β 0 = n [δ i X i exp(β 0 + β 1 Z i )] i=1 = (d 0 + d 1 ) (t 0 e β 0 + t 1 e β 0+β 1 ) log L β 1 = n Z i [δ i X i exp(β 0 + β 1 Z i )] i=1 = d 1 t 1 e β 0+β 1 20

21 This implies: ˆλ1 = e ˆβ 0 + ˆβ 1 =? ˆλ 0 = e ˆβ 0 =? ˆβ 0 =? ˆβ 1 =? 21

22 Important Result: The maximum likelihood estimates (MLE s) of the hazard rates under the exponential model are the number of events divided by the person-years of follow-up! (this result will be relied on heavily when we discuss study design) 22

23 Exponential Regression: Means and Medians Mean Survival Time For the exponential distribution, E(T ) = 1/λ. Control Group: T 0 = 1/ˆλ 0 = 1/ exp( ˆβ 0 ) Treatment Group: T 1 = 1/ˆλ 1 = 1/ exp( ˆβ 0 + ˆβ 1 ) 23

24 Median Survival Time This is the value M at which S(t) = e λt = 0.5, so M = median = log(0.5) λ Control Group: ˆM 0 = log(0.5) ˆλ 0 = log(0.5) exp( ˆβ 0 ) Treatment Group: ˆM 1 = log(0.5) ˆλ 1 = log(0.5) exp( ˆβ 0 + ˆβ 1 ) 24

25 Exponential Regression: Variance Estimates and Test Statistics We can also calculate the variances of the MLE s as simple functions of the number of failures: var( ˆβ 0 ) = 1 d 0 var( ˆβ 1 ) = 1 d d 1 25

26 So our test statistics are formed as: For testing H o : β 0 = 0: For testing H o : β 1 = 0: χ 2 w = χ 2 w = ( ˆβ0 ) 2 var( ˆβ 0 ) = [log(d 0/t 0 )] 2 1/d 0 = ( ˆβ1 ) 2 var( ˆβ 1 ) [ log( d 1/t 1 d 0 /t 0 ) 1 d d 1 How would we form confidence intervals for the hazard ratio? ] 2 26

27 The Likelihood Ratio Test Statistic: (An alternative to the Wald test) A likelihood ratio test is based on 2 times the log of the ratio of the likelihoods under the null and alternative. We reject H 0 if 2 log(lr) > χ 2 1,0.05, where LR = L(H 1) L(H 0 ) = L( λ 0, λ 1 ) L( λ) 27

28 For a sample of n independent exponential random variables with parameter λ, the Likelihood is: n L = [λ δ i exp( λx i )] i=1 = λ d exp( λ x i ) = λ d exp( λn x) where d is the number of deaths or failures. The log-likelihood is l = d log(λ) λn x and the MLE is λ = d/(n x) 28

29 2-Sample Case: LR test calculations Data: Group 0: d 0 failures among the n 0 females mean failure time is x 0 = ( n 0 i X i )/n 0 Group 1: d 1 failures among the n 1 males mean failure time is x 1 = ( n 1 i X i )/n 1 Under the alternative hypothesis: L = λ d 1 1 exp( λ 1n 1 x 1 ) λ d 0 0 exp( λ 0n 0 x 0 ) log(l) = d 1 log(λ 1 ) λ 1 n 1 x 1 + d 0 log(λ 0 ) λ 0 n 0 x 0 29

30 The MLE s are: λ 1 = d 1 /(n 1 x 1 ) for males λ 0 = d 0 /(n 0 x 0 ) for females Under the null hypothesis: L = λ d 1+d 0 exp[ λ(n 1 x 1 + n 0 x 0 )] log(l) = (d 1 + d 0 ) log(λ) λ[n 1 x 1 + n 0 x 0 ] The corresponding MLE is λ = (d 1 + d 0 )/[n 1 x 1 + n 0 x 0 ] 30

31 A likelihood ratio test can be constructed by taking twice the difference of the log-likelihoods under the alternative and the null hypotheses: 2 [ ( ) d0 + d 1 (d 0 + d 1 ) log t 0 + t 1 ] d 1 log[d 1 /t 1 ] d 0 log[d 0 /t 0 ] 31

32 Nursing home example: For the females: n 0 = 1173 d 0 = 902 t 0 = x 0 = 265 For the males: n 1 = 418 d 1 = 367 t 1 = x 1 = 181 Plugging these values in, we get a LR test statistic of

33 Hand Calculations using events and follow-up: By adding up los for males to get t 1 and for females to get t 0, I obtained: d 0 = 902 (females) d 1 = 367 (males) t 0 = (female follow-up) t 1 = (male follow-up) This yields an estimated log HR: [ ] [ ] d1/t1 367/75457 ˆβ 1 = log = log d0/t0 902/ = log(1.6756) =

34 The estimated standard error is: var( ˆβ 1 1 ) = + 1 = d 1 d = So the Wald test becomes: χ 2 W = ˆβ 1 2 var( ˆβ 1 ) = ( ) = We can also calculate ˆβ 0 = log(d 0 /t 0 ) = 5.842, along with its standard error se( ˆβ 0 ) = (1/d0) =

35 Exponential Regression in STATA. use nurshome. stset los fail. streg gender, dist(exp) nohr failure _d: fail analysis time _t: los Iteration 0: log likelihood = Iteration 1: log likelihood = Iteration 2: log likelihood = Iteration 3: log likelihood = Iteration 4: log likelihood = Exponential regression -- log relative-hazard form No. of subjects = 1591 Number of obs = 1591 No. of failures = 1269 Time at risk = LR chi2(1) = Log likelihood = Prob > chi2 = _t Coef. Std. Err. z P> z [95% Conf. Interval] gender _cons Since Z = 8.337, the chi-square test is Z 2 =

36 The Weibull Regression Model At the beginning of the course, we saw that the survivorship function for a Weibull random variable is: and the hazard function is: S(t) = exp[ λ(t κ )] λ(t) = κ λ t (κ 1) The Weibull regression model assumes that for someone with covariates Z i, the survivorship function is S(t; Z i ) = exp[ Ψ(Z i )(t κ )] where Ψ(Z i ) is defined as in exponential regression to be: Ψ(Z i ) = exp[β 0 + Z i1 β 1 + Z i2 β Z ip β p ] For the 2-sample problem, we have: Ψ(Z i ) = exp[β 0 + Z i1 β 1 ] 36

37 Weibull MLEs for the 2-sample problem: Log-likelihood: log L = n i=1 δ i log [ κ exp(β 0 + β 1 Z i )X κ 1 ] n i Xi κ exp(β 0 + β 1 Z i ) i=1 exp( ˆβ 0 ) = d 0 /t 0 κ exp( ˆβ 0 + ˆβ 1 ) = d 1 /t 1 κ where t jκ = ˆλ 0 (t) = ˆκ exp( ˆβ 0 ) tˆκ 1 n j i=1 X ˆκ i among n j subjects ˆλ1 (t) = ˆκ exp( ˆβ 0 + ˆβ 1 ) tˆκ 1 ĤR = ˆλ 1 (t)/ˆλ 0 (t) = exp( ˆβ 1 ) ( ) d1 /t 1 κ = exp d 0 /t 0 κ 37

38 Weibull Regression: Means and Medians Mean Survival Time For the Weibull distribution, E(T ) = λ ( 1/κ) Γ[(1/κ) + 1]. Control Group: T 0 = ˆλ ( 1/ˆκ) 0 Γ[(1/ˆκ) + 1] Treatment Group: T 1 = ˆλ ( 1/ˆκ) 1 Γ[(1/ˆκ) + 1] 38

39 Median Survival Time For the Weibull distribution, M = median = [ ] 1/κ log(0.5) λ Control Group: ˆM 0 = [ log(0.5) ˆλ 0 ] 1/ˆκ Treatment Group: ˆM 1 = [ log(0.5) ˆλ 1 ] 1/ˆκ where ˆλ 0 = exp( ˆβ 0 ) and ˆλ 1 = exp( ˆβ 0 + ˆβ 1 ). 39

40 Note: the symbol Γ is the gamma function. If x is an integer, then Γ(x) = (x 1)! In cases where x is not an integer, this function has to be evaluated numerically. In homework and labs, I will supply this value to you. The Weibull regression model is very easy to fit: In stata: Just specify dist(weibull) instead of dist(exp) within the streg command In sas: use model option dist=weibull within the proc lifereg procedure Note: to get more information on these modeling procedures, use the online help facilities. For example, in Stata, you can type:.help streg 40

41 Weibull in Stata:. streg gender, dist(weibull) nohr failure _d: analysis time _t: fail los Fitting constant-only model: Iteration 0: log likelihood = Iteration 1: log likelihood = Iteration 2: log likelihood = Iteration 3: log likelihood = Iteration 4: log likelihood = Fitting full model: Iteration 0: log likelihood = Iteration 1: log likelihood = Iteration 2: log likelihood = Iteration 3: log likelihood = Iteration 4: log likelihood =

42 Weibull regression -- log relative-hazard form No. of subjects = 1591 Number of obs = 1591 No. of failures = 1269 Time at risk = LR chi2(1) = Log likelihood = Prob > chi2 = _t Coef. Std. Err. z P> z [95% Conf. Interval] gender _cons /ln_p p /p

43 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 u r v i v a l L e n g t h o f S t a y ( d a y s ) 43

44 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 u r v i v a l Which do you think fits best? L e n g t h o f S t a y ( d a y s ) 44

45 Other useful plots for evaluating fit to exponential and Weibull models log(ŝ(t)) vs t log[ log(ŝ(t))] vs log(t) Why are these useful? If T is exponential, then S(t) = exp( λt)) so log(s(t)) = λt and Λ(t) = λ t a straight line in t with slope λ and intercept=0 45

46 If T is Weibull, then S(t) = exp( (λt) κ ) so log(s(t)) = λt κ then Λ(t) = λt κ and log( log(s(t))) = log(λ) + κ log(t) a straight line in log(t) with slope κ and intercept log(λ). 46

47 So we can calculate our estimated Λ(t) and plot it versus t, and if it seems to form a straight line, then the exponential distribution is probably appropriate for our dataset. Plots for nursing home data: ˆΛ(t) vs t N e g a t i v e L o g S D F L O S 47

48 Or we can plot log ˆΛ(t) versus log(t), and if it seems to form a straight line, then the Weibull distribution is probably appropriate for our dataset. Plots for nursing home data: log[ log(ŝ(t))] vs log(t) L o g N e g a t i v e L o g S D F L o g o f L O S 48

49 Comparison of Methods for the Two-sample problem: Data: Z i Subjects Events Follow-up Group 0: Z i = 0 n 0 d 0 t 0 = n 0 i=1 X i Group 1: Z i = 1 n 1 d 1 t 1 = n 1 i=1 X i In General: λ z (t) = λ(t, Z = z) for z = 0 or 1. The hazard rate depends on the value of the covariate Z. In this case, we are assuming that we only have a single covariate, and it is binary (Z = 1 or Z = 0) 49

50 Reading from Collett: Section(s) Description 4.1.1, Exponential properties Weibull properties 4.3.1, Exponential ML estimation Weibull ML estimation 4.5 General Weibull regression 4.6 Model selection - Weibull regression 4.7 Weibull/AFT model connection Ch.6 AFT - Other parametric models 50

51 MODELS Exponential Regression: λ z (t) = exp(β 0 + β 1 Z) λ 0 = exp(β 0 ) λ 1 = exp(β 0 + β 1 ) HR = exp(β 1 ) 51

52 Weibull Regression: λ z (t) = κ exp(β 0 + β 1 Z) t κ 1 λ 0 = κ exp(β 0 ) t κ 1 λ 1 = κ exp(β 0 + β 1 ) t κ 1 HR = exp(β 1 ) Proportional Hazards Model: λ z (t) = λ 0 (t) exp(β 1 ) λ 0 = λ 0 (t) KM? λ 1 = λ 0 (t) exp(β 1 ) HR = exp(β 1 ) 52

53 Remarks Exponential model is a special case of the Weibull model with κ = 1 (note: Collett uses γ instead of κ) Exponential and Weibull models are both special cases of the Cox PH model. How can you show this? If either the exponential model or the Weibull model is valid, then these models will tend to be more efficient than PH (smaller s.e. s of estimates). This is because they assume a particular form for λ 0 (t), rather than estimating it at every death time. 53

54 For the Exponential model, the hazards are constant over time, given the value of the covariate Z i : Z i = 0 ˆλ 0 = exp( ˆβ 0 ) Z i = 1 ˆλ 0 = exp( ˆβ 0 + ˆβ 1 ) For the Weibull model, we have to estimate the hazard as a function of time, given the estimates of β 0, β 1 and κ: Z i = 0 ˆλ 0 (t) = ˆκ exp( ˆβ 0 ) tˆκ 1 Z i = 1 ˆλ 1 (t) = ˆκ exp( ˆβ 0 + ˆβ 1 ) tˆκ 1 However, the ratio of the hazards is still just exp( ˆβ 1 ), since the other terms cancel out. 54

55 Here s what the estimated hazards look like for the nursing home data: H a z a r d R a t e E x p o n e n t i a l H a z a r d : F e m a l e E x p o n e n t i a l H a z a r d : M a l e W e i b u l l H a z a r d : F e m a l e W e i b u l l H a z a r d : M a l e L e n g t h o f s t a y ( d a y s ) 55

56 Proportional Hazards Model: To get the MLE s for this model, we have to maximize the Cox partial likelihood iteratively. There are not closed form expressions like above. L(β) = = [ n i=1 [ n i=1 e βz i l R(X i ) eβz l ] δi e β 0+β 1 Z i l R(X i ) eβ 0+β 1 Z l ] δi 56

57 Comparison with Proportional Hazards Model. stcox gender, nohr failure _d: fail analysis time _t: los Iteration 0: log likelihood = Iteration 1: log likelihood = Iteration 2: log likelihood = Iteration 3: log likelihood = Refining estimates: Iteration 0: log likelihood = Cox regression -- Breslow method for ties No. of subjects = 1591 Number of obs = 1591 No. of failures = 1269 Time at risk = LR chi2(1) = Log likelihood = Prob > chi2 = _t _d Coef. Std. Err. z P> z [95% Conf. Interval] gender For the PH model, ˆβ 1 = and ĤR = e0.394 =

58 Comparison with the Logrank and Wilcoxon Tests. sts test gender failure _d: analysis time _t: fail los Log-rank test for equality of survivor functions Events gender observed expected Total chi2(1) = Pr>chi2 =

59 . sts test gender, wilcoxon failure _d: analysis time _t: fail los Wilcoxon (Breslow) test for equality of survivor functions Events Sum of gender observed expected ranks Total chi2(1) = Pr>chi2 =

60 Comparison of Hazard Ratios and Test Statistics for effect of Gender Wald Model/Method λ 0 λ 1 HR log(hr) se(log HR) Statistic Exponential Weibull t = t = t = Logrank Wilcoxon Cox PH Ties=Breslow Ties=Discrete Ties=Efron Ties=Exact Score (Discrete)

61 Comparison of Mean and Median Survival Times by Gender Mean Survival Median Survival Model/Method Female Male Female Male Exponential Weibull Kaplan-Meier Cox PH (Kalbfleisch/Prentice) 61

62 The Accelerated Failure Time Model The general form of an accelerated failure time (AFT) model is: where log(t i ) = β AF T Z i + σϵ log(t i ) is the log of a survival time β AF T is the vector of AFT model parameters corresponding to the covariate vector Z i ϵ is a random error term σ is a scale factor In other words, we can model the log-survival times as a linear function of the covariates! The streg command in stata (without the exponential or weibull option) uses this log-linear model for fitting parametric models. 62

63 By choosing different distributions for ϵ, we can obtain different parametric distributions: Exponential Weibull Gamma Log-logistic Normal Lognormal We can compare the predicted survival under any of these parametric distributions to the KM estimated survival to see which one seems to fit best. Once we decide on a certain class of model (say, Gamma), we can evaluate the contributions of covariates by finding the MLE s, and constructing Wald, Score, or LR tests of the covariate effects. 63

64 We can motivate the AFT model by first demonstrating the following two relationships: 1. For the Exponential Model: If the failure times T i = T (Z i ) follow an exponential distribution, i.e., S i (t) = e λ it with λ i = exp(βz i ), then log(t i ) = βz i + ϵ where ϵ follows an extreme value distribution (which just means that e ϵ follows a unit exponential distribution). 64

65 2. For the Weibull Model: If the failure times T i = T (Z i ) follow a Weibull distribution, i.e., S i (t) = e λ it κ with λ i = exp(βz i ), then log(t i ) = σβz i + σϵ where ϵ again follows an extreme value distribution, and σ = 1/κ. In other words, both the Exponential and Weibull model can be written in the form of a log-linear model for the survival times, if we choose the right distribution for ϵ. 65

66 The log-linear form for the exponential can be derived by: (1) Creating a new variable T 0 = T Z exp(βz i ) ( (2) Taking the log of T Z, yielding log(t Z ) = log Step (1): For an exponential model, recall that: T 0 exp(βz i ) ) S i (t) = P r(t Z t) = e λt, with λ = exp(βz i ) It follows that T 0 exp(1): S 0 (t) = P r(t 0 t) = P r(t Z exp(βz) t) = P r(t Z t exp( βz)) = exp [ λ t exp( βz)] = exp [ exp(βz) t exp( βz)] = exp( t) 66

67 Step (2): Now take the log of the survival time: ( ) T 0 log(t Z ) = log exp(βz i ) = log(t 0 ) log (exp(βz i )) = βz i + log(t 0 ) = βz i + ϵ where ϵ = log(t 0 ) follows the extreme value distribution. 67

68 Relationship between Exponential and Weibull If T Z has a Weibull distribution, i.e., S(t) = e λtκ with λ = exp(βz i ), then you can show that the new variable T Z = T κ Z follows an exponential distribution with parameter exp(βz i ). Based on the previous page, we can therefore write: log(t ) = βz + ϵ (where ϵ has an extreme value distribution.) 68

69 But since log(t ) = log(t κ ) = κ log(t ), we can write: log(t ) = log(t )/κ = (1/κ) ( βz i + ϵ) = σβz i + σϵ where σ = 1/κ. 69

70 This motivates the following general definition of the Accelerated Failure Time Model by: log(t i ) = β AF T Z i + σϵ where ϵ is a random error term, σ is a scale factor, Y is the log of a survival random variable, and β AF T = σβ e where β e came from the hazard λ = exp(βz). 70

71 The defining feature of an AFT model is: S(t; Z) = S i (t) = S 0 (ϕ t) That is, the effect of covariates is to accelerate (stretch) or decelerate (shrink) the time-scale. Effect of AFT on hazard: λ i (t) = ϕ λ 0 (ϕt) 71

72 One way to interpret the AFT model is via its effect on median survival times. If S i (t) = 0.5, then S 0 (ϕt) = 0.5. This means: Interpretation: M i = ϕm 0 For ϕ < 1, there is an acceleration of the endpoint (if M 0 = 2yrs in control and ϕ = 0.5, then M i = 1yr. For ϕ > 1, there is a stretching or delay in endpoint In general, the lifetime of individual i is ϕ times what they would have experienced in the reference group Since ϕ must be positive and a function of the covariates, we model ϕ = exp(βz i ). 72

73 When does Proportional hazards = AFT? According to the proportional hazards model: S(t) = S 0 (t) exp(βz i) and according to the accelerated failure time model: S(t) = S 0 (t exp(βz i )) Say T i W eibull(λ, κ). Then λ(t) = λκt (κ 1) 73

74 Under the AFT model: λ i (t) = ϕ λ 0 (ϕt) = e βz i λ 0 (e βz i t) = e βz i λ 0 κ ( e βz i t ) (κ 1) = ( e βz i ) κ λ0 κt (κ 1) = ( e βz i ) κ λ0 (t) But this looks just like the PH model: λ i (t) = exp(β Z i ) λ 0 (t) It turns out that the Weibull distribution (and exponential, since this is just a special case of a Weibull with κ = 1) is the only one for which the accelerated failure time and proportional hazards models coincide. 74

75 Special cases of AFT models Exponential regression: σ = 1, ϵ following the extreme value distribution. Weibull regression: σ arbitrary, ϵ following the extreme value distribution. Lognormal regression: σ arbitrary, ϵ following the normal distribution. 75

76 Examples in stata: Using the streg command, one has the following options of distributions for the log-survival times:. streg trt, dist(lognormal) exponential weibull gompertz lognormal loglogistic gamma 76

77 . streg gender, dist(exponential) nohr _t Coef. Std. Err. z P> z [95% Conf. Interval] gender streg gender, dist(weibull) nohr _t Coef. Std. Err. z P> z [95% Conf. Interval] gender /p streg gender, dist(lognormal) _t Coef. Std. Err. z P> z [95% Conf. Interval] gender _cons sigma

78 . streg gender, dist(gamma) _t Coef. Std. Err. z P> z [95% Conf. Interval] gender _cons sigma This gives a good idea of the sensitivity of the test of gender to the choice of model. It is also easy to get predicted survival curves under any of the parametric models using the following:. streg gender, dist(gamma). stcurv, survival The options hazard and cumhaz can also be substituted for survival above to obtain plots. 78