Lecture 12. Multivariate Survival Data Statistics Survival Analysis. Presented March 8, 2016

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1 Statistics Survival Analysis Presented March 8, 2016 Dan Gillen Department of Statistics University of California, Irvine 12.1

2 Examples Clustered or correlated survival times Disease onset in family members, litter mates Survival of multiple skin grafts on same patient Duration of multiple catheterizations on same patient Recurrent events of same type Recurrent episodes of herpes infection Recurrence of tumor in bladder cancer patients Recurrent attacks of asthma, angina, etc. Recurrent events of different types Death following relapse in cancer patients 12.2

3 Two Basic Approaches Frailty (random effects) models: Covariate effects apply at cluster level Interpretation of regression coefficients is controversial Standard software not yet available Marginal ( population average") models: Covariate effects apply at population level (Usually) preferred interpretation Can use standard Cox PH software, but need to adjust covariances for correlation within clusters 12.3

4 (K & M Ex 13.1) 16 burn patients treated with 1-4 skin allografts Binary indicator of HLA match: 1=good, 0=bad Some observations censored by death of the patient > allograft[1:10,] patient time rejection match

5 (K & M Ex 13.1) Note that some patients have a combination of good and bad HLA matched allografts Now compare the naive and robust analyses... First, fit the standard Cox PH model, ignoring correlation > fit1 <- coxph( Surv(time, rejection) ~ match, data=allograft ) > summary(fit1) n= 34 coef exp(coef) se(coef) z Pr(> z ) match * exp(coef) exp(-coef) lower.95 upper.95 match

6 (K & M Ex 13.1) Now account for correlation by clustering on patient id > fit2 <- coxph( Surv(time, rejection) ~ match + cluster(patient), data=allograft ) > summary(fit2) n= 34 coef exp(coef) se(coef) robust se z Pr(> z ) match ** exp(coef) exp(-coef) lower.95 upper.95 match Rsquare= (max possible= 0.99 ) Likelihood ratio test= 6.84 on 1 df, p= Wald test = 7.78 on 1 df, p= Score (logrank) test = 6.76 on 1 df, p= , Robust = 5.36 p= (Note: the likelihood ratio and score tests assume independence of observations within a cluster, the Wald and robust score tests do not). 12.6

7 (K & M Ex 13.1) Notice that there is only a slight change in the standard errors here The reduction in the standard error suggests that some information about HLA effect came from the within patient comparisons For patients with both good and bad matches, they can serve as their own controls This just like a paired t-test We have increased precision because we are adjusting for potential precision/confounding variables This will not be the case for a variable that only changes at the cluster level (eg. sex or age) 12.7

8 Events of the same type Single baseline hazard function Time usually measured from same origin for all events Recurrent events: use left truncation on preceding event time Specify entry time in R to be time of previous event Hazard for the k th event in cluster i given by: λ ik (t x i ) = λ 0 (t) exp{x ik (t)β} 12.8

9 Events of distinct types Separate baseline hazard for each event type Time measured from same origin or from time of preceding event Use of left truncation for recurrent events of the same type Hazard for k th event in cluster i given by: λ ik (t x i ) = λ k0 (t) exp{x ik (t)β} 12.9

10 Distinct clusters assumed statistically independent and treating all observations as independent, the partial likelihood is given by L(β) = n K i i=1 k=1 { exp(β T x ki (t ki )) S (0) (β, t ki ) } δki with K i the number of events in the i-th cluster and S (0) (β, t ki ) = j R ki exp(β T x j (t ki )). The score function is given by U(β) = logl(β) β = n i=1 K i k=1 { } δ ki x ki (t ki ) S(1) (β, t ki ) S (0) (β, t ki ) with S (1) (β, t ki ) = j R ki x j (t ki ) exp(β T x j (t ki ))

11 Robust (sandwich, empirical) estimator of var( ˆβ) The observed information matrix is given by I (β) = 2 logl(β) β 2 n K i { S (2) (β, t ki ) = δ ki S (0) (β, t ki ) S(1) (β, t ki ) 2 } S (0) (β, t ki ) 2 i=1 k=1 with a 2 = aa T and S (2) (β, t ki ) = j R ki x j (t ki ) 2 exp(β T x j (t ki )). is given by finding β such that U( β) =

12 Robust (sandwich, empirical) estimator of var( ˆβ) In this case, it can be shown that where with B(β) = U(β) N (0, B(β)) n i=1 K i K i k=1 l=1 W ki (β) = δ ki { x ki (t ki ) S(1) (β, t ki ) S (0) (β, t ki ) n j=1 K j l=1 } δ lj Y ki (t lj )e βt x ki (t lj ) S (0) (β, t ki ) W ki (β)w li (β) T { } x ki (t lj ) S(1) (β, t ki ) S (0) (β, t ki ) 12.12

13 Robust (sandwich, empirical) estimator of var( ˆβ) From the above, a first order Taylor expansion yields β N (β, D) where D = I ( β) 1 B( β)i ( β) 1 When K = 1 and under correct model specification, I ( β) 1 =B( β) 1 and D =I ( β)

14 Robust (sandwich, empirical) estimator of var( ˆβ) Score" and Wald tests using the robust variance estimate are then given by: H0 : β = 0 U T (0)B 1 (0)U(0) χ 2 p H0 : Lβ = 0 (L β) T (LDL T ) 1 (L β) χ 2 r where L is a r p contrast matrix 12.14

15 85 patients with superficial bladder tumors Randomly assigned to thiotep (rx==2) or placebo (rx==1) Multiple recurrences per patient Considered up to 4 recurrences (enum) If fewer than 4, start and stop times set equal Covariates of interest (other than treatment): number of tumors inititially (1-8) size of tumors initially (1-8 cm) Observations censored by death or end of study A quick look at the data

16 Patient 1 Patient 1 had no recurrences after 1 month and was then lost to follow-up > vabladder[ is.element(vabladder$id, c(1,5,9,32)), ] id rx number size start stop event enum Patient 5 Patient 5 had 1 recurrences at 6 months and was then followed recurrence free to 10 months

17 Patient 9 Patient 9 had 2 recurrences at 12 months and 16 months Patient 32 Patient 32 had 4 recurrences at 9, 17, 22, and 24 months

18 Distribution of recurrences Patient ID Time on Study 12.18

19 Major issues/decisions Same hazard function for each recurrence, or different? Measure time from zero or preceding recurrence? Subjects at risk for each recurrence from zero ( marginal-time model) or from preceding recurrence (recurrent event (or gaptime" model)? Marginal-time" model (time from origin for each event) Gaptime" model (time from preceding event) Covariate effects same for each recurrence, or different? 12.19

20 Time to each recurrence: Marginal vs. Gaptime Marginal-time model survival curves Survival Recurrence 1 Recurrence 2 Recurrence 3 Recurrence Time from study start (mths) Recurrence 1 85 (0) 37 (39) 14 (47) 2 (47) Recurrence 2 46 (0) 23 (21) 8 (29) 0 (29) Recurrence 3 27 (0) 20 (7) 4 (20) 0 (22) Recurrence 4 20 (0) 18 (3) 5 (12) 0 (14) Total 178 (0) 98 (70) 31 (108) 2 (112) 12.20

21 Time to each recurrence: Marginal vs. Gaptime Gap-time survival curves Survival Recurrence 1 Recurrence 2 Recurrence 3 Recurrence Time from study start (mths) Recurrence 1 85 (0) 37 (39) 14 (47) 2 (47) Recurrence 2 46 (0) 12 (26) 3 (29) 0 (29) Recurrence 3 27 (0) 3 (20) 0 (22) 0 (22) Recurrence 4 20 (0) 2 (14) 0 (14) 0 (14) Total 178 (0) 54 (99) 17 (112) 2 (112) 12.21

22 Examination of treatment effect Notice how the ordering of the survival curves change. Is this intuitive? Let s graphically examine the treatment effect under the marginal-time model... Survival Recurrence 1 Rx Group 1 Rx Group 2 Survival Recurrence 2 Rx Group 1 Rx Group Time from study start (mths) Time from study start (mths) Recurrence 3 Recurrence 4 Survival Rx Group 1 Rx Group 2 Survival Rx Group 1 Rx Group Time from study start (mths) Time from study start (mths) 12.22

23 Analysis 1 Separate baseline hazard and covariate effects for each recurrence Keep the original time axis, but use left truncation to re-enter patients after each recurrence First Recurrence: > recfit1 <- coxph( Surv( start, stop, event) ~ number + size + rx, data=vabladder, subset=enum==1 ) > summary(recfit1) n= 85 coef exp(coef) se(coef) z Pr(> z ) number ** size rx exp(coef) exp(-coef) lower.95 upper.95 number size rx

24 Analysis 1 Second Recurrence: > recfit2 <- coxph( Surv( start, stop, event) ~ number + size + rx, data=vabladder, subset=enum==2 ) > summary(recfit2) n= 46 coef exp(coef) se(coef) z Pr(> z ) number size rx exp(coef) exp(-coef) lower.95 upper.95 number size rx

25 Analysis 1 Third Recurrence: > recfit3 <- coxph( Surv( start, stop, event) ~ number + size + rx, data=vabladder, subset=enum==3 ) > summary(recfit3) n= 27 coef exp(coef) se(coef) z Pr(> z ) number size rx exp(coef) exp(-coef) lower.95 upper.95 number size rx

26 Analysis 1 Fourth Recurrence: > recfit4 <- coxph( Surv( start, stop, event) ~ number + size + rx, data=vabladder, subset=enum==4 ) > summary(recfit4) n= 20 coef exp(coef) se(coef) z Pr(> z ) number size rx exp(coef) exp(-coef) lower.95 upper.95 number size rx

27 Analysis 2 Separate baseline hazard and common covariate effects for each recurrence Stratify by recurrence (enum) and use robust standard errors > gaptimefit <- coxph( Surv( start, stop, event) ~ number + size + rx + strata(enum) + cluster(id), data=vabladder, subset=enum<5 ) > summary( gaptimefit ) n= 178 coef exp(coef) se(coef) robust se z Pr(> z ) number * size rx exp(coef) exp(-coef) lower.95 upper.95 number size rx Likelihood ratio test= 6.51 on 3 df, p= Wald test = 7.26 on 3 df, p=0.064 Score (logrank) test = 6.91 on 3 df, p=0.0747, Robust = 8.83 p=

28 Now plot the baseline survival for each recurrence... Survival Recurrence 1 Recurrence 2 Recurrence 3 Recurrence Time from study start (mths) 12.28

29 Analysis 3 Common baseline hazard and common covariate effects for each recurrence Again, we need to use robust standard errors > gaptimefit2 <- coxph( Surv( start, stop, event) ~ number + size + rx + cluster(id), data=vabladder, subset=enum<5 ) > summary( gaptimefit2 ) n= 178 coef exp(coef) se(coef) robust se z Pr(> z ) number ** size rx exp(coef) exp(-coef) lower.95 upper.95 number size rx Rsquare= (max possible= ) Likelihood ratio test= 17.5 on 3 df, p= Wald test = 11.5 on 3 df, p= Score (logrank) test = 19.5 on 3 df, p= , Robust = 11.3 p=

30 This is sometimes referred to as the Anderson-Gill model This model has the most straightfoward interpretation (and is most commonly used) Need to question whether the assumption of a common baseline hazard and covariate effect is reasonable 12.30

31 - Comparison of Layouts Analysis Time Interval Event Stratum Recurrent - 1 & 2 (0,9] 1 1 (9,13] 1 2 (13,28] 1 3 (28,31] 0 4 Recurrent - 3 (0,9] 1 1 (Anderson-Gill; (9,13] 1 1 Counting Process) (13,28] 1 1 (28,31] 0 1 Marginal - 1 & 2 (0,9] 1 1 (0,13] 1 2 (0,28] 1 3 (0,31] 0 4 Marginal - 3 (0,9] 1 1 (0,13] 1 1 (0,28] 1 1 (0,31]