1. Workshop 1 Alternative models for clustered survival data

2. Log-linear model representation in parametric survival models In most packages (SAS, R) survival models (and their estimates) are parametrized as log linear models If the error term e ij has extreme value distribution, then this model corresponds to –PH Weibull model with –AFT Weibull model with

3. The first data set bivariate survival data Time to reconstitution of blood-milk barrier after mastitis –Two quarters are infected with E. coli –One quarter treated locally, other quarter not –Blood milk-barrier destroyed –Milk Na + increases –Time to normal Na + level

4. Time to reconstitution data Cow number123…99100 Heifer110…10 Treatment1.96.50*4.78…0.664.93 Placebo0.416.50*2.62…0.986.50*

5. Reading in the data library(survival) bloodmilk<-read.table("c://bivariate.dat",header=T) cows<-bloodmilk$cows y<-bloodmilk$y uncens<-bloodmilk$uncens trt<-bloodmilk$trt heifer<-bloodmilk$heifer table(trt,uncens) uncens trt 0 1 0 28 72 1 19 81

6. Types of models fitted Treatment effect –Parametric models with constant hazard Unadjusted model Marginal model Fixed effects model –Parametric models based on Weibull distribution Unadjusted model and coefficient interpretation –Models with unspecified baseline hazard Stratified model Semiparametric marginal model Heifer effect Fixed effects model with constant hazard Stratified model

7. Exponential unadjusted model res.unadjust<-survreg(Surv(y,uncens)~trt,dist="exponential",data=bloodmilk) summary(res.unadjust) b.unadjust<- -res.unadjust$coef[2] v.unadjust<- sqrt(res.unadjust$var[2,2]) l.unadjust<-exp(-res.unadjust$coef[1]) Call: survreg(formula = Surv(y, uncens) ~ trt, data = bloodmilk, dist = "exponential") Value Std. Error z p (Intercept) 1.531 0.118 12.99 1.33e-38 trt -0.176 0.162 -1.09 2.77e-01 > l.unadjust (Intercept) 0.2162472

8. Exponential marginal model ncows<-length(levels(as.factor(cows))) bdel<-rep(NA,ncows) for (i in 1:ncows){ temp<-bloodmilk[bloodmilk$cows!=i,] bdel[i]<- survreg(Surv(y,uncens)~trt,data=temp,dist="exponential")$coeff[2]} sqrt(sum((bdel-b.unadjust)^2)) 0.1535

9. Exponential fixed effects model res.fixed<- survreg(Surv(y,uncens)~trt+as.factor(cows),dist="exponential",data=bloodmilk) summary(res.fixed) b.fix<- -res.fixed$coef[2] v.fix<- sqrt(res.fixed$var[2,2]) Call:survreg(formula = Surv(y, uncens) ~ trt + as.factor(cows), data = bloodmilk, dist = "exponential") Value Std. Error z p (Intercept) 2.10e+01 3331.692 0.00629 0.995 trt -1.85e-01 0.190 -0.96993 0.332 as.factor(cows)2 -1.88e+01 3331.692 -0.00564 0.995 … as.factor(cows)100 -1.86e+01 3331.692 -0.00557 0.996

10. Weibull unadjusted model (1) res.unadjustw<-survreg(Surv(y,uncens)~trt,dist="weibull",data=bloodmilk) summary(res.unadjustw) b.unadjustw<- -res.unadjustw$coef[2]/res.unadjustw$scale v.unadjustw<- sqrt(res.unadjustw$var[2,2]) l.unadjustw<-exp(-res.unadjustw$coef[1]/res.unadjustw$scale) r.unadjustw<-1/res.unadjustw$scale Call:survreg(formula = Surv(y, uncens) ~ trt, data = bloodmilk, dist = "weibull") Value Std. Error z p (Intercept) 1.533 0.1222 12.542 4.38e-36 trt -0.180 0.1681 -1.072 2.84e-01 Log(scale) 0.036 0.0695 0.518 6.04e-01 Scale= 1.04 b.unadjustw trt 0.17387 v.unadjustw 0.1681 l.unadjustw (Intercept) 0.2278757 r.unadjustw 0.9646458

11. Weibull unadjusted model (2) likelihood.unadjust.weib.exptransf<-function(p){ cumhaz<-exp(trt*p[2])*y^(exp(p[3]))*exp(p[1]) lnhaz<-uncens*(trt*p[2]+log(exp(p[1]))+log(exp(p[3]))+(exp(p[3])-1)*log(y) ) lik<- sum(cumhaz)-sum(lnhaz)} initial<-c(log(0.23),0.2,log(1)) res<-nlm(likelihood.unadjust.weib.exptransf,initial) l.unadjustw2<-exp(res$estimate[1]) b.unadjustw2<-res$estimate[2] r.unadjustw2<-exp(res$estimate[3]) 0.228 0.174 0.965

12. Weibull unadjusted model (3) likelihood.unadjust.weib<-function(p){ cumhaz<-exp(trt*p[2])*y^(p[3])*p[1] lnhaz<-uncens*(trt*p[2]+log(p[1])+log(p[3])+(p[3]-1)*log(y)) lik<- sum(cumhaz)-sum(lnhaz)} initial<-c(exp(res$estimate[1]),res$estimate[2],exp(res$estimate[3])) res.unadjustw2<-nlm(likelihood.unadjust.weib,initial,iterlim=1,hessian=T) l.unadjustw2<-res.unadjustw2$estimate[1] b.unadjustw2<-res.unadjustw2$estimate[2] r.unadjustw2<-res.unadjustw2$estimate[3] v.unadjustw2<-sqrt(solve(res.unadjustw2$hessian)[2,2]) 0.228 0.174 0.965 0.162

13. Stratified model res.strat<-coxph(Surv(y,uncens)~trt+strata(cows)) summary(res.strat) b.strat<-res.strat$coef[1] v.strat<-res.strat$coef[3] Call: coxph(formula = Surv(y, uncens) ~ trt + strata(cows)) n= 200 coef exp(coef) se(coef) z p exp(-coef) lower.95 upper.95 trt 0.131 1.14 0.209 0.625 0.53 0.878 0.757 1.72

14. Semiparametric marginal model (1) res.semimarg<-coxph(Surv(y,uncens)~trt+cluster(cows)) summary(res.semimarg) b.semimarg<-summary(res.semimarg)$coef[1] v.semimarg<-summary(res.semimarg)$coef[4] Call: coxph(formula = Surv(y, uncens) ~ trt + cluster(cows)) n= 200 coef exp(coef) se(coef) robust se z p lower.95 upper.95 trt 0.161 1.17 0.162 0.145 1.11 0.27 0.883 1.56 Rsquare= 0.005 (max possible= 0.999 ) Likelihood ratio test= 0.98 on 1 df, p=0.322 Wald test = 1.22 on 1 df, p=0.269 Score (logrank) test = 0.98 on 1 df, p=0.322, Robust = 1.23 p=0.268

15. Semiparametric marginal model (2) b.unadjust<-coxph(Surv(y,uncens)~trt)$coeff[1] ncows<-length(levels(as.factor(cows))) bdel<-rep(NA,ncows) for (i in 1:ncows){ temp<-bloodmilk[bloodmilk$cows!=i,] bdel[i]<- -coxph(Surv(y,uncens)~trt,data=temp)$coeff[1] } sqrt(sum((-bdel-b.unadjust)^2)) [1] 0.1468473

16. Fixed effects model with heifer, heifer first summary(survreg(Surv(y,uncens)~heifer+as.factor(cows),dist="exponential", data=bloodmilk)) Call:survreg(formula = Surv(y, uncens) ~ heifer + as.factor(cows),data=bloodmilk, dist = "exponential") Value Std. Error z p (Intercept) 2.09e+01 4723 4.42e-03 0.996 heifer -2.01e+01 6680 -3.01e-03 0.998 as.factor(cows)2 1.21e+00 4723 2.56e-04 1.000... as.factor(cows)100 -1.85e+01 4723 -3.92e-03 0.997

17. Fixed effects model with heifer, cow first summary(survreg(Surv(y,uncens)~as.factor(cows)+heifer,dist="exponential", data=bloodmilk)) Call:survreg(formula = Surv(y, uncens) ~ as.factor(cows) + as.factor(heifer),data=bloodmilk,dist="exponential") Value Std. Error z p (Intercept) 2.09e+01 4723 0.00442 0.996 as.factor(cows)2 -1.89e+01 4723 -0.00399 0.997 as.factor(cows)100 -1.85e+01 4723 -0.00392 0.997 as.factor(heifer)1 0.00e+00 6680 0.00000 1.000

18. Stratified model with heifer > summary(coxph(Surv(y,uncens)~heifer+strata(cows),data=bloodmilk)) Call:coxph(formula = Surv(y, uncens) ~ heifer + strata(cows), data = bloodmilk) n= 200 coef exp(coef) se(coef) z p exp(-coef) lower.95 upper.95 heifer NA NA 0 NA NA NA NA NA Rsquare= 0 (max possible= 0.471 ) Likelihood ratio test= 0 on 0 df, p=NaN Wald test = NA on 0 df, p=NA Score (logrank) test = 0 on 0 df, p=NaN Warning messages: 1: X matrix deemed to be singular; variable 1 in: coxph(Surv(y, uncens) ~ heifer + strata(cows), data = bloodmilk) 2: NaNs produced in: pchisq(q, df, lower.tail, log.p) 3: NaNs produced in: pchisq(q, df, lower.tail, log.p)

19. The data set: Time to first insemination Database of regional Dairy Herd Improvement Association (DHIA) –Milk recording service –Artificial insemination –Select sample –Subset of 2567 cows from 49 dairy farms

20. Fixed covariates data set insemfix.dat

21. Effect of initial ureum concentration Fit the following models –Parametric models with constant hazard Unadjusted model Fixed effects model Marginal model –Models with unspecified baseline hazard Stratified model Semiparametric marginal model Fixed effects model