Workshop 1 Alternative models for clustered survival data

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

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

Time to reconstitution data Cow number123…99100 Heifer110…10 Treatment *4.78… Placebo *2.62… *

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

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

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) e-38 trt e-01 > l.unadjust (Intercept)

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))

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 trt -1.85e as.factor(cows) e … as.factor(cows) e

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) e-36 trt e-01 Log(scale) e-01 Scale= 1.04 b.unadjustw trt v.unadjustw l.unadjustw (Intercept) r.unadjustw

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])

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])

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

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 Rsquare= (max possible= ) 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

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]

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 e heifer -2.01e e as.factor(cows)2 1.21e e as.factor(cows) e e

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 as.factor(cows) e as.factor(cows) e as.factor(heifer)1 0.00e

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= ) 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)

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

Fixed covariates data set insemfix.dat

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