mets (version 1.2.3.1)

survival.twostage: Twostage survival model for multivariate survival data

Description

Fits Clayton-Oakes or bivariate Plackett models for bivariate survival data using marginals that are on Cox or addtive form. The dependence can be modelled via

1. Regression design on dependence parameter.

2. Random effects, additive gamma model.

Can also fit standard frailty model with the two.stage=0 that considers the frailty model with additive hazard condtional on the random effects $$ \lambda_{ij} = (V_{ij}^T Z) (X_{ij}^T \alpha(t)) $$ The baseline \(\alpha(t)\) is profiled out using marginal modelling adjusted for the random effects structure as in Eriksson and Scheike (2015). One advantage of the standard frailty model is that one can deal with competing risks for this model.

For all models the standard errors do not reflect this uncertainty of the baseline estimates, and might therefore be a bit to small. To remedy this one can do bootstrapping or use survival.twostage.fullse function when possible.

If clusters contain more than two times, the algoritm uses a composite likelihood based on the pairwise bivariate models. Can also fit a additive gamma random effects model described in detail below.

We allow a regression structure for the indenpendent gamma distributed random effects and their variances that may depend on cluster covariates. So $$ \theta = z_j^T \alpha $$ where \(z\) is specified by theta.des The reported standard errors are based on the estimated information from the likelihood assuming that the marginals are known.

Can also fit a structured additive gamma random effects model, such as the ACE, ADE model for survival data.

Given the gamma distributed random effects it is assumed that the survival functions are indpendent, and that the marginal survival functions are on additive form (or Cox form) $$ P(T > t| x) = S(t|x)= exp( -x^T A(t) ) $$

Now random.design specificies the random effects for each subject within a cluster. This is a matrix of 1's and 0's with dimension n x d. With d random effects. For a cluster with two subjects, we let the random.design rows be \(v_1\) and \(v_2\). Such that the random effects for subject 1 is $$v_1^T (Z_1,...,Z_d)$$, for d random effects. Each random effect has an associated parameter \((\lambda_1,...,\lambda_d)\). By construction subjects 1's random effect are Gamma distributed with mean \(\lambda_j/v_1^T \lambda\) and variance \(\lambda_j/(v_1^T \lambda)^2\). Note that the random effect \(v_1^T (Z_1,...,Z_d)\) has mean 1 and variance \(1/(v_1^T \lambda)\). It is here asssumed that \(lamtot=v_1^T \lambda\) is fixed over all clusters as it would be for the ACE model below. The lamtot parameter may be specified separately for some sets of the parameter is the additive.gamma.sum (ags) matrix is specified and then lamtot for the j'th random effect is \(ags_j^T \lambda\).

Based on these parameters the relative contribution (the heritability, h) is equivalent to the expected values of the random effects \(\lambda_j/v_1^T \lambda\)

The DEFAULT parametrization uses the variances of the random effecs $$ \theta_j = \lambda_j/(v_1^T \lambda)^2 $$ For alternative parametrizations one can specify how the parameters relate to \(\lambda_j\) with the function

Given the random effects the survival distributions with a cluster are independent and on the form $$ P(T > t| x,z) = exp( -Z \cdot Laplace^{-1}(lamtot,lamtot,S(t|x)) ) $$ with the inverse laplace of the gamma distribution with mean 1 and variance lamtot.

The parameters \((\lambda_1,...,\lambda_d)\) are related to the parameters of the model by a regression construction \(pard\) (d x k), that links the \(d\) \(\lambda\) parameters with the (k) underlying \(\theta\) parameters $$ \lambda = theta.des \theta $$ here using theta.des to specify these low-dimension association. Default is a diagonal matrix.

The case.control option that can be used with the pair specification of the pairwise parts of the estimating equations. Here it is assumed that the second subject of each pair is the proband.

Usage

survival.twostage(margsurv,data=sys.parent(),score.method="fisher.scoring",
                  Nit=60,detail=0,clusters=NULL,
                  silent=1,weights=NULL, control=list(),theta=NULL,theta.des=NULL,
                  var.link=1,iid=1,step=0.5,notaylor=0,model="clayton.oakes",
                  marginal.trunc=NULL,marginal.survival=NULL,marginal.status=NULL,
                  strata=NULL,
                  se.clusters=NULL,max.clust=NULL,numDeriv=0,random.design=NULL,
                  pairs=NULL,pairs.rvs=NULL,
                  numDeriv.method="simple",
                  additive.gamma.sum=NULL,var.par=1,two.stage=1,cr.models=NULL,
                  case.control=0, ascertained=0, shut.up=0)

Arguments

margsurv

Marginal model

data

data frame

score.method

Scoring method "fisher.scoring", "nlminb", "optimize", "nlm"

Nit

Number of iterations

detail

Detail

clusters

Cluster variable

silent

Debug information

weights

Weights

control

Optimization arguments

theta

Starting values for variance components

theta.des

design for dependence parameters, when pairs are given this is could be a (pairs) x (numer of parameters) x (max number random effects) matrix

var.link

Link function for variance

iid

Calculate i.i.d. decomposition

step

Step size

notaylor

Taylor expansion

model

model

marginal.trunc

marginal left truncation probabilities

marginal.survival

optional vector of marginal survival probabilities

marginal.status

related to marginal survival probabilities

strata

strata for fitting, see example

se.clusters

for clusters for se calculation with iid

max.clust

max se.clusters for se calculation with iid

numDeriv

to get numDeriv version of second derivative, otherwise uses sum of squared score

random.design

random effect design for additive gamma model, when pairs are given this is a (pairs) x (2) x (max number random effects) matrix, see pairs.rvs below

pairs

matrix with rows of indeces (two-columns) for the pairs considered in the pairwise composite score, useful for case-control sampling when marginal is known.

pairs.rvs

for additive gamma model and random.design and theta.des are given as arrays, this specifice number of random effects for each pair.

numDeriv.method

uses simple to speed up things and second derivative not so important.

additive.gamma.sum

for two.stage=0, this is specification of the lamtot in the models via a matrix that is multiplied onto the parameters theta (dimensions=(number random effects x number of theta parameters), when null then sums all parameters.

var.par

is 1 for the default parametrization with the variances of the random effects, var.par=0 specifies that the \(\lambda_j\)'s are used as parameters.

two.stage

to fit two-stage model, if 0 then will fit hazard model with additive gamma structure (WIP)

cr.models

competing risks models for two.stage=0, should be given as a list with models for each cause

case.control

assumes case control structure for "pairs" with second column being the probands, when this options is used the twostage model is profiled out via the paired estimating equations for the survival model.

ascertained

if the pair are sampled only when there is an event. This is in contrast to case.control sampling where a proband is given. This can be combined with control probands. Pair-call of twostage is needed and second column of pairs are the first jump time with an event for ascertained pairs, or time of control proband.

shut.up

to make the program more silent in the context of iterative procedures for case-control and ascertained sampling

References

Estimating heritability for cause specific mortality based on twins studies Scheike, Holst, Hjelmborg (2014), LIDA

Measuring early or late dependence for bivariate twin data Scheike, Holst, Hjelmborg (2015), LIDA

Twostage modelling of additive gamma frailty models for survival data. Scheike and Holst, working paper

Additive Gamma frailty models for competing risks data, Biometrics (2015) Eriksson and Scheike (2015),

Shih and Louis (1995) Inference on the association parameter in copula models for bivariate survival data, Biometrics, (1995).

Glidden (2000), A Two-Stage estimator of the dependence parameter for the Clayton Oakes model, LIDA, (2000).

Examples

# NOT RUN {
library("timereg")
library("survival")
data(diabetes)

# Marginal Cox model  with treat as covariate
margph <- coxph(Surv(time,status)~treat,data=diabetes)
### Clayton-Oakes, from timereg
fitco1<-two.stage(margph,data=diabetes,theta=1.0,detail=0,Nit=40,clusters=diabetes$id)
summary(fitco1)
### Plackett model
fitp <- survival.twostage(margph,data=diabetes,theta=3.0,Nit=40,
               clusters=diabetes$id,var.link=1,model="plackett")
summary(fitp)
### Clayton-Oakes
fitco2 <- survival.twostage(margph,data=diabetes,theta=0.0,detail=0,
                 clusters=diabetes$id,var.link=1,model="clayton.oakes")
summary(fitco2)
fitco3 <- survival.twostage(margph,data=diabetes,theta=1.0,detail=0,
                 clusters=diabetes$id,var.link=0,model="clayton.oakes")
summary(fitco3)

### without covariates using Aalen for marginals
marg <- aalen(Surv(time,status)~+1,data=diabetes,n.sim=0,max.clust=NULL,robust=0)
fitpa <- survival.twostage(marg,data=diabetes,theta=1.0,detail=0,Nit=40,
                clusters=diabetes$id,score.method="optimize")
summary(fitpa)

fitcoa <- survival.twostage(marg,data=diabetes,theta=1.0,detail=0,Nit=40,clusters=diabetes$id,
                 var.link=1,model="clayton.oakes")
summary(fitcoa)

### Piecewise constant cross hazards ratio modelling
########################################################

d <- subset(simClaytonOakes(2000,2,0.5,0,stoptime=2,left=0),!truncated)
udp <- piecewise.twostage(c(0,0.5,2),data=d,score.method="optimize",
                          id="cluster",timevar="time",
                          status="status",model="clayton.oakes",silent=0)
summary(udp)

# }
# NOT RUN {
 ## Reduce Ex.Timings
### Same model using the strata option, a bit slower
########################################################
## makes the survival pieces for different areas in the plane 
##ud1=surv.boxarea(c(0,0),c(0.5,0.5),data=d,id="cluster",timevar="time",status="status")
##ud2=surv.boxarea(c(0,0.5),c(0.5,2),data=d,id="cluster",timevar="time",status="status")
##ud3=surv.boxarea(c(0.5,0),c(2,0.5),data=d,id="cluster",timevar="time",status="status")
##ud4=surv.boxarea(c(0.5,0.5),c(2,2),data=d,id="cluster",timevar="time",status="status")

## everything done in one call 
ud <- piecewise.data(c(0,0.5,2),data=d,timevar="time",status="status",id="cluster")
ud$strata <- factor(ud$strata); 
ud$intstrata <- factor(ud$intstrata)

## makes strata specific id variable to identify pairs within strata
## se's computed based on the id variable across strata "cluster"
ud$idstrata <- ud$id+(as.numeric(ud$strata)-1)*2000

marg2 <- aalen(Surv(boxtime,status)~-1+factor(num):factor(intstrata),
               data=ud,n.sim=0,robust=0)
tdes <- model.matrix(~-1+factor(strata),data=ud)
fitp2 <- survival.twostage(marg2,data=ud,se.clusters=ud$cluster,clusters=ud$idstrata,
                score.method="fisher.scoring",model="clayton.oakes",
                theta.des=tdes,step=0.5)
summary(fitp2)

### now fitting the model with symmetry, i.e. strata 2 and 3 same effect
ud$stratas <- ud$strata; 
ud$stratas[ud$strata=="0.5-2,0-0.5"] <- "0-0.5,0.5-2"
tdes2 <- model.matrix(~-1+factor(stratas),data=ud)
fitp3 <- survival.twostage(marg2,data=ud,clusters=ud$idstrata,se.cluster=ud$cluster,
                score.method="fisher.scoring",model="clayton.oakes",
                theta.des=tdes2,step=0.5)
summary(fitp3)

### same model using strata option, a bit slower 
fitp4 <- survival.twostage(marg2,data=ud,clusters=ud$cluster,se.cluster=ud$cluster,
                score.method="fisher.scoring",model="clayton.oakes",
                theta.des=tdes2,step=0.5,strata=ud$strata)
summary(fitp4)
# }
# NOT RUN {
### structured random effects model additive gamma ACE 
### simulate structured two-stage additive gamma ACE model
data <- simClaytonOakes.twin.ace(2000,2,1,0,3)
out <- twin.polygen.design(data,id="cluster")
pardes <- out$pardes
des.rv <- out$des.rv
aa <- aalen(Surv(time,status)~+1,data=data,robust=0)
ts <- survival.twostage(aa,data=data,clusters=data$cluster,detail=0,
	       theta=c(2,1)/10,var.link=0,step=0.5,
	       random.design=des.rv,theta.des=pardes)
summary(ts)

### case control sampling of data, call via pairs

data2 <- fast.reshape(data,id="cluster")
ncase <- 400; ncont <- 100
controls <- which(data2$status2==0)
cases <-    which(data2$status2==1)
controls<-sort(sample(controls,min(ncont,length(controls))))
cases <- sort( sample(cases,   min(ncase,length(cases))))
clustco <- data2$cluster[controls]
clustca <- data2$cluster[cases]
ss <- data$cluster %in% c(clustco,clustca)
datacc <- data[ss,]

mm <- familycluster.index(datacc$cluster)
pairs <- mm$pairs
head(pairs)
## second column of pairs represent probands 
kinship <- rep(1,nrow(pairs))
kinship[datacc$zyg[pairs[,1]]=="DZ"] <- 0.5
dout <- make.pairwise.design(pairs,kinship,type="ace")
## additive model specified via formula-list
cr.models <- list(Surv(time,status)~+1)

tscce <- survival.twostage(NULL,data=datacc,clusters=datacc$cluster,
           detail=0,theta=c(2,1)/10,var.link=0,step=1.0,
       pairs=pairs,
       random.design=dout$random.design,theta.des=dout$theta.des,
       pairs.rvs=dout$ant.rvs,
       case.control=1, marginal.status=datacc$status,
       cr.models=cr.models)
summary(tscce)

### see also pairwise*.r demos under inst for frailty, competing risks and
### case control sampling 
# }