survival.twostage: Twostage survival model for multivariate survival data

In mets: Analysis of Multivariate Event Times

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 form. The dependence can be modelled via

1. Regression design on dependence parameter.

2. Random effects, additive gamma model.

If clusters contain more than two subjects, we use a composite likelihood based on the pairwise bivariate models, for full MLE see twostageMLE.

The two-stage model is constructed such that given the gamma distributed random effects it is assumed that the survival functions are indpendent, and that the marginal survival functions are on Cox form (or additive form)

P(T > t| x) = S(t|x)= exp( -exp(x^T β) A_0(t) )

One possibility is to model the variance within clusters via a regression design, and then one can specify a regression structure for the independent gamma distributed random effect for each cluster, such that the variance is given by

θ = h( z_j^T α)

where z is specified by theta.des, and a possible link function var.link=1 will will use the exponential link h(x)=exp(x), and var.link=0 the identity link h(x)=x. The reported standard errors are based on the estimated information from the likelihood assuming that the marginals are known (unlike the twostageMLE and for the additive gamma model below).

Can also fit a structured additive gamma random effects model, such as the ACE, ADE model for survival data. In this case the 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 (λ_1,...,λ_d). By construction subjects 1's random effect are Gamma distributed with mean λ_j/v_1^T λ and variance λ_j/(v_1^T λ)^2. Note that the random effect v_1^T (Z_1,...,Z_d) has mean 1 and variance 1/(v_1^T λ). It is here asssumed that lamtot=v_1^T λ is fixed within clusters as it would be for the ACE model below.

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

The DEFAULT parametrization (var.par=1) uses the variances of the random effecs

θ_j = λ_j/(v_1^T λ)^2

For alternative parametrizations one can specify how the parameters relate to λ_j with the argument var.par=0.

For both types of models the basic model assumptions are that given the random effects of the clusters the survival distributions within 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 1/lamtot.

The parameters (λ_1,...,λ_d) are related to the parameters of the model by a regression construction pard (d x k), that links the d λ parameters with the (k) underlying θ parameters

λ = theta.des θ

here using theta.des to specify these low-dimension association. Default is a diagonal matrix. This can be used to make structural assumptions about the variances of the random-effects as is needed for the ACE model for example.

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 = parent.frame(),
  method = "nr",
  detail = 0,
  clusters = NULL,
  silent = 1,
  weights = NULL,
  theta = NULL,
  theta.des = NULL,
  var.link = 1,
  baseline.iid = 1,
  model = "clayton.oakes",
  marginal.trunc = NULL,
  marginal.survival = NULL,
  strata = NULL,
  se.clusters = NULL,
  numDeriv = 1,
  random.design = NULL,
  pairs = NULL,
  dim.theta = NULL,
  numDeriv.method = "simple",
  additive.gamma.sum = NULL,
  var.par = 1,
  no.opt = FALSE,
  ...
)

Arguments

margsurv	Marginal model
data	data frame
method	Scoring method "nr", for lava NR optimizer
detail	Detail
clusters	Cluster variable
silent	Debug information
weights	Weights
theta	Starting values for variance components
theta.des	design for dependence parameters, when pairs are given the indeces of the theta-design for this pair, is given in pairs as column 5
var.link	Link function for variance: exp-link.
baseline.iid	to adjust for baseline estimation, using phreg function on same data.
model	model
marginal.trunc	marginal left truncation probabilities
marginal.survival	optional vector of marginal survival probabilities
strata	strata for fitting, see example
se.clusters	for clusters for se calculation with iid
numDeriv	to get numDeriv version of second derivative, otherwise uses sum of squared scores for each pair
random.design	random effect design for additive gamma model, when pairs are given the indeces of the pairs random.design rows are given as columns 3:4
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.
dim.theta	dimension of the theta parameter for pairs situation.
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 λ_j's are used as parameters.
no.opt	for not optimizng
...	Additional arguments to maximizer NR of lava. and ascertained sampling

Author(s)

Thomas Scheike

References

Twostage estimation of additive gamma frailty models for survival data. Scheike (2019), work in progress

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).

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

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

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

Examples

data(diabetes)

# Marginal Cox model  with treat as covariate
margph <- phreg(Surv(time,status)~treat+cluster(id),data=diabetes)
### Clayton-Oakes, MLE
fitco1<-twostageMLE(margph,data=diabetes,theta=1.0)
summary(fitco1)

### Plackett model
mph <- phreg(Surv(time,status)~treat+cluster(id),data=diabetes)
fitp <- survival.twostage(mph,data=diabetes,theta=3.0,Nit=40,
               clusters=diabetes$id,var.link=1,model="plackett")
summary(fitp)

### Clayton-Oakes
fitco2 <- survival.twostage(mph,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 but with stratafied
marg <- phreg(Surv(time,status)~+strata(treat)+cluster(id),data=diabetes)
fitpa <- survival.twostage(marg,data=diabetes,theta=1.0,
                clusters=diabetes$id,model="clayton.oakes")
summary(fitpa)

fitcoa <- survival.twostage(marg,data=diabetes,theta=1.0,clusters=diabetes$id,
                 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,method="optimize",
                          id="cluster",timevar="time",
                          status="status",model="clayton.oakes",silent=0)
summary(udp)

 ## 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 <- timereg::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,
                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,
                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,
                model="clayton.oakes",theta.des=tdes2,step=0.5,strata=ud$strata)
summary(fitp4)


 ## Reduce Ex.Timings
### structured random effects model additive gamma ACE
### simulate structured two-stage additive gamma ACE model
data <- simClaytonOakes.twin.ace(4000,2,1,0,3)
out <- twin.polygen.design(data,id="cluster")
pardes <- out$pardes
pardes
des.rv <- out$des.rv
head(des.rv)
aa <- phreg(Surv(time,status)~x+cluster(cluster),data=data,robust=0)
ts <- survival.twostage(aa,data=data,clusters=data$cluster,detail=0,
	       theta=c(2,1),var.link=0,step=0.5,
	       random.design=des.rv,theta.des=pardes)
summary(ts)

mets documentation built on Jan. 17, 2023, 5:12 p.m.