Fits Clayton-Oakes or bivariate Plackett models for bivariate survival data using marginals that are on Cox form. The dependence can be modelled via
Regression design on dependence parameter.
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) $$ P(T > t| x) = S(t|x)= exp( -exp(x^T \beta) A_0(t) ) $$
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 $$ \theta = h( z_j^T \alpha) $$ 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 \((\lambda_1,...,\lambda_d)\). 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 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: \(\lambda_j/v_1^T \lambda\)
The DEFAULT parametrization (var.par=1) 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 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 \((\lambda_1,...,\lambda_d)\) 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. 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.
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,
...
)Marginal model
data frame
Scoring method "nr", for lava NR optimizer
Detail
Cluster variable
Debug information
Weights
Starting values for variance components
design for dependence parameters, when pairs are given the indeces of the theta-design for this pair, is given in pairs as column 5
Link function for variance: exp-link.
to adjust for baseline estimation, using phreg function on same data.
model
marginal left truncation probabilities
optional vector of marginal survival probabilities
strata for fitting, see example
for clusters for se calculation with iid
to get numDeriv version of second derivative, otherwise uses sum of squared scores for each pair
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
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.
dimension of the theta parameter for pairs situation.
uses simple to speed up things and second derivative not so important.
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.
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.
for not optimizng
Additional arguments to maximizer NR of lava. and ascertained sampling
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),
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)
#> Dependence parameter for Clayton-Oakes model
#> Variance of Gamma distributed random effects
#> $estimates
#> Coef. SE z P-val Kendall tau SE
#> dependence1 0.9526614 0.3543033 2.68883 0.007170289 0.322645 0.08127892
#>
#> $type
#> NULL
#>
#> attr(,"class")
#> [1] "summary.mets.twostage"
### Plackett model
mph <- phreg(Surv(time,status)~treat+cluster(id),data=diabetes)
fitp <- survival.twostage(mph,data=diabetes,theta=3.0,Nit=40,
clusters=diabetes$id,var.link=1,model="plackett")
summary(fitp)
#> Dependence parameter for Odds-Ratio (Plackett) model
#> With log-link
#> $estimates
#> log-Coef. SE z P-val Spearman Corr. SE
#> dependence1 1.14188 0.3026537 3.772891 0.0001613666 0.3648216 0.08869229
#>
#> $or
#> Estimate Std.Err 2.5% 97.5% P-value
#> dependence1 3.133 0.9481 1.274 4.991 0.0009528
#>
#> $type
#> [1] "plackett"
#>
#> attr(,"class")
#> [1] "summary.mets.twostage"
### Clayton-Oakes
fitco2 <- survival.twostage(mph,data=diabetes,theta=0.0,detail=0,
clusters=diabetes$id,var.link=1,model="clayton.oakes")
summary(fitco2)
#> Dependence parameter for Clayton-Oakes model
#> Variance of Gamma distributed random effects
#> With log-link
#> $estimates
#> log-Coef. SE z P-val Kendall tau SE
#> dependence1 -0.0484957 0.3718487 -0.1304178 0.8962359 0.322645 0.08126576
#>
#> $vargam
#> Estimate Std.Err 2.5% 97.5% P-value
#> dependence1 0.9527 0.3542 0.2584 1.647 0.007161
#>
#> $type
#> [1] "clayton.oakes"
#>
#> attr(,"class")
#> [1] "summary.mets.twostage"
fitco3 <- survival.twostage(margph,data=diabetes,theta=1.0,detail=0,
clusters=diabetes$id,var.link=0,model="clayton.oakes")
summary(fitco3)
#> Dependence parameter for Clayton-Oakes model
#> Variance of Gamma distributed random effects
#> $estimates
#> Coef. SE z P-val Kendall tau SE
#> dependence1 0.9526614 0.3543 2.688855 0.007169754 0.322645 0.08127816
#>
#> $type
#> [1] "clayton.oakes"
#>
#> attr(,"class")
#> [1] "summary.mets.twostage"
### without covariates but with stratafied
marg <- phreg(Surv(time,status)~+strata(treat)+cluster(id),data=diabetes)
fitpa <- survival.twostage(marg,data=diabetes,theta=1.0,
clusters=diabetes$id,model="clayton.oakes")
summary(fitpa)
#> Dependence parameter for Clayton-Oakes model
#> Variance of Gamma distributed random effects
#> With log-link
#> $estimates
#> log-Coef. SE z P-val Kendall tau SE
#> dependence1 -0.05684062 0.3721207 -0.1527478 0.8785971 0.320824 0.08108359
#>
#> $vargam
#> Estimate Std.Err 2.5% 97.5% P-value
#> dependence1 0.9447 0.3516 0.2557 1.634 0.007203
#>
#> $type
#> [1] "clayton.oakes"
#>
#> attr(,"class")
#> [1] "summary.mets.twostage"
fitcoa <- survival.twostage(marg,data=diabetes,theta=1.0,clusters=diabetes$id,
model="clayton.oakes")
summary(fitcoa)
#> Dependence parameter for Clayton-Oakes model
#> Variance of Gamma distributed random effects
#> With log-link
#> $estimates
#> log-Coef. SE z P-val Kendall tau SE
#> dependence1 -0.05684062 0.3721207 -0.1527478 0.8785971 0.320824 0.08108359
#>
#> $vargam
#> Estimate Std.Err 2.5% 97.5% P-value
#> dependence1 0.9447 0.3516 0.2557 1.634 0.007203
#>
#> $type
#> [1] "clayton.oakes"
#>
#> attr(,"class")
#> [1] "summary.mets.twostage"
### Piecewise constant cross hazards ratio modelling
########################################################
d <- subset(simClaytonOakes(2000,2,0.5,0,stoptime=2,left=0),!truncated)
udp <- piecewise.twostage(c(0,0.5,2),data=d,method="optimize",
id="cluster",timevar="time",
status="status",model="clayton.oakes",silent=0)
#> Data-set 1 out of 4
#> Number of joint events: 542 of 2000
#> Data-set 2 out of 4
#> Number of joint events: 244 of 1188
#> Data-set 3 out of 4
#> Number of joint events: 265 of 1213
#> Data-set 4 out of 4
#> Number of joint events: 613 of 943
summary(udp)
#> [1] 1
#> Dependence parameter for Clayton-Oakes model
#> log-coefficient for dependence parameter (SE)
#> 0 - 0.5 0.5 - 2
#> 0 - 0.5 0.724 (0.065) 0.754 (0.088)
#> 0.5 - 2 0.626 (0.097) 0.59 (0.065)
#>
#> Kendall's tau (SE)
#> 0 - 0.5 0.5 - 2
#> 0 - 0.5 0.508 (0.016) 0.515 (0.022)
#> 0.5 - 2 0.483 (0.024) 0.474 (0.016)
#>
## Reduce Ex.Timings
### Same model using the strata option, a bit slower
########################################################
## makes the survival pieces for different areas in the plane
##ud1=surv.boxarea(c(0,0),c(0.5,0.5),data=d,id="cluster",timevar="time",status="status")
##ud2=surv.boxarea(c(0,0.5),c(0.5,2),data=d,id="cluster",timevar="time",status="status")
##ud3=surv.boxarea(c(0.5,0),c(2,0.5),data=d,id="cluster",timevar="time",status="status")
##ud4=surv.boxarea(c(0.5,0.5),c(2,2),data=d,id="cluster",timevar="time",status="status")
## everything done in one call
ud <- piecewise.data(c(0,0.5,2),data=d,timevar="time",status="status",id="cluster")
ud$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)
#> Dependence parameter for Clayton-Oakes model
#> Variance of Gamma distributed random effects
#> With log-link
#> $estimates
#> log-Coef. SE z P-val
#> factor(strata)0-0.5,0-0.5 0.8974824 0.06601337 13.595465 0.000000e+00
#> factor(strata)0-0.5,0.5-2 0.5007013 0.10511215 4.763496 1.902677e-06
#> factor(strata)0.5-2,0-0.5 0.3849839 0.10900019 3.531956 4.124975e-04
#> factor(strata)0.5-2,0.5-2 0.5423840 0.06434127 8.429799 0.000000e+00
#> Kendall tau SE
#> factor(strata)0-0.5,0-0.5 0.5509068 0.01633227
#> factor(strata)0-0.5,0.5-2 0.4520365 0.02603623
#> factor(strata)0.5-2,0-0.5 0.4235631 0.02661320
#> factor(strata)0.5-2,0.5-2 0.4623804 0.01599426
#>
#> $vargam
#> Estimate Std.Err 2.5% 97.5% P-value
#> dependence1 2.453 0.1620 2.136 2.771 7.758e-52
#> dependence2 1.650 0.1734 1.310 1.990 1.841e-21
#> dependence3 1.470 0.1602 1.156 1.784 4.545e-20
#> dependence4 1.720 0.1107 1.503 1.937 1.799e-54
#>
#> $type
#> [1] "clayton.oakes"
#>
#> attr(,"class")
#> [1] "summary.mets.twostage"
### now fitting the model with symmetry, i.e. strata 2 and 3 same effect
ud$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)
#> Dependence parameter for Clayton-Oakes model
#> Variance of Gamma distributed random effects
#> With log-link
#> $estimates
#> log-Coef. SE z P-val
#> factor(stratas)0-0.5,0-0.5 0.8974824 0.06601337 13.595465 0.000000e+00
#> factor(stratas)0-0.5,0.5-2 0.4398282 0.08492830 5.178818 2.232961e-07
#> factor(stratas)0.5-2,0.5-2 0.5423840 0.06434127 8.429799 0.000000e+00
#> Kendall tau SE
#> factor(stratas)0-0.5,0-0.5 0.5509068 0.01633227
#> factor(stratas)0-0.5,0.5-2 0.4370068 0.02089507
#> factor(stratas)0.5-2,0.5-2 0.4623804 0.01599426
#>
#> $vargam
#> Estimate Std.Err 2.5% 97.5% P-value
#> dependence1 2.453 0.1620 2.136 2.771 7.758e-52
#> dependence2 1.552 0.1318 1.294 1.811 5.274e-32
#> dependence3 1.720 0.1107 1.503 1.937 1.799e-54
#>
#> $type
#> [1] "clayton.oakes"
#>
#> attr(,"class")
#> [1] "summary.mets.twostage"
### same model using strata option, a bit slower
fitp4 <- survival.twostage(marg2,data=ud,clusters=ud$cluster,se.cluster=ud$cluster,
model="clayton.oakes",theta.des=tdes2,step=0.5,strata=ud$strata)
summary(fitp4)
#> Dependence parameter for Clayton-Oakes model
#> Variance of Gamma distributed random effects
#> With log-link
#> $estimates
#> log-Coef. SE z P-val
#> factor(stratas)0-0.5,0-0.5 0.8974824 0.06601337 13.595465 0.000000e+00
#> factor(stratas)0-0.5,0.5-2 0.4398282 0.08492830 5.178818 2.232961e-07
#> factor(stratas)0.5-2,0.5-2 0.5423840 0.06434127 8.429799 0.000000e+00
#> Kendall tau SE
#> factor(stratas)0-0.5,0-0.5 0.5509068 0.01633227
#> factor(stratas)0-0.5,0.5-2 0.4370068 0.02089507
#> factor(stratas)0.5-2,0.5-2 0.4623804 0.01599426
#>
#> $vargam
#> Estimate Std.Err 2.5% 97.5% P-value
#> dependence1 2.453 0.1620 2.136 2.771 7.758e-52
#> dependence2 1.552 0.1318 1.294 1.811 5.274e-32
#> dependence3 1.720 0.1107 1.503 1.937 1.799e-54
#>
#> $type
#> [1] "clayton.oakes"
#>
#> attr(,"class")
#> [1] "summary.mets.twostage"
## Reduce Ex.Timings
### structured random effects model additive gamma ACE
### simulate structured two-stage additive gamma ACE model
data <- simClaytonOakes.twin.ace(4000,2,1,0,3)
out <- twin.polygen.design(data,id="cluster")
pardes <- out$pardes
pardes
#> [,1] [,2]
#> [1,] 1.0 0
#> [2,] 0.5 0
#> [3,] 0.5 0
#> [4,] 0.5 0
#> [5,] 0.0 1
des.rv <- out$des.rv
head(des.rv)
#> MZ DZ DZns1 DZns2 env
#> 1 1 0 0 0 1
#> 2 1 0 0 0 1
#> 3 1 0 0 0 1
#> 4 1 0 0 0 1
#> 5 1 0 0 0 1
#> 6 1 0 0 0 1
aa <- phreg(Surv(time,status)~x+cluster(cluster),data=data,robust=0)
ts <- survival.twostage(aa,data=data,clusters=data$cluster,detail=0,
theta=c(2,1),var.link=0,step=0.5,
random.design=des.rv,theta.des=pardes)
summary(ts)
#> Dependence parameter for Clayton-Oakes model
#> Variance of Gamma distributed random effects
#> $estimates
#> Coef. SE z P-val Kendall tau SE
#> dependence1 2.0686552 0.1488964 13.893256 0.000000e+00 0.5084371 0.01798922
#> dependence2 0.9452688 0.1186084 7.969663 1.554312e-15 0.3209448 0.02734611
#>
#> $type
#> [1] "clayton.oakes"
#>
#> $h
#> Estimate Std.Err 2.5% 97.5% P-value
#> [1,] 0.6864 0.04003 0.6079 0.7648 6.776e-66
#> [2,] 0.3136 0.04003 0.2352 0.3921 4.701e-15
#>
#> $vare
#> NULL
#>
#> $vartot
#> Estimate Std.Err 2.5% 97.5% P-value
#> p1 3.014 0.09764 2.823 3.205 3.157e-209
#>
#> attr(,"class")
#> [1] "summary.mets.twostage"