Fits additive gamma 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.
If clusters contain more than two times, we use 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.
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 $$ P(T > t| x,z) = exp( -Z A(t) \exp( Z^t beta)) $$
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.
survival.iterative(
margsurv,
data = parent.frame(),
method = "nr",
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,
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,
cr.models = NULL,
case.control = 0,
ascertained = 0,
shut.up = 0
)Marginal model
data frame
Scoring method "nr", "nlminb", "optimize", "nlm"
Number of iterations
Detail
Cluster variable
Debug information
Weights
Optimization arguments
Starting values for variance components
design for dependence parameters, when pairs are given this is could be a (pairs) x (numer of parameters) x (max number random effects) matrix
Link function for variance
Calculate i.i.d. decomposition
Step size
model
marginal left truncation probabilities
optional vector of marginal survival probabilities
related to marginal survival probabilities
strata for fitting, see example
for clusters for se calculation with iid
max se.clusters for se calculation with iid
to get numDeriv version of second derivative, otherwise uses sum of squared score
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
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.
for additive gamma model and random.design and theta.des are given as arrays, this specifice number of random effects for each pair.
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.
competing risks models for two.stage=0, should be given as a list with models for each cause
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.
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.
to make the program more silent in the context of iterative procedures for case-control and ascertained sampling
Thomas Scheike