binomial.twostage: Fits Clayton-Oakes or bivariate Plackett (OR) models for...

In mets: Analysis of Multivariate Event Times

Fits Clayton-Oakes or bivariate Plackett (OR) models for binary data using marginals that are on logistic form. If clusters contain more than two times, the algoritm uses a compososite likelihood based on all pairwise bivariate models.

Description

The pairwise pairwise odds ratio model provides an alternative to the alternating logistic regression (ALR).

Usage

binomial.twostage(
  margbin,
  data = parent.frame(),
  method = "nr",
  detail = 0,
  clusters = NULL,
  silent = 1,
  weights = NULL,
  theta = NULL,
  theta.des = NULL,
  var.link = 0,
  var.par = 1,
  var.func = NULL,
  iid = 1,
  notaylor = 1,
  model = "plackett",
  marginal.p = NULL,
  beta.iid = NULL,
  Dbeta.iid = NULL,
  strata = NULL,
  max.clust = NULL,
  se.clusters = NULL,
  numDeriv = 0,
  random.design = NULL,
  pairs = NULL,
  dim.theta = NULL,
  additive.gamma.sum = NULL,
  pair.ascertained = 0,
  case.control = 0,
  no.opt = FALSE,
  twostage = 1,
  beta = NULL,
  ...
)

Arguments

margbin	Marginal binomial model
data	data frame
method	Scoring method "nr", for lava NR optimizer
detail	Detail
clusters	Cluster variable
silent	Debug information
weights	Weights for log-likelihood, can be used for each type of outcome in 2x2 tables.
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
var.par	parametrization
var.func	when alternative parametrizations are used this function can specify how the paramters are related to the \lambda_j's.
iid	Calculate i.i.d. decomposition when iid>=1, when iid=2 then avoids adding the uncertainty for marginal paramters for additive gamma model (default).
notaylor	Taylor expansion
model	model
marginal.p	vector of marginal probabilities
beta.iid	iid decomposition of marginal probability estimates for each subject, if based on GLM model this is computed.
Dbeta.iid	derivatives of marginal model wrt marginal parameters, if based on GLM model this is computed.
strata	strata for fitting: considers only pairs where both are from same strata
max.clust	max clusters
se.clusters	clusters for iid decomposition for roubst standard errors
numDeriv	uses Fisher scoring aprox of second derivative if 0, otherwise numerical derivatives
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 theta when pairs and pairs specific design is given. That is when pairs has 6 columns.
additive.gamma.sum	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. Default is a matrix of 1's
pair.ascertained	if pairs are sampled only when there are events in the pair i.e. Y1+Y2>=1.
case.control	if data is case control data for pair call, and here 2nd column of pairs are probands (cases or controls)
no.opt	for not optimizing
twostage	default twostage=1, to fit MLE use twostage=0
beta	is starting value for beta for MLE version
...	for NR of lava

Details

The reported standard errors are based on a cluster corrected score equations from the pairwise likelihoods assuming that the marginals are known. This gives correct standard errors in the case of the Odds-Ratio model (Plackett distribution) for dependence, but incorrect standard errors for the Clayton-Oakes types model (that is also called "gamma"-frailty). For the additive gamma version of the standard errors are adjusted for the uncertainty in the marginal models via an iid deomposition using the iid() function of lava. For the clayton oakes model that is not speicifed via the random effects these can be fixed subsequently using the iid influence functions for the marginal model, but typically this does not change much.

For the Clayton-Oakes version of the model, given the gamma distributed random effects it is assumed that the probabilities are indpendent, and that the marginal survival functions are on logistic form

logit(P(Y=1|X)) = \alpha + x^T \beta

therefore conditional on the random effect the probability of the event is

logit(P(Y=1|X,Z)) = exp( -Z \cdot Laplace^{-1}(lamtot,lamtot,P(Y=1|x)) )

Can also fit a structured additive gamma random effects model, such 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 DEFAULT parametrization uses the variances of the random effecs (var.par=1)

\theta_j = \lambda_j/(v_1^T \lambda)^2

For alternative parametrizations (var.par=0) one can specify how the parameters relate to \lambda_j with the function

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

Given the random effects the probabilities are independent and on the form

logit(P(Y=1|X)) = exp( - Laplace^{-1}(lamtot,lamtot,P(Y=1|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.

Author(s)

Thomas Scheike

References

Two-stage binomial modelling

Examples

data(twinstut)
twinstut0 <- subset(twinstut, tvparnr<4000)
twinstut <- twinstut0
twinstut$binstut <- (twinstut$stutter=="yes")*1
theta.des <- model.matrix( ~-1+factor(zyg),data=twinstut)
margbin <- glm(binstut~factor(sex)+age,data=twinstut,family=binomial())
bin <- binomial.twostage(margbin,data=twinstut,var.link=1,
         clusters=twinstut$tvparnr,theta.des=theta.des,detail=0)
summary(bin)

twinstut$cage <- scale(twinstut$age)
theta.des <- model.matrix( ~-1+factor(zyg)+cage,data=twinstut)
bina <- binomial.twostage(margbin,data=twinstut,var.link=1,
		         clusters=twinstut$tvparnr,theta.des=theta.des)
summary(bina)

theta.des <- model.matrix( ~-1+factor(zyg)+factor(zyg)*cage,data=twinstut)
bina <- binomial.twostage(margbin,data=twinstut,var.link=1,
		         clusters=twinstut$tvparnr,theta.des=theta.des)
summary(bina)

 ## Reduce Ex.Timings
## refers to zygosity of first subject in eash pair : zyg1
## could also use zyg2 (since zyg2=zyg1 within twinpair's))
out <- easy.binomial.twostage(stutter~factor(sex)+age,data=twinstut,
                          response="binstut",id="tvparnr",var.link=1,
	             	      theta.formula=~-1+factor(zyg1))
summary(out)

## refers to zygosity of first subject in eash pair : zyg1
## could also use zyg2 (since zyg2=zyg1 within twinpair's))
desfs<-function(x,num1="zyg1",num2="zyg2")
    c(x[num1]=="dz",x[num1]=="mz",x[num1]=="os")*1

out3 <- easy.binomial.twostage(binstut~factor(sex)+age,
      data=twinstut,response="binstut",id="tvparnr",var.link=1,
      theta.formula=desfs,desnames=c("mz","dz","os"))
summary(out3)


### use of clayton oakes binomial additive gamma model
###########################################################
 ## Reduce Ex.Timings
data <- simbinClaytonOakes.family.ace(10000,2,1,beta=NULL,alpha=NULL)
margbin <- glm(ybin~x,data=data,family=binomial())
margbin

head(data)
data$number <- c(1,2,3,4)
data$child <- 1*(data$number==3)

### make ace random effects design
out <- ace.family.design(data,member="type",id="cluster")
out$pardes
head(out$des.rv)

bints <- binomial.twostage(margbin,data=data,
     clusters=data$cluster,detail=0,var.par=1,
     theta=c(2,1),var.link=0,
     random.design=out$des.rv,theta.des=out$pardes)
summary(bints)

data <- simbinClaytonOakes.twin.ace(10000,2,1,beta=NULL,alpha=NULL)
out  <- twin.polygen.design(data,id="cluster",zygname="zygosity")
out$pardes
head(out$des.rv)
margbin <- glm(ybin~x,data=data,family=binomial())

bintwin <- binomial.twostage(margbin,data=data,
     clusters=data$cluster,var.par=1,
     theta=c(2,1),random.design=out$des.rv,theta.des=out$pardes)
summary(bintwin)
concordanceTwinACE(bintwin)

mets documentation built on May 29, 2024, 3:51 a.m.