# binomial.twostage: Fits Clayton-Oakes or bivariate Plackett (OR) models for... In mets: Analysis of Multivariate Event Times

 binomial.twostage R Documentation

## 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.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,
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 λ_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)) = α + x^T β

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 (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 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)

θ_j = λ_j/(v_1^T λ)^2

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

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

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 (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.

Thomas Scheike

### References

Two-stage binomial modelling

### Examples

data(twinstut)
twinstut0 <- subset(twinstut, tvparnr<11000)
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())
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)
clusters=twinstut$tvparnr,theta.des=theta.des) summary(bina) ## refers to zygosity of first subject in eash pair : zyg1 ## could also use zyg2 (since zyg2=zyg1 within twinpair's)) out <- easy.binomial.twostage(stutter~factor(sex)+age,data=twinstut, response="binstut",id="tvparnr",var.link=1, theta.formula=~-1+factor(zyg1)) summary(out) ## refers to zygosity of first subject in eash pair : zyg1 ## could also use zyg2 (since zyg2=zyg1 within twinpair's)) desfs<-function(x,num1="zyg1",num2="zyg2") c(x[num1]=="dz",x[num1]=="mz",x[num1]=="os")*1 out3 <- easy.binomial.twostage(binstut~factor(sex)+age, data=twinstut,response="binstut",id="tvparnr",var.link=1, theta.formula=desfs,desnames=c("mz","dz","os")) summary(out3) ### use of clayton oakes binomial additive gamma model ########################################################### ## Reduce Ex.Timings data <- simbinClaytonOakes.family.ace(10000,2,1,beta=NULL,alpha=NULL) margbin <- glm(ybin~x,data=data,family=binomial()) margbin head(data) data$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 Oct. 2, 2022, 5:06 p.m.