wtsc {weightedScores}R Documentation

THE WEIGHTED SCORES EQUATIONS WITH INPUTS OF THE WEIGHT MATRICES AND THE DATA

Description

The weighted scores equations with inputs of the weight matrices and the data.

Usage

wtsc(param,WtScMat,xdat,ydat,id,tvec,margmodel,link)
wtsc.ord(param,WtScMat,xdat,ydat,id,tvec,link)

Arguments

param

The vector of regression and not regression parameters.

WtScMat

A list containing the following components. omega: The array with the Ωi,i=1,,n\Omega_i,\,i=1,\ldots,n matrices; delta: The array with the Δi,i=1,,n\Delta_i,\,i=1,\ldots,n matrices; X: The array with the Xi,i=1,,nX_i,\,i=1,\ldots,n matrices.

xdat

(x1,x2,,xn)(\mathbf{x}_1 , \mathbf{x}_2 , \ldots , \mathbf{x}_n )^\top, where the matrix xi,i=1,,n\mathbf{x}_i,\,i=1,\ldots,n for a given unit will depend on the times of observation for that unit (jij_i) and will have number of rows jij_i, each row corresponding to one of the jij_i elements of yiy_i and pp columns where pp is the number of covariates including the unit first column to account for the intercept (except for ordinal regression where there is no intercept). This xdat matrix is of dimension (N×p),(N\times p), where N=i=1njiN =\sum_{i=1}^n j_i is the total number of observations from all units.

ydat

(y1,y2,,yn)(y_1 , y_2 , \ldots , y_n )^\top, where the response data vectors yi,i=1,,ny_i,\,i=1,\ldots,n are of possibly different lengths for different units. In particular, we now have that yiy_i is (ji×1j_i \times 1), where jij_i is the number of observations on unit ii. The total number of observations from all units is N=i=1njiN =\sum_{i=1}^n j_i. The ydat are the collection of data vectors yi,i=1,,ny_i, i = 1,\ldots,n one from each unit which summarize all the data together in a single, long vector of length NN.

id

An index for individuals or clusters.

tvec

A vector with the time indicator of individuals or clusters.

margmodel

Indicates the marginal model. Choices are “poisson” for Poisson, “bernoulli” for Bernoulli, and “nb1” , “nb2” for the NB1 and NB2 parametrization of negative binomial in Cameron and Trivedi (1998).

link

The link function. Choices are “log” for the log link function, “logit” for the logit link function, and “probit” for the probit link function.

Details

The weighted scores estimating equations, with Wi,workingW_{i,\rm working} based on a working discretized MVN, have the form:

g1=g1(a)=i=1nXiTWi,working1si(a)=0, g_1= g_1( a)=\sum_{i=1}^n X_i^T\, W_{i,{\rm working}}^{-1}\, s_i( a)=0,

where Wi,working1=ΔiΩi,working1=Δi(a~)Ωi(a~,R~)1 W_{i,\rm working}^{-1}=\Delta_i\Omega_{i,\rm working}^{-1}= \Delta_i({\tilde a})\Omega_i({\tilde a},{\tilde R})^{-1} is based on the covariance matrix of si(a) s_i( a) computed from the fitted discretized MVN model with estimated parameters a~,R~{\tilde a}, {\tilde R}.

Note that wtsc.ord is a variant of the code for ordinal (probit and logistic) regression.

Value

The weighted scores equations.

References

Nikoloulopoulos, A.K., Joe, H. and Chaganty, N.R. (2011) Weighted scores method for regression models with dependent data. Biostatistics, 12, 653–665. doi: 10.1093/biostatistics/kxr005.

Nikoloulopoulos, A.K. (2016) Correlation structure and variable selection in generalized estimating equations via composite likelihood information criteria. Statistics in Medicine, 35, 2377–2390. doi: 10.1002/sim.6871.

Nikoloulopoulos, A.K. (2017) Weighted scores method for longitudinal ordinal data. Arxiv e-prints, <arXiv:1510.07376>. https://arxiv.org/abs/1510.07376.

See Also

solvewtsc, weightMat, godambe, wtsc.wrapper


[Package weightedScores version 0.9.5.3 Index]