Correlated Bernoulli observations

[Grace Wahba]

TR re multivariate Bernoulli observations:  available via 
http://www.stat.wisc.edu/~wahba -> TRLIST


Smoothing Spline ANOVA for Multivariate Bernoulli Observations, 
            With Application to Ophthalmalogy Data

Fangyu Gao, Grace Wahba, Ronald Klein, MD and Barbara Klein, MD
     UW-Madison Statistics Dept TR 1009, July 15, 1999, submitted

We combine a Smoothing Spline ANOVA model and a log-linear
model to build a partly flexible model for multivariate
correlated Bernoulli response data, where the joint distribution 
of the components of the Bernoulli response vector may depend   
on a complex set of predictor variables. The joint distribution 
conditioning on the predictor variables is estimated via a SS-ANOVA
variational problem. The log odds ratio
is used to measure the association between  outcome
variables. A numerical scheme based on the block
one-step SOR-Newton-Ralphson algorithm is proposed
to obtain an approximate solution for the variational
problem. We extend $GACV$ (Generalized Approximate 
Cross Validation) to the case of multivariate Bernoulli
responses. Its randomized version is fast and stable to
compute and is used to adaptively select smoothing 
parameters in each block one-step SOR iteration.
Approximate Bayesian confidence intervals are obtained for
the flexible estimates of the conditional logit functions.
Simulation studies are conducted to check the performance
of the proposed method. Finally, the model is applied to 
two-eyes observational data from the Beaver Dam Eye Study
to examine the association of pigmentary abnormalities 
and various covariates. The results are applicable to 
a variety of problems where the response of interest 
is a vector of 0's and 1's that exhibit pairwise and 
higher order correlations.