A unified approach for analyzing exchangeable binary data with applications to developmental toxicity studies.

Abstract

In this article, we present a general procedure to analyze exchangeable binary data that may also be viewed as realizations of binomial mixtures. Our approach unifies existing models and is practical and computationally easy. Resulting from completely monotonic functions, we introduce a rich family of parametric parsimonious binomial mixtures, including the incomplete Beta-, Gamma-, Normal-, and Poisson-binomial, generalizing the Beta-binomial. We show that the family is closed under convex linear combinations, products, and composites. We also give the moments and the Markov property of the family. With such distributions, we can perform statistical inference on correlated binary data and, in particular, overdispersed data. We propose a regression procedure that generalizes logistic regression. We provide a forward model selection procedure. We run a small simulation to validate the inclusion of the binomial distribution. Finally, we apply the proposed procedure to analyze the 2, 4, 5-Trichlorophenoxyacetic acid and E2 data and compare the results with existing procedures.

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A Unified Approach for Analyzing Exchangeable Binary
Data with Applications to Clinical and Developmental
Toxicity Studies
Xin Danga, Hanxiang Penga,∗, Xueqin Wangb
aDepartment of Mathematics
University of Mississippi
University, MS 38677
∗Corresponding author
Email: mmpeng@olemiss.edu
bDepartment of Epidemiology and Public Health
Yale University School of Medicine
New Haven, CT 06520-8034,
March 26, 2008

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Abstract
In this article, we present an approach to analyze data that are exchangeable (i.e. realiza-
tions of binomial mixtures). Our approach unifies the existing approaches and is practicable
and computationally easy. Resulted from completely monotonic functions, we give a class
of parsimonious binomial mixtures, including the incomplete Gamma-, Normal-, Poisson-,
and Beta-Binomial, generalizing the Beta-Binomial. We show that the class is closed under
convex linear combination, product, and two types of composites. We point out that the
folded logistic link by Bowman and George(1995) is incorrect. The moments and Markov
property are described. With such numerous mixtures, we can perform statistical inference
on correlated binary data, in particular, over-dispersed data. We propose a regression pro-
cedure which generalizes the logistic regression. We give a stepwise forward procedure about
how a possible best model from the class can be achieved. A small simulation is run to
validate the inclusion of the Binomial. The proposed procedure is applied to the 2, 4, 5-T
and the E2 data and compared favorably with the existing procedures.
Key words and phrases: Beta-binomial, Binomial mixture, Complete monotonicity, Ex-
changeability, Link, Logistic regression, Over-dispersion.

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1Introduction
Binomial distribution is widely used in modeling binary response data in many different
areas of science. One key assumption in Binomial distribution is that the binary responses
are assumed to be independent. The independence, however, may not be valid and non-
independence usually leads to a variance that is greater than the nominal variance. This is
known as over-dispersion (or extra-binomial variation). The perception of exchangeability,
intensely studied over the past century, is meant to capture the notion of symmetry in a
collection of random variables and is often used as an alternative to independence.
In the familial correlated data such as a developmental toxicity experiment in animal
studies, the fetuses from a litter will respond more similarly to a stimulus than from different
litters. Consider a litter of m fetuses with binary responses B1,...,Bmwhere Bi= 1 or 0
denoting death or no death, tumor or no tumor. It cannot be assumed that these binary
responses in a litter are independent. One way to simplify the analysis is to assume that they
are exchangeable. In the risk analysis in Finance such as CreditRisk+(Credit Suisse Financial
Products, 1997), the default probability of a company is assumed to depend on a set of
common economic factors; given these common factors, defaults of the individual obligors
are conditionally independent. Consider a portfolio of m obligors with concentration on the
binary outcomes B1, ..., Bmof default and non-default of the obligors. One way in practice
to formalize the notion of a homogeneous group is to assume that these binary outcomes are
exchangeable. For a full description, see e.g. Frey and McNeil(2003). Specifically B1,...,Bm
are exchangeable if for every {0,1}-valued variables b1,...,bm, one has
P(B1= b1,...,Bm= bm) = P(Bπ1= b1,...,Bπm= bm),
for every permutation π1...πm of 1...m. Let Y = B1+ ... + Bm be the total number of
“successes”. Under the exchangeability, the distribution of Y is given by
?m
k=0
where λ0 = 1, λk = P(B1 = 1,...,Bk = 1),1 ≤ k ≤ m are the marginal probabilities.
See e.g. Kendall(1967), George & Bowman(1995), or Chow & Teicher(1998). We write
Y ∼ EB(λ)(λ = {λk}) and refer it to as the Exchangeable Binomial Distribution(EB).
Applications are not limited to the aforementioned financial and familial data. A wide
variety can be found in other fields of science.
Conaway(1990); in clinical trails, see Fanaroff et al(1994); and in botany, see Example 1.3,
Collett(2003). A theoretical application is in the study of latent variable models, see Frey &
McNeil(2003).
P(Y = y) =
y
?m−y
?
(−1)k
?m − y
k
?
λy+k,y = 0,1,...,m,(1.1)
For example, in social science, see e.g.
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The literature on correlated binary data is extensive, see e.g. Collett(2003) or Joe(1997).
The statistical inference based on the conditional analysis does not use any information
about the distribution of the latent effects since it is lost in the conditioning. The EM
algorithm method by Mislevy(1985) is conceptually simple, but the computations may be
formidable because each E-step in the computation may require a numerical integration. The
empirical Bayes approach by Stiratelli, Laird, and Ware(1984) may also be computationally
difficult. The GEE approach (e.g. Liang, Qaqish and Zeger, 1992) only uses the first two
moments while higher order of correlation is approximated by a “working matrix”. In this
article, we present an approach to analyze correlated binary data by exploiting the notions
of exchangeability and complete monotonicity. Our approach uses all the information and
has mathematical simplicity and computational ease and unifies the existing approaches.
By virtue of the celebrated de Finnetti representation theorem, a sequence of exchange-
able binary random variables is a mixture of binomial distributions, while the mixture can
be characterized by a complete monotonic sequence. Using the relationship between ex-
changeability and complete monotonicity, Bowman and George(1995) initiated an approach
on correlated binary data and gave the folded logistic link, though the folded logistic link is
not a legitimate link as pointed out in Section 2, where we also give a modified version which
is a legitimate link. Xu and Prorok(2003) showed that the MLEs of the population param-
eters have no closed form in general and pointed out that the MLE’s can only be calculated
by numerical methods. They applied their results to a double-blind randomized clinical trial
data in comparing two antibiotics, cefaclor and amoxicillin and performed several simula-
tion studies. Kuk(2004) proposed the power link and established its complete monotonicity.
They also applied their approach in clinical and developmental toxicity studies and achieved
superior results to other exiting approaches such as GEE, Beta-Binomial, etc. It is to our
thinking that further study on this approach is worthwhile, for example, providing enough
completely monotone links that is of practicable use in performing real statistical inference;
giving a procedure about finding a possible best model. In this article, we further their
idea and attempt to present a unified theory and give a wide variety of complete monotonic
links. We introduce a rich class of parsimonious binomial mixtures (completely monotonic
links) including the incomplete Beta-, Gamma-, Normal-, and Poisson- Binomial. These
mixtures are called incomplete because their probability mass functions contain incomplete
special functions. For example, the incomplete Beta-Binomial include the incomplete Beta
function. Since the Beta function is a special case of the incomplete Beta function, the pop-
ularly used Beta-Binomial is a special case of the incomplete Beta-Binomial. Further, the
incomplete Beta-Binomial includes the Binomial as a special case, unlike the Beta-Binomial
which does not include the Binomial as a special case in its parameter domain. Indeed all the
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other introduced incomplete mixtures contain the Binomial as a special case, while existing
binomial mixtures usually do not; for example, the random effects model for binary data
by Conaway(1990), the logistic regression by nonparametric mixing by Follmann and Lam-
bert(1989), and the mixtures resulted from random variables (see section 2). This inclusion
of the Binomial is useful, for instance, regression based on these incomplete mixtures will
boil down to the logistic regression if data is resulted from independent binary responses.
We further demonstrate that the class is closed under convex linear combination, product
and two types of composites. These properties enormously enlarge the class. The probability
mass functions of the mixtures are quite simple both mathematically and computationally.
The likelihoods can be computed without numerical integration and parameter estimation
can be obtained by maximum likelihood using the R or SAS packages. In addition, we
present the Markov property of the stochastic binomial process, moments and correlations
of the mixtures. We give a stepwise procedure about how a possible best model can be
arrived from such numerous mixtures. A simulation is conducted to validate the inclusion
of the Binomial. The proposed procedure is applied to the 2, 4, 5-T and the E2 data. Com-
parisons are made with the Binomial, the Correlated Binomial, the Beta-Binomial, GEE’s,
Williams’ logistic regression(1982), etc. Our results indicate that the proposed procedure
improves these models and, in particular, the Gamma-Binomials outperformed the other
models considered in regression on the 2, 4, 5-T data.
The proposed parsimonious mixtures unify the existing approaches. In addition to the
aforemention approaches, additional examples include the usual Binomial, the Correlated Bi-
nomial(Kupper and Haseman, 1978), the Additive Model(Altham, 1978), and the generalized
logistic regression by nonparametric mixing(Follmann and Lambert, 1989). The model by
Conaway(1990), interestingly, turns out to be a special case of our incomplete Gamma Power
Binomial. Brooks(1997) explored finite mixture models for proportions. These models can
be recovered from the linear combinations of the proposed class. From this standpoint, our
work generalizes Brooks’ result to conclude that the products and two types of composites of
the CM links(see section 2) are also closed, see section 4. Some additional related results are
as follows. Stephanescu and Turnbull(2003) used the EM algorithm to model exchangeable
binary data.Wood(1992) investigated binomial mixtures and finite exchangeability. Meester
and MacKey(1994) introduced a parametric model for cluster correlated categorical data.
Prentice explored the binary regression using extended Beta-Binomial in 1986 and the cor-
related binary regression in 1988. An early work on extra-binomial variation was Williams’
procedure in 1982.
The rest of the article is organized as follows. The parsimonious mixtures are introduced
in section 2, followed by the methods about obtaining such mixtures and the Markov prop-
3