Association Models for a Multivariate Binary Response

Abstract

Models for a multivariate binary response are parameterized by univariate marginal probabilities and dependence ratios of all orders. The w-order dependence ratio is the joint success probability of w binary responses divided by the joint success probability assuming independence. This parameterization supports likelihood-based inference for both regression parameters, relating marginal probabilities to explanatory variables, and association model parameters, relating dependence ratios to simple and meaningful mechanisms. Five types of association models are proposed, where responses are (1) independent given a necessary factor for the possibility of a success, (2) independent given a latent binary factor, (3) independent given a latent beta distributed variable, (4) follow a Markov chain, and (5) follow one of two first-order Markov chains depending on the realization of a binary latent factor. These models are illustrated by reanalyzing three data sets, foremost a set of binary time series on auranofin therapy against arthritis. Likelihood-based approaches are contrasted with approaches based on generalized estimating equations. Association models specified by dependence ratios are contrasted with other models for a multivariate binary response that are specified by odds ratios or correlation coefficients.

ORTH: R and SAS software for regression models of correlated binary data based on orthogonalized residuals and alternating logistic regressions

“…Our first example is a medical practice data set analyzed using GEE1 by Preisser and Qaqish (1996). Our second example is based on data analyzed by Fitzmaurice and Lipsitz (1995 , Table 2) and later re-analyzed by Ekholm et al (2000) using a different approach.…”

Multi-Market Direction-of-Change Modeling Using Dependence Ratios

“…The expectation of the product at the end of this formula has the form i 1 i 2 ::: i` i 1 i 2 :::if or an appropriate combination i 1 ; i 2 ; :::; i`: For example, when m = 3; there are three dependence ratios of order 2, 12 ; 13 ; 23 ; and one dependence ratio of order 3; 123 ; the mapping looks as follows: Now, let previously considered variables additionally indexed by t correspond to observation t (t = 1; 2; :::; T ), containing the vector of returns x t = (x 1;t ; x 2;t ; ::; x m;t ) 0 and associated indicator vector I t = (I 1;t ; I 2;t ; ::; I m;t ) 0 : Note that Ekholm, Smith, and McDonald (1995) and Ekholm, McDonald and Smith (2000) parameterize the marginal probabilities as some functions of exogenous variables characterizing corresponding IID units. In our context, "units" are time periods having a di¤erent nature, so we keep the marginal probabilities constant for now and postpone parameterizations to the next section.…”

“…As pointed out in Ekholm, Smith, and McDonald (1995) and Ekholm, McDonald and Smith (2000), using the dependence ratios has several advantages over using other association structures such as conditional odds ratios frequently used in statistical processing of binary data (for example, in Fitzmaurice and Laird, 1993).…”

“…Another one is a hypothesis of horizontal homogeneity which assumes equal dependence within all subsets of markets of equal size. For example, when m = 3; this implies two restrictions 12 = 23 = 13 ; when m = 4; this implies eight restrictions More intricate hypotheses implying existence of latent factors underlying the joint distribution of indicators are described in Ekholm, McDonald and Smith (2000). One of their association models, called "a latent binary factor", seems to be relevant to …nancial markets.…”

“…As noted in the previous section, in Ekholm, Smith, and McDonald (1995) and Ekholm, McDonald and Smith (2000) marginal probabilities and dependence ratios are handled separately. In particular, the authors suggest setting each marginal probability to some function (such as logistic) of associated covariates.…”