Lattice Conditional Independence Models Research Articles

Lattice conditional independence models and Hibi ideals

Lattice Conditional Independence models are a class of models developed first for the Gaussian case in which a distributive lattice classifies all the conditional independence statements. The main result is that these models can equivalently be described via a transitive directed acyclic graph (TDAG) in which, as is normal for causal models, the conditional independence is in terms of conditioning on ancestors in the graph. We demonstrate that a parallel stream of research in algebra, the theory of Hibi ideals, not only maps directly to the LCI models but gives a vehicle to generalise the theory from the linear Gaussian case. Given a distributive lattice (i) each conditional independence statement is associated with a Hibi relation defined on the lattice, (ii) the directed graph is given by chains in the lattice which correspond to chains of conditional independence, (iii) the elimination ideal of product terms in the chains gives the Hibi ideal and (iv) the TDAG can be recovered from a special bipartite graph constructed via the Alexander dual of the Hibi ideal. It is briefly demonstrated that there are natural applications to statistical log-linear models, time series, and Shannon information flow.

Inadmissibility of the Maximum Likekihood Estimator of Normal Covariance Matrices with the Lattice Conditional Independence

Lattice conditional independence (LCI) models introduced by S. A. Andersson and M. D. Perlman (1993, Ann. Statist.21, 1318–1358) have the pleasant feature of admitting explicit maximum likelihood estimators and likelihood ratio test statistics. This is because the likelihood function and parameter space for a LCI model can be factored into products of conditional likelihood functions and parameter spaces, where the standard multivariate techniques can be applied. In this paper we consider the problem of estimating the covariance matrices under LCI restriction in a decision theoretic setup. The Stein loss function is used in this study and, using the factorization mentioned above, minimax estimators are obtained. Since the maximum likelihood estimator has constant risk and is different from the minimax estimator, this shows that the maximum likelihood estimator under LCI restriction inadmissible. These results extend those obtained by W. James and C. Stein (1960, in “Proceedings of the Fourth Berkeley Symposium on Mathematics, Statistics, and Probability,” Vol. 1, pp. 360–380, Univ. of California Press, Berkeley, CA) and D. K. Dey and C. Srinivasan (1985, Ann. Statist.13, 1581–1591) for estimating normal covariance matrices to the LCI models.

Testing lattice conditional independence models based on monotone missing data

Lattice conditional independence (LCI) models (Anderson and Perlman, 1991. Statist. Probab. Lett. 12, 465–486; 1993 Ann. Statist. 21, 1318–1358) can be applied to the analysis of missing data problems with non-monotone missing patterns. Closed-form maximum likelihood estimates can always be obtained under the LCI models naturally determined by the observed data patterns. In practice, it is important to test the appropriateness of LCI models. In the present paper, we derive explicit likelihood ratio tests for testing LCI models based on a monotone subset of the observed data.

Some model-restricted shrinkage estimators for contingency tables with missing data

For contingency tables with extensive missing data, the unrestricted MLE under the saturated model, computed by the EM algorithm, is generally unsatisfactory. In this case, it may be better to fit a simpler model by imposing some restrictions on the parameter space. Perlman and Wu (1999) propose lattice conditional independence (LCI) models for contingency tables with arbitrary missing data patterns. When this LCI model fits well, the restricted MLE under the LCI model is more accurate than the unrestricted MLE under the saturated model, but not in general. Here we propose certain empirical Bayes (EB) estimators that adaptively combine the best features of the restricted and unrestricted MLEs. These EB estimators appear to be especially useful when the observed data is sparse, even in cases where the suitability of the LCI model is uncertain. We also study a restricted EM algorithm (called the ER algorithm) with similar desirable features.

Efficiency of lattice conditional independence models for multinormal samples with non-monotone missing data

For multivariate normal data with non-monotone (i.e. arbitrary) missing data patterns, lattice conditional independence (LCI) models determined by the observed data patterns can be used to obtain closed-form MLEs (Andersson and Perlman, 1991, 1993). In this paper, three procedures — LCI models, the EM algorithm, and the complete-data method — are compared by means of a Monte Carlo experiment. When the LCI model is accepted by the LR test, the LCI estimate is more efficient than those based on the EM algorithm and the complete-data method. When the LCI model is not accepted, the LCI estimate may lose efficiency but still may be more efficient than the EM estimate if the observed data is sparse. When the LCI model appears too restrictive, it may be possible to obtain a less restrictive LCI model by.discarding only a small portion of the incomplete observations. LCI models appear to be especially useful when the observed data is sparse, even in cases where the suitability of the LCI model is uncertain.

Lattice conditional independence models for seemingly unrelated regressions

Seemingly unrelated regressions (SUR) models appear frequently in econometrics and in the analyses of repeated measures designs and longitudinal data. It is known that iterative algorithms are generally required to obtain the MLEs of the regression parameters. Under a minimal set of lattice conditional independence (LCI) restrictions imposed on the covariance structure, however, closed-form MLEs can be obtained by standard linear regression techniques (Andersson and Perlman, 1993, 1994, 1998). In this paper, simulation is used to study the efficiency of these LCI model-based estimators. We also propose two possible improvements of the usual two-stage estimators for the regression parameters.

Lattice conditional independence models for contingency tables with non-monotone missing data patterns

In the analysis of non-monotone missing data patterns in multinomial distributions for contingency tables, it is known that explicit MLEs of the unknown parameters cannot be obtained. Iterative procedures such as the EM-algorithm are therefore required to obtain the MLEs. These iterative procedures, however, may offer several potential difficulties. Andersson and Perlman [Ann. Statist. 21 (1993) 1318–1358] introduced lattice conditional independence (LCI) models for multivariate normal distributions, which can be applied to the analysis of non-monotone missing observations in continuous data (Andersson and Perlman, Statist. Probab. Lett. 12 (1991) 465–486). In this paper, we show that LCI models may also be applied to the analysis of categorical data with non-monotone missing data patterns. Under a parsimonious set of LCI assumptions naturally determined by the observed data pattern, the likelihood function for the observed data can be factored as in the monotone case and explicit MLEs can be obtained for the unknown parameters. Furthermore, the LCI assumptions can be tested by explicit likelihood ratio tests.

Estimation and testing for lattice conditional independence models on Euclidean Jordan algebras

In this paper we generalize the major results of Andersson and Perlman on LCI models to the setting of symmetric cones and give an explicit closed form formula for the estimate of the covariance matrix in the generalized LCI models that we define. To this end, we replace the cone $H_I^+(\mathbb{R})$ sitting inside the Jordan algebra of symmetric real $I \times I$-matrices by the symmetric cone $\Omega$ of an Euclidean Jordan algebra $V$. We introduce the Andersson-Perlman cone $\Omega(\mathscr{K}\subseteq\Omega$ which generalizes $\mathscr{P}(\mathscr{K})\subseteq H_I^+(\mathscr{R})$. We prove several characterizations and properties of $\Omega(\mathscr{K})$ which allows us to recover, though with different proofs, the main results of Andersson and Perlman regarding $\mathscr{P}(\mathscr{K})$. The new lattice conditional independence models are defined, assuming that the Euclidean Jordan algebra $V$ has a symmetric representation. Using standard results from the theory of Jordan algebras, we can reduce the general model to the case where $V$ is the Jordan algebra of Hermitian matrices over the real, complex or quaternionic numbers, and $\Omega$ is the corresponding cone of positive-definite matrices. Our main statistical result is a closed-form formula for the estimate of the covariance matrix in the generalized LCI model. We also give the likelihood ratio test for testing a given model versus another one, nested within the first.

A graphical characterization of lattice conditional independence models

Lattice conditional independence (LCI) models for multivariate normal data recently have been introduced for the analysis of nondmonotone missing data patterns and of nonnested dependent linear regression models (\equiv seemingly unrelated regressions). It is shown here that the class of LCI models coincides with a subclass of the class of graphical Markov models determined by acyclic digraphs (ADGs), namely, the subclass of transitive ADG models. An explicit graphdtheoretic characterization of those ADGs that are Markov equivalent to some transitive ADG is obtained. This characterization allows one to determine whether a specific ADG D is Markov equivalent to some transitive ADG, hence to some LCI model, in polynomial time, without an exhaustive search of the (possibly superexponentially large) equivalence class [D]. These results do not require the existence or positivity of joint densities.

Testing Lattice Conditional Independence Models

The lattice conditional independence (LCI) model N( K ) is defined to be the set of all normal distributions N(0, Σ) on R I such that for every pair L, M ∈ K , x L and x M are conditionally independent given x L ∩ M . Here K is a ring of subsets (hence a distributive lattice) of the finite index set I such that ∅ I ∈ K , while for K ∈ K , x K is the coordinate projection of x ∈ R I onto R K . These LCI models have especially tractable statistical properties and arise naturally in the analysis of non-monotone multivariate missing data patterns and non-nested dependent linear regression models ≡ seemingly unrelated regressions. The present paper treats the problem of testing one LCI model against another, i.e., testing N( K ) vs N( M ) when M is a subring of K . The likelihood ratio test statistic is derived, together with its central distribution, and several examples are presented.

Unbiasedness of the Likelihood Ratio Test for Lattice Conditional Independence Models

The lattice conditional independence (LCI) model N( K ) is defined to be the set of all normal distributions N(0, Σ) on R I such that for every pair L, M ∈ K , x L and x M are conditionally independent given x L ∩ M . Here K is a ring of subsets (hence a distributive lattice) of the finite index set I such that ∅ I ∈ K , while for K ∈ K , x K is the coordinate projection of x ∈ R I onto R K . Andersson and Perlman in the preceding paper derived the likelihood ratio (LR) statistic λ for testing one LCI model against another, i.e., for testing N( K ) vs N( M ) based on a random sample from N(0, Σ), where M is a subring of K . In the present paper the strict unbiasedness of the LR test is established, and related results regarding the distribution of the maximum likelihood estimator of Σ under the LCI model N( K ) are presented.

Lattice Models for Conditional Independence in a Multivariate Normal Distribution

The lattice conditional independence model $\mathbf{N}(\mathscr{K})$ is defined to be the set of all normal distributions on $\mathbb{R}^I$ such that for every pair $L, M \in \mathscr{K}, x_L$ and $x_M$ are conditionally independent given $x_{L \cap M}$. Here $\mathscr{K}$ is a ring of subsets of the finite index set $I$ and, for $K \in \mathscr{K}, x_K$ is the coordinate projection of $x \in \mathbb{R}^I$ onto $\mathbb{R}^K$. Statistical properties of $\mathbf{N}(\mathscr{K})$ may be studied, for example, maximum likelihood inference, invariance and the problem of testing $H_0: \mathbf{N}(\mathscr{K})$ vs. $H: \mathbf{N}(\mathscr{M})$ when $\mathscr{M}$ is a subring of $\mathscr{K}$. The set $J(\mathscr{K})$ of join-irreducible elements of $\mathscr{K}$ plays a central role in the analysis of $\mathbf{N}(\mathscr{K})$. This class of statistical models occurs in the analysis of nonnested multivariate missing data patterns and nonnested dependent linear regression models.