Yuchung J. Wang

Stochastic Blockmodels for Directed Graphs

Abstract Holland and Leinhardt (1981) proposed the p 1 model for the analysis of binary directed graph data in network studies. Such a model provides information about the “attractiveness” and “expansiveness” of the individual nodes in the network, as well as the tendency of a pair of nodes to reciprocate relational ties. When the nodes are a priori partitioned into subgroups based on attributes such as race and sex, the density of ties from one subgroup to another can differ considerably from that relating another pair of subgroups, thus creating a situation called blocking in social networks. The p 1 model completely ignores this extra piece of information and is, therefore, unable to explain the block structure. Blockmodels that are simple extensions of the p 1 model are proposed specifically for such data. An iterative scaling algorithm is presented for fitting the model parameters by maximum likelihood. The methodology is illustrated in detail on two empirical examples.

Deterministic lot-size inventory models with shortages and deterioration for fluctuating demand

We establish various inventory replenishment policies. We then analytically identify the best alternative among them based on the minimum total relevant costs. Finally, we prove that the relevant cost is convex with the number of replenishments. Consequently, the search for the optimal replenishment number is reduced to finding a local minimum.

Canonical representation of conditionally specified multivariate discrete distributions

Most work on conditionally specified distributions has focused on approaches that operate on the probability space, and the constraints on the probability space often make the study of their properties challenging. We propose decomposing both the joint and conditional discrete distributions into characterizing sets of canonical interactions, and we prove that certain interactions of a joint distribution are shared with its conditional distributions. This invariance opens the door for checking the compatibility between conditional distributions involving the same set of variables. We formulate necessary and sufficient conditions for the existence and uniqueness of discrete conditional models, and we show how a joint distribution can be easily computed from the pool of interactions collected from the conditional distributions. Hence, the methods can be used to calculate the exact distribution of a Gibbs sampler. Furthermore, issues such as how near compatibility can be reconciled are also discussed. Using mixed parametrization, we show that the proposed approach is based on the canonical parameters, while the conventional approaches are based on the mean parameters. Our advantage is partly due to the invariance that holds only for the canonical parameters.

LOCALLY DEPENDENT LATENT TRAIT MODEL FOR POLYTOMOUS RESPONSES WITH APPLICATION TO INVENTORY OF HOSTILITY

Psychological tests often involve item clusters that are designed to solicit responses to behavioral stimuli. The dependency between individual responses within clusters beyond that which can be explained by the underlying trait sometimes reveals structures that are of substantive interest. The paper describes two general classes of models for this type of locally dependent responses. Specifically, the models include a generalized log-linear representation and a hybrid parameterization model for polytomous data. A compact matrix notation designed to succinctly represent the system of complex multivariate polytomous responses is presented. The matrix representation creates the necessary formulation for the locally dependent kernel for polytomous item responses. Using polytomous data from an inventory of hostility, we provide illustrations as to how the locally dependent models can be used in psychological measurement.

Designing Outer Array Points

Taguchis product-array design consists of two portions: an inner array containing the design factors and an outer array containing noise factors. The function of the outer array is very different from that of the inner array, however. The outer array i..

Gibbs ensembles for nearly compatible and incompatible conditional models

Gibbs sampler has been used exclusively for compatible conditionals that converge to a unique invariant joint distribution. However, conditional models are not always compatible. In this paper, a Gibbs sampling-based approach - Gibbs ensemble -is proposed to search for a joint distribution that deviates least from a prescribed set of conditional distributions. The algorithm can be easily scalable such that it can handle large data sets of high dimensionality. Using simulated data, we show that the proposed approach provides joint distributions that are less discrepant from the incompatible conditionals than those obtained by other methods discussed in the literature. The ensemble approach is also applied to a data set regarding geno-polymorphism and response to chemotherapy in patients with metastatic colorectal.

A simple algorithm for checking compatibility among discrete conditional distributions

A distribution is said to be conditionally specified when only its conditional distributions are known or available. The very first issue is always compatibility: does there exist a joint distribution capable of reproducing all of the conditional distributions? We review five methods-mostly for two or three variables-published since 2002, and we conclude that these methods are either mathematically too involved and/or are too difficult (and in many cases impossible) to generalize to a high dimension. The purpose of this paper is to propose a general algorithm that can efficiently verify compatibility in a straightforward fashion. Our method is intuitively simple and general enough to deal with any full-conditional specifications. Furthermore, we illustrate the phenomenon that two theoretically equivalent conditional models can be different in terms of compatibilities, or can result in different joint distributions. The implications of this phenomenon are also discussed.

Multivariate normal integrals and contingency tables with ordered categories

Ak-dimensional multivariate normal distribution is made discrete by partitioning thek-dimensional Euclidean space with rectangular grids. The collection of probability integrals over the partitioned cubes is ak-dimensional contingency table with ordered categories. It is shown that loglinear model with main effects plus two-way interactions provides an accurate approximation for thek-dimensional table. The complete multivariate normal integral table is computed via the iterative proportional fitting algorithm from bivariate normal integral tables. This approach imposes no restriction on the correlation matrix. Comparisons with other numerical integration algorithms are reported. The approximation suggests association models for discretized multivariate normal distributions and contingency tables with ordered categories.

A strategy for designing telescoping models for analyzing multiway contingency tables using mixed parameters

In the analysis of cross-classified data, sociologists often focus on building flexible models for the marginal distributions of a selected set of variables. One strategy for achieving flexible modeling is to design models for telescoping marginal distributions. As an illustration of telescoping distributions, consider a joint distribution of four crossclassifying variables: occupational attainment, education level, race, and gender. A set of telescoping distributions would be the univariate occupation distribution, the bivariate occupation by education, the trivariate occupation by education by race, and the entire joint distribution. A methodology that enables the telescoping modeling strategy is mixed parameterization, which has its roots in the statistics and sociology literatures. In this article, the authors develop a scheme of multilevel mixed parameterization that can be applied to a hierarchy of marginal distributions of reducing dimension. An example from the General Social Survey illustrates mixed parameters for telescoping models.

Compatibility of discrete conditional distributions with structural zeros

A general algorithm is provided for determining the compatibility among full conditionals of discrete random variables with structural zeros. The algorithm is scalable and it can be implemented in a fairly straightforward manner. A MATLAB program is included in the Appendix and therefore, it is now feasible to check the compatibility of multi-dimensional conditional distributions with constrained supports. Rather than the linear equations in the restricted domain of Arnold et al. (2002) [11] Tian et al. (2009) [16], the approach is odds-oriented and it is a discrete adaptation of the compatibility check of Besag (1994) [17]. The method naturally leads to the calculation of a compatible joint distribution or, in the absence of compatibility, a nearly compatible joint distribution. Besags [5] factorization of a joint density in terms of conditional densities is used to justify the algorithm.