Conditional estimation of exponential random graph models from snowball sampling designs

A complete survey of a network in a large population may be prohibitively difficult and costly. So it is important to estimate models for networks using data from various network sampling designs, such as link-tracing designs. We focus here on snowball sampling designs, designs in which the members of an initial sample of network members are asked to nominate their network partners, their network partners are then traced and asked to nominate their network partners, and so on. We assume an exponential random graph model (ERGM) of a particular parametric form and outline a conditional maximum likelihood estimation procedure for obtaining estimates of ERGM parameters. This procedure is intended to complement the likelihood approach developed by  Handcock and Gile (2010) by providing a practical means of estimation when the size of the complete network is unknown and/or the complete network is very large. We report the outcome of a simulation study with a known model designed to assess the impact of initial sample size, population size, and number of sampling waves on properties of the estimates. We conclude with a discussion of the potential applications and further developments of the approach.     

Operating on Functions with Variable Domains

The sum, difference, product and quotient of two functions with different domains are usually defined only on their common domain. This paper extends these definitions so that the sum and other operations are essentially defined anywhere that at least one of the components is defined. This idea is applied to propositions and events, expressed as indicator functions, to define conditional propositions and conditional events as three-valued indicator functions that are undefined when their condition is false. Extended operations of and, or, not and conditioning are then defined on these conditional events with variable conditions. The probabilities of the disjunction (or) and of the conjunction (and) of two conditionals are expressed in terms of the conditional probabilities of the component conditionals. In a special case, these are shown to be weighted averages of the component conditional probabilities where the weights are the relative probabilities of the various conditions. Next, conditional random variables are defined to be random variables X whose domain has been restricted by a condition on a second random variable Y. The extended sum, difference, product and conditioning operations on functions are then applied to these conditional random variables. The expectation of a random variable and the conditional expectation of a conditional random variable are recounted. Theorem 1 generalizes the standard result that the conditional expectation of the sum of two conditional random variables with disjoint and exhaustive conditions is a weighted sum of the conditional expectations of the component conditional random variables. Because of the extended operations, the theorem is true for arbitrary conditions. Theorem 2 gives a formula for the expectation of the product of two conditional random variables. After the definition of independence of two random variables is extended to accommodate the extended operations, it is applied to the formula of Theorem 2 to simplify the expectation of a product of conditional random variables. Two examples end the paper. The first concerns a work force of n workers of different output levels and work shifts. The second example involves two radars with overlapping surveillance regions and different detection error rates. One radar's error rate is assumed to be sensitive to fog and the other radar's error rate is assumed to be sensitive to air traffic density. The combined error rate over the combined surveillance region given heavy fog and moderate air traffic is computed.     

Thurstonian modeling of ranking data via mean and covariance structure analysis

Although Thurstonian models provide an attractive representation of choice behavior, they have not been extensively used in ranking applications since only recently efficient estimation methods for these models have been developed. These, however, require the use of special-purpose estimation programs, which limits their applicability. Here we introduce a formulation of Thurstonian ranking models that turns an idiosyncratic estimation problem into an estimation problem involving mean and covariance structures with dichotomous indicators. Well-known standard solutions for the latter can be readily applied to this specific problem, and as a result any Thurstonian model for ranking data can be fitted using existing general purpose software for mean and covariance structure analysis. Although the most popular programs for covariance structure analysis (e.g., LISREL and EQS) cannot be presently used to estimate Thurstonian ranking models, other programs such as MECOSA already exist that can be straightforwardly used to estimate these models.This paper is based on the author's doctoral dissertation. Ulf Böckenholt was my advisor. The author is indebted to Ulf Böckenholt for his comments on a previous version of this paper and to Gerhard Arminger for his extensive support on the use of MECOSA. The final stages of this research took place while the author was at the Department of Statistics and Econometrics, Universidad Carlos III de Madrid. Conversations with my colleague there, Adolfo Hernández, helped to greatly improve this paper.     

Monte Carlo studies in nonmetric scaling

In response to Arabie several random ranking studies are compared and discussed. Differences are typically very small, however it is noted that those studies which used arbitrary configurations tend to produce slightly higher stress values. The choice of starting configuration is discussed and we suggest that the use of a principal components decomposition of the doubly centered matrix of dissimilarities, or some transformation thereof, will yield an initial configuration which is superior to a randomly chosen one.This research was supported by the National Research Council of Canada (Grant No. A8351) and by the National Institute of Mental Health (Grant Nos. MH10006 and MH26504). The authorship order has been determined by Monte Carlo methods.     

A modified beta binomial model with applications to multiple choice and taste tests

A modified beta binomial model is presented for use in analyzing ramdom guessing multiple choice tests and certain forms of taste tests. Detection probabilities for each item are distributed beta across the population subjects. Properties for the observable distribution of correct responses are derived. Two concepts of true score estimates are presented. One, analogous to Duncan's empirical Bayes posterior mean score, is appropriate for assessing the subject's performance on that particular test. The second is more suitable for predicting outcomes on similar tests.This research was made possible by a grant from the Center for Food Policy Research, Graduate School of Business, Columbia University.