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It is increasingly common to collect pre-post data with pseudonyms or self-constructed identifiers. On survey responses from sensitive populations, identifiers may be made optional to encourage higher response rates. The ability to match responses between pre- and post-intervention phases for every participant may be impossible in such applications, leaving practitioners with a choice between the paired t-test on the matched samples and the two-sample t-test on all samples for evaluating mean differences. We demonstrate the inadequacies with both approaches, as the former test requires discarding unmatched data, while the latter test ignores correlation and assumes independence. In cases with a subset of matched samples, an opportunity to achieve limited inference about the correlation exists. We propose a novel technique for such `partially matched' data, which we refer to as the Quantile-based t-test for correlated samples, to assess mean differences using a conservative estimate of the correlation between responses based on the matched subset. Critically, our approach does not discard unmatched samples, nor does it assume independence. Our results demonstrate that the proposed method yields nominal Type I error probability while affording more power than existing approaches. Practitioners can readily adopt our approach with basic statistical programming software., Comment: 17 pages, 3 figures

The COVID-19 infection fatality rate (IFR) is the proportion of individuals infected with SARS-CoV-2 who subsequently die. As COVID-19 disproportionately affects older individuals, age-specific IFR estimates are imperative to facilitate comparisons of the impact of COVID-19 between locations and prioritize distribution of scare resources. However, there lacks a coherent method to synthesize available data to create estimates of IFR and seroprevalence that vary continuously with age and adequately reflect uncertainties inherent in the underlying data. In this paper we introduce a novel Bayesian hierarchical model to estimate IFR as a continuous function of age that acknowledges heterogeneity in population age structure across locations and accounts for uncertainty in the estimates due to seroprevalence sampling variability and the imperfect serology test assays. Our approach simultaneously models test assay characteristic, serology, and death data, where the serology and death data are often available only for binned age groups. Information is shared across locations through hierarchical modeling to improve estimation of the parameters with limited data. Modeling data from 26 developing country locations during the first year of the COVID-19 pandemic, we found seroprevalence did not change dramatically with age, and the IFR at age 60 was above the high-income country benchmark for most locations.

In this manuscript we consider random objects being measured in multiple metric spaces, which may arise when those objects may be measured in multiple distinct ways. In this new multivariate setting, we define a Fr\'echet covariance and Fr\'echet correlation in two metric spaces, and a Fr\'echet covariance matrix and Fr\'echet correlation matrix in an arbitrary number of metric spaces. We prove consistency for the sample Fr\'echet covariance, and propose several tests to compare the means and covariance matrices between two or more groups. Lastly, we investigate the power and Type I error of each test under a variety of scenarios.

Community discovery is the general process of attaining assortative communities from a network: collections of nodes that are densely connected within yet sparsely connected to the rest of the network. While community discovery has been well studied, few such techniques exist for heterogeneous networks, which contain different types of nodes and possibly different connectivity patterns between the node types. In this paper, we introduce a framework called ECoHeN, which \textbf{e}xtracts \textbf{co}mmunities from a \textbf{he}terogeneous \textbf{n}etwork in a statistically meaningful way. Using a heterogeneous configuration model as a reference distribution, ECoHeN identifies communities that are significantly more densely connected than expected given the node types and connectivity of its membership. Specifically, the ECoHeN algorithm extracts communities one at a time through a dynamic set of iterative updating rules, is guaranteed to converge, and imposes no constraints on the type composition of extracted communities. To our knowledge this is the first discovery method that distinguishes and identifies both homogeneous and heterogeneous, possibly overlapping, community structure in a network. We demonstrate the performance of ECoHeN through simulation and in application to a political blogs network to identify collections of blogs which reference one another more than expected considering the ideology of its' members., Comment: 31 pages, 8 figures, 1 table, 1 algorithm

Network regression models, where the outcome comprises the valued edge in a network and the predictors are actor or dyad-level covariates, are used extensively in the social and biological sciences. Valid inference relies on accurately modeling the residual dependencies among the relations. Frequently homogeneity assumptions are placed on the errors which are commonly incorrect and ignore critical, natural clustering of the actors. In this work, we present a novel regression modeling framework that models the errors as resulting from a community-based dependence structure and exploits the subsequent exchangeability properties of the error distribution to obtain parsimonious standard errors for regression parameters.

The configuration model is a standard tool for uniformly generating random graphs with a specified degree sequence, and is often used as a null model to evaluate how much of an observed network's structure can be explained by its degree structure alone. A Markov chain Monte Carlo (MCMC) algorithm, based on a degree-preserving double-edge swap, provides an asymptotic solution to sample from the configuration model. However, accurately and efficiently detecting this Markov chain's convergence on its stationary distribution remains an unsolved problem. Here, we provide a solution to detect convergence and sample from the configuration model. We develop an algorithm, based on the assortativity of the sampled graphs, for estimating the gap between effectively independent MCMC states, and a computationally efficient gap-estimation heuristic derived from analyzing a corpus of 509 empirical networks. We provide a convergence detection method based on the Dickey-Fuller Generalized Least Squares test, which we show is more accurate and efficient than three alternative Markov chain convergence tests., Comment: Same as version v3 but with corrected white spaces between paragraphs

Data sets in the form of binary matrices are ubiquitous across scientific domains, and researchers are often interested in identifying and quantifying noteworthy structure. One approach is to compare the observed data to that which might be obtained under a null model. Here we consider sampling from the space of binary matrices which satisfy a set of marginal row and column sums. Whereas existing sampling methods have focused on uniform sampling from this space, we introduce modified versions of two elementwise swapping algorithms which sample according to a non-uniform probability distribution defined by a weight matrix, which gives the relative probability of a one for each entry. We demonstrate that values of zero in the weight matrix, i.e. structural zeros, are generally problematic for swapping algorithms, except when they have special monotonic structure. We explore the properties of our algorithms through simulation studies, and illustrate the potential impact of employing a non-uniform null model using a classic bird habitation dataset.

Undirected, binary network data consist of indicators of symmetric relations between pairs of actors. Regression models of such data allow for the estimation of effects of exogenous covariates on the network and for prediction of unobserved data. Ideally, estimators of the regression parameters should account for the inherent dependencies among relations in the network that involve the same actor. To account for such dependencies, researchers have developed a host of latent variable network models, however, estimation of many latent variable network models is computationally onerous and which model is best to base inference upon may not be clear. We propose the Probit Exchangeable (PX) model for undirected binary network data that is based on an assumption of exchangeability, which is common to many of the latent variable network models in the literature. The PX model can represent the first two moments of any exchangeable network model. We leverage the EM algorithm to obtain an approximate maximum likelihood estimator of the PX model that is extremely computationally efficient. Using simulation studies, we demonstrate the improvement in estimation of regression coefficients of the proposed model over existing latent variable network models. In an analysis of purchases of politically-aligned books, we demonstrate political polarization in purchase behavior and show that the proposed estimator significantly reduces runtime relative to estimators of latent variable network models, while maintaining predictive performance.

Longitudinal bipartite relational data characterize the evolution of relations between pairs of actors, where actors are of two distinct types and relations exist only between disparate types. A common goal is to understand the temporal dependencies, specifically which actor relations incite later actor relations. There are two existing approaches to this problem. The first approach projects the bipartite data in each time period to a unipartite network and uses existing unipartite network models. Unfortunately, information is lost in calculating the projection and generative models for networks obtained through this process are scarce. The second approach represents dependencies using two unipartite \emph{influence networks}, corresponding to the two actor types. Existing models taking this approach are bilinear in the influence networks, creating challenges in computation and interpretation. We propose a novel generative model that permits estimation of weighted, directed influence networks and does not suffer from these shortcomings. The proposed model is linear in the influence networks, permitting inference using off-the-shelf software tools. We prove our estimator is consistent under cases of model misspecification and nearly asymptotically equivalent to the bilinear estimator. We demonstrate the performance of the proposed model in simulation studies and an analysis of weekly international state interactions.

D\'ail \'Eireann is the principal chamber of the Irish parliament. The 31st D\'ail \'Eireann is the principal chamber of the Irish parliament. The 31st D\'ail was in session from March 11th, 2011 to February 6th, 2016. Many of the members of the D\'ail were active on social media and many were Twitter users who followed other members of the D\'ail. The pattern of following amongst these politicians provides insights into political alignment within the D\'ail. We propose a new model, called the generalized latent space stochastic blockmodel, which extends and generalizes both the latent space model and the stochastic blockmodel to study social media connections between members of the D\'ail. The probability of an edge between two nodes in a network depends on their respective class labels as well as latent positions in an unobserved latent space. The proposed model is capable of representing transitivity, clustering, as well as disassortative mixing. A Bayesian method with Markov chain Monte Carlo sampling is proposed for estimation of model parameters. Model selection is performed using the WAIC criterion and models of different number of classes or dimensions of latent space are compared. We use the model to study Twitter following relationships of members of the D\'ail and interpret structure found in these relationships. We find that the following relationships amongst politicians is mainly driven by past and present political party membership. We also find that the modeling outputs are informative when studying voting within the D\'ail., Comment: 31 pages, 9 figures

Relational arrays represent measures of association between pairs of actors, often in varied contexts or over time. Trade flows between countries, financial transactions between individuals, contact frequencies between school children in classrooms, and dynamic protein-protein interactions are all examples of relational arrays. Elements of a relational array are often modeled as a linear function of observable covariates. Uncertainty estimates for regression coefficient estimators -- and ideally the coefficient estimators themselves -- must account for dependence between elements of the array (e.g. relations involving the same actor) and existing estimators of standard errors that recognize such relational dependence rely on estimating extremely complex, heterogeneous structure across actors. This paper develops a new class of parsimonious coefficient and standard error estimators for regressions of relational arrays. We leverage an exchangeability assumption to derive standard error estimators that pool information across actors and are substantially more accurate than existing estimators in a variety of settings. This exchangeability assumption is pervasive in network and array models in the statistics literature, but not previously considered when adjusting for dependence in a regression setting with relational data. We demonstrate improvements in inference theoretically, via a simulation study, and by analysis of a data set involving international trade., Comment: Accepted at Biometrika

Relational event data, which consist of events involving pairs of actors over time, are now commonly available at the finest of temporal resolutions. Existing continuous-time methods for modeling such data are based on point processes and directly model interaction "contagion," whereby one interaction increases the propensity of future interactions among actors, often as dictated by some latent variable structure. In this article, we present an alternative approach to using temporal-relational point process models for continuous-time event data. We characterize interactions between a pair of actors as either spurious or that resulting from an underlying, persistent connection in a latent social network. We argue that consistent deviations from expected behavior, rather than solely high frequency counts, are crucial for identifying well-established underlying social relationships. This study aims to explore these latent network structures in two contexts: one comprising of college students and another involving barn swallows.

Many existing statistical and machine learning tools for social network analysis focus on a single level of analysis. Methods designed for clustering optimize a global partition of the graph, whereas projection based approaches (e.g. the latent space model in the statistics literature) represent in rich detail the roles of individuals. Many pertinent questions in sociology and economics, however, span multiple scales of analysis. Further, many questions involve comparisons across disconnected graphs that will, inevitably be of different sizes, either due to missing data or the inherent heterogeneity in real-world networks. We propose a class of network models that represent network structure on multiple scales and facilitate comparison across graphs with different numbers of individuals. These models differentially invest modeling effort within subgraphs of high density, often termed communities, while maintaining a parsimonious structure between said subgraphs. We show that our model class is projective, highlighting an ongoing discussion in the social network modeling literature on the dependence of inference paradigms on the size of the observed graph. We illustrate the utility of our method using data on household relations from Karnataka, India.

Random graph null models have found widespread application in diverse research communities analyzing network datasets, including social, information, and economic networks, as well as food webs, protein-protein interactions, and neuronal networks. The most popular family of random graph null models, called configuration models, are defined as uniform distributions over a space of graphs with a fixed degree sequence. Commonly, properties of an empirical network are compared to properties of an ensemble of graphs from a configuration model in order to quantify whether empirical network properties are meaningful or whether they are instead a common consequence of the particular degree sequence. In this work we study the subtle but important decisions underlying the specification of a configuration model, and investigate the role these choices play in graph sampling procedures and a suite of applications. We place particular emphasis on the importance of specifying the appropriate graph labeling (stub-labeled or vertex-labeled) under which to consider a null model, a choice that closely connects the study of random graphs to the study of random contingency tables. We show that the choice of graph labeling is inconsequential for studies of simple graphs, but can have a significant impact on analyses of multigraphs or graphs with self-loops. The importance of these choices is demonstrated through a series of three vignettes, analyzing network datasets under many different configuration models and observing substantial differences in study conclusions under different models. We argue that in each case, only one of the possible configuration models is appropriate. While our work focuses on undirected static networks, it aims to guide the study of directed networks, dynamic networks, and all other network contexts that are suitably studied through the lens of random graph null models., Comment: To appear in SIAM Review, June 2018. Code available at github.com/joelnish/double-edge-swap-mcmc. v3 fixed minor typos

Network modeling techniques provide a means for quantifying social structure in populations of individuals. Data used to define social connectivity are often expensive to collect and based on case-specific, ad hoc criteria. Moreover, in applications involving animal social networks, collection of these data is often opportunistic and can be invasive. Frequently, the social network of interest for a given population is closely related to the way individuals move. Thus telemetry data, which are minimally-invasive and relatively inexpensive to collect, present an alternative source of information. We develop a framework for using telemetry data to infer social relationships among animals. To achieve this, we propose a Bayesian hierarchical model with an underlying dynamic social network controlling movement of individuals via two mechanisms: an attractive effect, and an aligning effect. We demonstrate the model and its ability to accurately identify complex social behavior in simulation, and apply our model to telemetry data arising from killer whales. Using auxiliary information about the study population, we investigate model validity and find the inferred dynamic social network is consistent with killer whale ecology and expert knowledge.

In data fusion analysts seek to combine information from two databases comprised of disjoint sets of individuals, in which some variables appear in both databases and other variables appear in only one database. Most data fusion techniques rely on variants of conditional independence assumptions. When inappropriate, these assumptions can result in unreliable inferences. We propose a data fusion technique that allows analysts to easily incorporate auxiliary information on the dependence structure of variables not observed jointly; we refer to this auxiliary information as glue. With this technique, we fuse two marketing surveys from the book publisher HarperCollins using glue from the online, rapid-response polling company CivicScience. The fused data enable estimation of associations between people's preferences for authors and for learning about new books. The analysis also serves as a case study on the potential for using online surveys to aid data fusion.

Network analysis is often focused on characterizing the dependencies between network relations and node-level attributes. Potential relationships are typically explored by modeling the network as a function of the nodal attributes or by modeling the attributes as a function of the network. These methods require specification of the exact nature of the association between the network and attributes, reduce the network data to a small number of summary statistics, and are unable provide predictions simultaneously for missing attribute and network information. Existing methods that model the attributes and network jointly also assume the data are fully observed. In this article we introduce a unified approach to analysis that addresses these shortcomings. We use a latent variable model to obtain a low dimensional representation of the network in terms of node-specific network factors and use a test of dependence between the network factors and attributes as a surrogate for a test of dependence between the network and attributes. We propose a formal testing procedure to determine if dependencies exists between the network factors and attributes. We also introduce a joint model for the network and attributes, for use if the test rejects, that can capture a variety of dependence patterns and be used to make inference and predictions for missing observations.

The United Nations (UN) Population Division is considering producing probabilistic projections for the total fertility rate (TFR) using the Bayesian hierarchical model of Alkema et al. (2011), which produces predictive distributions of TFR for individual countries. The UN is interested in publishing probabilistic projections for aggregates of countries, such as regions and trading blocs. This requires joint probabilistic projections of future country-specific TFRs, taking account of the correlations between them. We propose an extension of the Bayesian hierarchical model that allows for probabilistic projection of TFR for any set of countries. We model the correlation between country forecast errors as a linear function of time invariant covariates, namely whether the countries are contiguous, whether they had a common colonizer after 1945, and whether they are in the same UN region. The resulting correlation model is incorporated into the Bayesian hierarchical model's error distribution. We produce predictive distributions of TFR for 1990-2010 for each of the UN's primary regions. We find that the proportions of the observed values that fall within the prediction intervals from our method are closer to their nominal levels than those produced by the current model. Our results suggest that a significant proportion of the correlation between forecast errors for TFR in different countries is due to countries' geographic proximity to one another, and that if this correlation is accounted for, the quality of probabilitistic projections of TFR for regions and other aggregates is improved.

Human mortality data sets can be expressed as multiway data arrays, the dimensions of which correspond to categories by which mortality rates are reported, such as age, sex, country and year. Regression models for such data typically assume an independent error distribution or an error model that allows for dependence along at most one or two dimensions of the data array. However, failing to account for other dependencies can lead to inefficient estimates of regression parameters, inaccurate standard errors and poor predictions. An alternative to assuming independent errors is to allow for dependence along each dimension of the array using a separable covariance model. However, the number of parameters in this model increases rapidly with the dimensions of the array and, for many arrays, maximum likelihood estimates of the covariance parameters do not exist. In this paper, we propose a submodel of the separable covariance model that estimates the covariance matrix for each dimension as having factor analytic structure. This model can be viewed as an extension of factor analysis to array-valued data, as it uses a factor model to estimate the covariance along each dimension of the array. We discuss properties of this model as they relate to ordinary factor analysis, describe maximum likelihood and Bayesian estimation methods, and provide a likelihood ratio testing procedure for selecting the factor model ranks. We apply this methodology to the analysis of data from the Human Mortality Database, and show in a cross-validation experiment how it outperforms simpler methods. Additionally, we use this model to impute mortality rates for countries that have no mortality data for several years. Unlike other approaches, our methodology is able to estimate similarities between the mortality rates of countries, time periods and sexes, and use this information to assist with the imputations., Comment: Published in at http://dx.doi.org/10.1214/13-AOAS694 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)