arXiv subjects

Peter D. Hoff

Publications and source records attributed to Peter D. Hoff.

At least 37 records · Page 2Linked to original sources

arXiv2018

Additive and multiplicative effects network models

Network datasets typically exhibit certain types of statistical dependencies, such as within-dyad correlation, row and column heterogeneity, and third-order dependence patterns such as transitivity and clustering. The first two of these can be well-represented statistically with a social relations model, a type of additive random effects model originally developed for continuous dyadic data. Third-order patterns can be represented with multiplicative random effects models, which are related to matrix decompositions commonly used for matrix-variate data analysis. Additionally, these multiplicative random effects models generalize other popular latent variable network models, such as the stochastic blockmodel and the latent space model. In this article we review a general regression framework for the analysis of network data that combines these two types of random effects and accommodates a variety of network data types, including continuous, binary and ordinal network relations.

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arXiv2018

Joint Mean and Covariance Modeling of Multiple Health Outcome Measures

Health exams determine a patient's health status by comparing the patient's measurement with a population reference range, a 95% interval derived from a homogeneous reference population. Similarly, most of the established relation among health problems are assumed to hold for the entire population. We use data from the 2009 - 2010 National Health and Nutrition Examination Survey (NHANES) on four major health problems in the U.S. and apply a joint mean and covariance model to study how the reference ranges and associations of those health outcomes could vary among subpopulations. We discuss guidelines for model selection and evaluation, using standard criteria such as AIC in conjunction with posterior predictive checks. The results from the proposed model can help identify subpopulations in which more data need to be collected to refine the reference range and to study the specific associations among those health problems.

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arXiv2017

Adaptive Sign Error Control

In multiple testing scenarios, typically the sign of a parameter is inferred when its estimate exceeds some significance threshold in absolute value. Typically, the significance threshold is chosen to control the experimentwise type I error rate, family-wise type I error rate or the false discovery rate. However, controlling these error rates does not explicitly control the sign error rate. In this paper, we propose two procedures for adaptively selecting an experimentwise significance threshold in order to control the sign error rate. The first controls the sign error rate conservatively, without any distributional assumptions on the parameters of interest. The second is an empirical Bayes procedure, and achieves optimal performance asymptotically when a model for the distribution of the parameters is correctly specified. We also discuss an adaptive procedure to minimize the sign error rate when the experimentwise type I error rate is held fixed.

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arXiv2017

Multiplicative Coevolution Regression Models for Longitudinal Networks and Nodal Attributes

We introduce a simple and extendable coevolution model for the analysis of longitudinal network and nodal attribute data. The model features parameters that describe three phenomena: homophily, contagion and autocorrelation of the network and nodal attribute process. Homophily here describes how changes to the network may be associated with between-node similarities in terms of their nodal attributes. Contagion refers to how node-level attributes may change depending on the network. The model we present is based upon a pair of intertwined autoregressive processes. We obtain least-squares parameter estimates for continuous-valued fully-observed network and attribute data. We also provide methods for Bayesian inference in several other cases, including ordinal network and attribute data, and models involving latent nodal attributes. These model extensions are applied to an analysis of international relations data and to data from a study of teen delinquency and friendship networks.

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arXiv2017

Exact adaptive confidence intervals for linear regression coefficients

We propose an adaptive confidence interval procedure (CIP) for the coefficients in the normal linear regression model. This procedure has a frequentist coverage rate that is constant as a function of the model parameters, yet provides smaller intervals than the usual interval procedure, on average across regression coefficients. The proposed procedure is obtained by defining a class of CIPs that all have exact $1-α$ frequentist coverage, and then selecting from this class the procedure that minimizes a prior expected interval width. Such a procedure may be described as "frequentist, assisted by Bayes" or FAB. We describe an adaptive approach for estimating the prior distribution from the data so that exact non-asymptotic $1-α$ coverage is maintained. Additionally, in a "$p$ growing with $n$" asymptotic scenario, this adaptive FAB procedure is asymptotically Bayes-optimal among $1-α$ frequentist CIPs.

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arXiv2017

Lasso, fractional norm and structured sparse estimation using a Hadamard product parametrization

Using a multiplicative reparametrization, I show that a subclass of $L_q$ penalties with $q\leq 1$ can be expressed as sums of $L_2$ penalties. It follows that the lasso and other norm-penalized regression estimates may be obtained using a very simple and intuitive alternating ridge regression algorithm. As compared to a similarly intuitive EM algorithm for $L_q$ optimization, the proposed algorithm avoids some numerical instability issues and is also competitive in terms of speed. Furthermore, the proposed algorithm can be extended to accommodate sparse high-dimensional scenarios, generalized linear models, and can be used to create structured sparsity via penalties derived from covariance models for the parameters. Such model-based penalties may be useful for sparse estimation of spatially or temporally structured parameters.

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arXiv2016

Adaptive multigroup confidence intervals with constant coverage

Confidence intervals for the means of multiple normal populations are often based on a hierarchical normal model. While commonly used interval procedures based on such a model have the nominal coverage rate on average across a population of groups, their actual coverage rate for a given group will be above or below the nominal rate, depending on the value of the group mean. Alternatively, a coverage rate that is constant as a function of a group's mean can be simply achieved by using a standard $t$-interval, based on data only from that group. The standard $t$-interval, however, fails to share information across the groups and is therefore not adaptive to easily obtained information about the distribution of group-specific means. In this article we construct confidence intervals that have a constant frequentist coverage rate and that make use of information about across-group heterogeneity, resulting in constant-coverage intervals that are narrower than standard $t$-intervals on average across groups. Such intervals are constructed by inverting biased tests for the mean of a normal population. Given a prior distribution on the mean, Bayes-optimal biased tests can be inverted to form Bayes-optimal confidence intervals with frequentist coverage that is constant as a function of the mean. In the context of multiple groups, the prior distribution is replaced by a model of across-group heterogeneity. The parameters for this model can be estimated using data from all of the groups, and used to obtain confidence intervals with constant group-specific coverage that adapt to information about the distribution of group means.

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arXiv2015

Limitations on detecting row covariance in the presence of column covariance

Many inference techniques for multivariate data analysis assume that the rows of the data matrix are realizations of independent and identically distributed random vectors. Such an assumption will be met, for example, if the rows of the data matrix are multivariate measurements on a set of independently sampled units. In the absence of an independent random sample, a relevant question is whether or not a statistical model that assumes such row exchangeability is plausible. One method for assessing this plausibility is a statistical test of row covariation. Maintenance of a constant type I error rate regardless of the column covariance or matrix mean can be accomplished with a test that is invariant under an appropriate group of transformations. In the context of a class of elliptically contoured matrix regression models (such as matrix normal models), I show that there are no non-trivial invariant tests if the number of rows is not sufficiently larger than the number of columns. Furthermore, I show that even if the number of rows is large, there are no non-trivial invariant tests that have power to detect arbitrary row covariance in the presence of arbitrary column covariance. However, we can construct biased tests that have power to detect certain types of row covariance that may be encountered in practice.

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arXiv2015

A Pivot-Based Improvement to Sandwich-Based Confidence Intervals

The current standard for confidence interval construction in the context of a possibly misspecified model is to use an interval based on the sandwich estimate of variance. These intervals provide asymptotically correct coverage, but small-sample coverage is known to be poor. By eliminating a plug-in assumption, we derive a pivot-based method for confidence interval construction under possibly misspecified models. When compared against confidence intervals generated by the sandwich estimate of variance, this method provides more accurate coverage of the pseudo-true parameter at small sample sizes. This is shown in the results of several simulation studies. Asymptotic results show that our pivot-based intervals have large sample efficiency equal to that of intervals based on the sandwich estimate of variance.

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arXiv2015

Multilinear tensor regression for longitudinal relational data

A fundamental aspect of relational data, such as from a social network, is the possibility of dependence among the relations. In particular, the relations between members of one pair of nodes may have an effect on the relations between members of another pair. This article develops a type of regression model to estimate such effects in the context of longitudinal and multivariate relational data, or other data that can be represented in the form of a tensor. The model is based on a general multilinear tensor regression model, a special case of which is a tensor autoregression model in which the tensor of relations at one time point are parsimoniously regressed on relations from previous time points. This is done via a separable, or Kronecker-structured, regression parameter along with a separable covariance model. In the context of an analysis of longitudinal multivariate relational data, it is shown how the multilinear tensor regression model can represent patterns that often appear in relational and network data, such as reciprocity and transitivity.

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arXiv2015

Dyadic data analysis with amen

Dyadic data on pairs of objects, such as relational or social network data, often exhibit strong statistical dependencies. Certain types of second-order dependencies, such as degree heterogeneity and reciprocity, can be well-represented with additive random effects models. Higher-order dependencies, such as transitivity and stochastic equivalence, can often be represented with multiplicative effects. The "amen" package for the R statistical computing environment provides estimation and inference for a class of additive and multiplicative random effects models for ordinal, continuous, binary and other types of dyadic data. The package also provides methods for missing, censored and fixed-rank nomination data, as well as longitudinal dyadic data. This tutorial illustrates the "amen" package via example statistical analyses of several of these different data types.

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arXiv2015

Relax, Tensors Are Here: Dependencies in International Processes

Previous models of international conflict have suffered two shortfalls. They tended not to embody dynamic changes, focusing rather on static slices of behavior over time. These models have also been empirically evaluated in ways that assumed the independence of each country, when in reality they are searching for the interdependence among all countries. We illustrate a solution to these two hurdles and evaluate this new, dynamic, network based approach to the dependencies among the ebb and flow of daily international interactions using a newly developed, and openly available, database of events among nations.

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arXiv2014

A higher-order LQ decomposition for separable covariance models

We develop a higher order generalization of the LQ decomposition and show that this decomposition plays an important role in likelihood-based estimation and testing for separable, or Kronecker structured, covariance models, such as the multilinear normal model. This role is analogous to that of the LQ decomposition in likelihood inference for the multivariate normal model. Additionally, this higher order LQ decomposition can be used to construct an alternative version of the popular higher order singular value decomposition for tensor-valued data. We also develop a novel generalization of the polar decomposition to tensor-valued data.

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arXiv2014

Hierarchical array priors for ANOVA decompositions of cross-classified data

ANOVA decompositions are a standard method for describing and estimating heterogeneity among the means of a response variable across levels of multiple categorical factors. In such a decomposition, the complete set of main effects and interaction terms can be viewed as a collection of vectors, matrices and arrays that share various index sets defined by the factor levels. For many types of categorical factors, it is plausible that an ANOVA decomposition exhibits some consistency across orders of effects, in that the levels of a factor that have similar main-effect coefficients may also have similar coefficients in higher-order interaction terms. In such a case, estimation of the higher-order interactions should be improved by borrowing information from the main effects and lower-order interactions. To take advantage of such patterns, this article introduces a class of hierarchical prior distributions for collections of interaction arrays that can adapt to the presence of such interactions. These prior distributions are based on a type of array-variate normal distribution, for which a covariance matrix for each factor is estimated. This prior is able to adapt to potential similarities among the levels of a factor, and incorporate any such information into the estimation of the effects in which the factor appears. In the presence of such similarities, this prior is able to borrow information from well-estimated main effects and lower-order interactions to assist in the estimation of higher-order terms for which data information is limited.

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arXiv2014

Separable factor analysis with applications to mortality data

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.

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arXiv2014

Information bounds for Gaussian copulas

Often of primary interest in the analysis of multivariate data are the copula parameters describing the dependence among the variables, rather than the univariate marginal distributions. Since the ranks of a multivariate dataset are invariant to changes in the univariate marginal distributions, rank-based estimators are natural candidates for semiparametric copula estimation. Asymptotic information bounds for such estimators can be obtained from an asymptotic analysis of the rank likelihood, that is, the probability of the multivariate ranks. In this article, we obtain limiting normal distributions of the rank likelihood for Gaussian copula models. Our results cover models with structured correlation matrices, such as exchangeable or circular correlation models, as well as unstructured correlation matrices. For all Gaussian copula models, the limiting distribution of the rank likelihood ratio is shown to be equal to that of a parametric likelihood ratio for an appropriately chosen multivariate normal model. This implies that the semiparametric information bounds for rank-based estimators are the same as the information bounds for estimators based on the full data, and that the multivariate normal distributions are least favorable.

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arXiv2013

Testing for nodal dependence in relational data matrices

Relational data are often represented as a square matrix, the entries of which record the relationships between pairs of objects. Many statistical methods for the analysis of such data assume some degree of similarity or dependence between objects in terms of the way they relate to each other. However, formal tests for such dependence have not been developed. We provide a test for such dependence using the framework of the matrix normal model, a type of multivariate normal distribution parameterized in terms of row- and column-specific covariance matrices. We develop a likelihood ratio test (LRT) for row and column dependence based on the observation of a single relational data matrix. We obtain a reference distribution for the LRT statistic, thereby providing an exact test for the presence of row or column correlations in a square relational data matrix. Additionally, we provide extensions of the test to accommodate common features of such data, such as undefined diagonal entries, a non-zero mean, multiple observations, and deviations from normality.

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arXiv2013

Testing and Modeling Dependencies Between a Network and Nodal Attributes

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.

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