arXiv subjects

Forrest W. Crawford

Publications and source records attributed to Forrest W. Crawford.

31 records · Page 2Linked to original sources

arXiv2015

Seeing the Unseen Network: Inferring Hidden Social Ties from Respondent-Driven Sampling

Learning about the social structure of hidden and hard-to-reach populations --- such as drug users and sex workers --- is a major goal of epidemiological and public health research on risk behaviors and disease prevention. Respondent-driven sampling (RDS) is a peer-referral process widely used by many health organizations, where research subjects recruit other subjects from their social network. In such surveys, researchers observe who recruited whom, along with the time of recruitment and the total number of acquaintances (network degree) of respondents. However, due to privacy concerns, the identities of acquaintances are not disclosed. In this work, we show how to reconstruct the underlying network structure through which the subjects are recruited. We formulate the dynamics of RDS as a continuous-time diffusion process over the underlying graph and derive the likelihood for the recruitment time series under an arbitrary recruitment time distribution. We develop an efficient stochastic optimization algorithm called RENDER (REspoNdent-Driven nEtwork Reconstruction) that finds the network that best explains the collected data. We support our analytical results through an exhaustive set of experiments on both synthetic and real data.

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arXiv2015

Identification of homophily and preferential recruitment in respondent-driven sampling

Respondent-driven sampling (RDS) is a link-tracing procedure for surveying hidden or hard-to-reach populations in which subjects recruit other subjects via their social network. There is significant research interest in detecting clustering or dependence of epidemiological traits in networks, but researchers disagree about whether data from RDS studies can reveal it. Two distinct mechanisms account for dependence in traits of recruiters and recruitees in an RDS study: homophily, the tendency for individuals to share social ties with others exhibiting similar characteristics, and preferential recruitment, in which recruiters do not recruit uniformly at random from their available alters. The different effects of network homophily and preferential recruitment in RDS studies have been a source of confusion in methodological research on RDS, and in empirical studies of the social context of health risk in hidden populations. In this paper, we give rigorous definitions of homophily and preferential recruitment and show that neither can be measured precisely in general RDS studies. We derive nonparametric identification regions for homophily and preferential recruitment and show that these parameters are not point identified unless the network takes a degenerate form. The results indicate that claims of homophily or recruitment bias measured from empirical RDS studies may not be credible. We apply our identification results to a study involving both a network census and RDS on a population of injection drug users in Hartford, CT.

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arXiv2015

Sex, lies and self-reported counts: Bayesian mixture models for heaping in longitudinal count data via birth-death processes

Surveys often ask respondents to report nonnegative counts, but respondents may misremember or round to a nearby multiple of 5 or 10. This phenomenon is called heaping, and the error inherent in heaped self-reported numbers can bias estimation. Heaped data may be collected cross-sectionally or longitudinally and there may be covariates that complicate the inferential task. Heaping is a well-known issue in many survey settings, and inference for heaped data is an important statistical problem. We propose a novel reporting distribution whose underlying parameters are readily interpretable as rates of misremembering and rounding. The process accommodates a variety of heaping grids and allows for quasi-heaping to values nearly but not equal to heaping multiples. We present a Bayesian hierarchical model for longitudinal samples with covariates to infer both the unobserved true distribution of counts and the parameters that control the heaping process. Finally, we apply our methods to longitudinal self-reported counts of sex partners in a study of high-risk behavior in HIV-positive youth.

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arXiv2015

The graphical structure of respondent-driven sampling

Respondent-driven sampling (RDS) is a chain-referral method for sampling members of a hidden or hard-to-reach population such as sex workers, homeless people, or drug users via their social network. Most methodological work on RDS has focused on inference of population means under the assumption that subjects' network degree determines their probability of being sampled. Criticism of existing estimators is usually focused on missing data: the underlying network is only partially observed, so it is difficult to determine correct sampling probabilities. In this paper, we show that data collected in ordinary RDS studies contain information about the structure of the respondents' social network. We construct a continuous-time model of RDS recruitment that incorporates the time series of recruitment events, the pattern of coupon use, and the network degrees of sampled subjects. Together, the observed data and the recruitment model place a well-defined probability distribution on the recruitment-induced subgraph of respondents. We show that this distribution can be interpreted as an exponential random graph model and develop a computationally efficient method for estimating the hidden graph. We validate the method using simulated data and apply the technique to an RDS study of injection drug users in St. Petersburg, Russia.

stat.ME↗

arXiv2015

Hidden population size estimation from respondent-driven sampling: a network approach

Estimating the size of stigmatized, hidden, or hard-to-reach populations is a major problem in epidemiology, demography, and public health research. Capture-recapture and multiplier methods have become standard tools for inference of hidden population sizes, but they require independent random sampling of target population members, which is rarely possible. Respondent-driven sampling (RDS) is a survey method for hidden populations that relies on social link tracing. The RDS recruitment process is designed to spread through the social network connecting members of the target population. In this paper, we show how to use network data revealed by RDS to estimate hidden population size. The key insight is that the recruitment chain, timing of recruitments, and network degrees of recruited subjects provide information about the number of individuals belonging to the target population who are not yet in the sample. We use a computationally efficient Bayesian method to integrate over the missing edges in the subgraph of recruited individuals. We validate the method using simulated data and apply the technique to estimate the number of people who inject drugs in St. Petersburg, Russia.

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arXiv2015

Nonparametric Identification for Respondent-Driven Sampling

Respondent-driven sampling is a survey method for hidden or hard-to-reach populations in which sampled individuals recruit others in the study population via their social links. The most popular estimator for for the population mean assumes that individual sampling probabilities are proportional to each subject's reported degree in a social network connecting members of the hidden population. However, it remains unclear under what circumstances these estimators are valid, and what assumptions are formally required to identify population quantities. In this short note we detail nonparametric identification results for the population mean when the sampling probability is assumed to be a function of network degree known to scale. Importantly, we establish general conditions for the consistency of the popular Volz-Heckathorn (VH) estimator. Our results imply that the conditions for consistency of the VH estimator are far less stringent than those suggested by recent work on diagnostics for RDS. In particular, our results do not require random sampling or the existence of a network connecting the population.

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arXiv2014

Markov counting models for correlated binary responses

We propose a class of continuous-time Markov counting processes for analyzing correlated binary data and establish a correspondence between these models and sums of exchangeable Bernoulli random variables. Our approach generalizes many previous models for correlated outcomes, admits easily interpretable parameterizations, allows different cluster sizes, and incorporates ascertainment bias in a natural way. We demonstrate several new models for dependent outcomes and provide algorithms for computing maximum likelihood estimates. We show how to incorporate cluster-specific covariates in a regression setting and demonstrate improved fits to well-known datasets from familial disease epidemiology and developmental toxicology.

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arXiv2014

On the distribution of interspecies correlation for Markov models of character evolution on Yule trees

Efforts to reconstruct phylogenetic trees and understand evolutionary processes depend fundamentally on stochastic models of speciation and mutation. The simplest continuous-time model for speciation in phylogenetic trees is the Yule process, in which new species are "born" from existing lineages at a constant rate. Recent work has illuminated some of the structural properties of Yule trees, but it remains mostly unknown how these properties affect sequence and trait patterns observed at the tips of the phylogenetic tree. Understanding the interplay between speciation and mutation under simple models of evolution is essential for deriving valid phylogenetic inference methods and gives insight into the optimal design of phylogenetic studies. In this work, we derive the probability distribution of interspecies covariance under Brownian motion and Ornstein-Uhlenbeck models of phenotypic change on a Yule tree. We compute the probability distribution of the number of mutations shared between two randomly chosen taxa in a Yule tree under discrete Markov mutation models. Our results suggest summary measures of phylogenetic information content, illuminate the correlation between site patterns in sequences or traits of related organisms, and provide heuristics for experimental design and reconstruction of phylogenetic trees.

q-bio.PE↗

arXiv2014

Birth-death processes

Many important stochastic counting models can be written as general birth-death processes (BDPs). BDPs are continuous-time Markov chains on the non-negative integers and can be used to easily parameterize a rich variety of probability distributions. Although the theoretical properties of general BDPs are well understood, traditionally statistical work on BDPs has been limited to the simple linear (Kendall) process, which arises in ecology and evolutionary applications. Aside from a few simple cases, it remains impossible to find analytic expressions for the likelihood of a discretely-observed BDP, and computational difficulties have hindered development of tools for statistical inference. But the gap between BDP theory and practical methods for estimation has narrowed in recent years. There are now robust methods for evaluating likelihoods for realizations of BDPs: finite-time transition, first passage, equilibrium probabilities, and distributions of summary statistics that arise commonly in applications. Recent work has also exploited the connection between continuously- and discretely-observed BDPs to derive EM algorithms for maximum likelihood estimation. Likelihood-based inference for previously intractable BDPs is much easier than previously thought and regression approaches analogous to Poisson regression are straightforward to derive. In this review, we outline the basic mathematical theory for BDPs and demonstrate new tools for statistical inference using data from BDPs. We give six examples of BDPs and derive EM algorithms to fit their parameters by maximum likelihood. We show how to compute the distribution of integral summary statistics and give an example application to the total cost of an epidemic. Finally, we suggest future directions for innovation in this important class of stochastic processes.

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arXiv2014

Combining List Experiment and Direct Question Estimates of Sensitive Behavior Prevalence

Survey respondents may give untruthful answers to sensitive questions when asked directly. In recent years, researchers have turned to the list experiment (also known as the item count technique) to overcome this difficulty. While list experiments may be less prone to bias than direct questioning, list experiments are also more susceptible to sampling variability. We show that researchers do not have to abandon direct questioning altogether in order to gain the advantages of list experimentation. We develop a nonparametric estimator of the prevalence of sensitive behaviors that combines list experimentation and direct questioning. We prove that this estimator is asymptotically more efficient than the standard difference-in-means estimator, and we provide a basis for inference using Wald-type confidence intervals. Additionally, leveraging information from the direct questioning, we derive two nonparametric placebo tests of the identifying assumptions for the list experiment. We demonstrate the effectiveness of our combined estimator and placebo tests with an original survey experiment.

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arXiv2012

Diversity, disparity, and evolutionary rate estimation for unresolved Yule trees

The branching structure of biological evolution confers statistical dependencies on phenotypic trait values in related organisms. For this reason, comparative macroevolutionary studies usually begin with an inferred phylogeny that describes the evolutionary relationships of the organisms of interest. The probability of the observed trait data can be computed by assuming a model for trait evolution, such as Brownian motion, over the branches of this fixed tree. However, the phylogenetic tree itself contributes statistical uncertainty to estimates of other evolutionary quantities, and many comparative evolutionary biologists regard the tree as a nuisance parameter. In this paper, we present a framework for analytically integrating over unknown phylogenetic trees in comparative evolutionary studies by assuming that the tree arises from a continuous-time Markov branching model called the Yule process. To do this, we derive a closed-form expression for the distribution of phylogenetic diversity, which is the sum of branch lengths connecting a set of taxa. We then present a generalization of phylogenetic diversity which is equivalent to the expected trait disparity in a set of taxa whose evolutionary relationships are generated by a Yule process and whose traits evolve by Brownian motion. We derive expressions for the distribution of expected trait disparity under a Yule tree. Given one or more observations of trait disparity in a clade, we perform fast likelihood-based estimation of the Brownian variance for unresolved clades. Our method does not require simulation or a fixed phylogenetic tree. We conclude with a brief example illustrating Brownian rate estimation for thirteen families in the Mammalian order Carnivora, in which the phylogenetic tree for each family is unresolved.

q-bio.PE↗

arXiv2011

Transition probabilities for general birth-death processes with applications in ecology, genetics, and evolution

A birth-death process is a continuous-time Markov chain that counts the number of particles in a system over time. In the general process with $n$ current particles, a new particle is born with instantaneous rate $λ_n$ and a particle dies with instantaneous rate $μ_n$. Currently no robust and efficient method exists to evaluate the finite-time transition probabilities in a general birth-death process with arbitrary birth and death rates. In this paper, we first revisit the theory of continued fractions to obtain expressions for the Laplace transforms of these transition probabilities and make explicit an important derivation connecting transition probabilities and continued fractions. We then develop an efficient algorithm for computing these probabilities that analyzes the error associated with approximations in the method. We demonstrate that this error-controlled method agrees with known solutions and outperforms previous approaches to computing these probabilities. Finally, we apply our novel method to several important problems in ecology, evolution, and genetics.

q-bio.PE↗

arXiv2011

Estimation for general birth-death processes

Birth-death processes (BDPs) are continuous-time Markov chains that track the number of "particles" in a system over time. While widely used in population biology, genetics and ecology, statistical inference of the instantaneous particle birth and death rates remains largely limited to restrictive linear BDPs in which per-particle birth and death rates are constant. Researchers often observe the number of particles at discrete times, necessitating data augmentation procedures such as expectation-maximization (EM) to find maximum likelihood estimates. The E-step in the EM algorithm is available in closed-form for some linear BDPs, but otherwise previous work has resorted to approximation or simulation. Remarkably, the E-step conditional expectations can also be expressed as convolutions of computable transition probabilities for any general BDP with arbitrary rates. This important observation, along with a convenient continued fraction representation of the Laplace transforms of the transition probabilities, allows novel and efficient computation of the conditional expectations for all BDPs, eliminating the need for approximation or costly simulation. We use this insight to derive EM algorithms that yield maximum likelihood estimation for general BDPs characterized by various rate models, including generalized linear models. We show that our Laplace convolution technique outperforms competing methods when available and demonstrate a technique to accelerate EM algorithm convergence. Finally, we validate our approach using synthetic data and then apply our methods to estimation of mutation parameters in microsatellite evolution.

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