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We present a probabilistic method for linking multiple datafiles. This task is not trivial in the absence of unique identifiers for the individuals recorded. This is a common scenario when linking census data to coverage measurement surveys for census coverage evaluation, and in general when multiple record-systems need to be integrated for posterior analysis. Our method generalizes the Fellegi-Sunter theory for linking records from two datafiles and its modern implementations. The multiple record linkage goal is to classify the record K-tuples coming from K datafiles according to the different matching patterns. Our method incorporates the transitivity of agreement in the computation of the data used to model matching probabilities. We use a mixture model to fit matching probabilities via maximum likelihood using the EM algorithm. We present a method to decide the record K-tuples membership to the subsets of matching patterns and we prove its optimality. We apply our method to the integration of three Colombian homicide record systems and we perform a simulation study in order to explore the performance of the method under measurement error and different scenarios. The proposed method works well and opens some directions for future research., Comment: Several changes with respect to previous version. Accepted in the Journal of the American Statistical Association

Traditional statistical methods for confidentiality protection of statistical databases do not scale well to deal with GWAS (genome-wide association studies) databases especially in terms of guarantees regarding protection from linkage to external information. The more recent concept of differential privacy, introduced by the cryptographic community, is an approach which provides a rigorous definition of privacy with meaningful privacy guarantees in the presence of arbitrary external information, although the guarantees come at a serious price in terms of data utility. Building on such notions, we propose new methods to release aggregate GWAS data without compromising an individual's privacy. We present methods for releasing differentially private minor allele frequencies, chi-square statistics and p-values. We compare these approaches on simulated data and on a GWAS study of canine hair length involving 685 dogs. We also propose a privacy-preserving method for finding genome-wide associations based on a differentially-private approach to penalized logistic regression.

Preserving the privacy of individual databases when carrying out statistical calculations has a long history in statistics and had been the focus of much recent attention in machine learning In this paper, we present a protocol for computing logistic regression when the data are held by separate parties without actually combining information sources by exploiting results from the literature on multi-party secure computation. We provide only the final result of the calculation compared with other methods that share intermediate values and thus present an opportunity for compromise of values in the combined database. Our paper has two themes: (1) the development of a secure protocol for computing the logistic parameters, and a demonstration of its performances in practice, and (2) and amended protocol that speeds up the computation of the logistic function. We illustrate the nature of the calculations and their accuracy using an extract of data from the Current Population Survey divided between two parties.

Rejoinder of "Bayesian Models and Methods in Public Policy and Government Settings" by S. E. Fienberg [arXiv:1108.2177], Comment: Published in at http://dx.doi.org/10.1214/11-STS331REJ the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Starting with the neo-Bayesian revival of the 1950s, many statisticians argued that it was inappropriate to use Bayesian methods, and in particular subjective Bayesian methods in governmental and public policy settings because of their reliance upon prior distributions. But the Bayesian framework often provides the primary way to respond to questions raised in these settings and the numbers and diversity of Bayesian applications have grown dramatically in recent years. Through a series of examples, both historical and recent, we argue that Bayesian approaches with formal and informal assessments of priors AND likelihood functions are well accepted and should become the norm in public settings. Our examples include census-taking and small area estimation, US election night forecasting, studies reported to the US Food and Drug Administration, assessing global climate change, and measuring potential declines in disability among the elderly., Comment: Published in at http://dx.doi.org/10.1214/10-STS331 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Discussion of "Network routing in a dynamic environment" by N.D. Singpurwalla [arXiv:1107.4852], Comment: Published in at http://dx.doi.org/10.1214/11-AOAS453A the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We study maximum likelihood estimation for the statistical model for undirected random graphs, known as the $\beta$-model, in which the degree sequences are minimal sufficient statistics. We derive necessary and sufficient conditions, based on the polytope of degree sequences, for the existence of the maximum likelihood estimator (MLE) of the model parameters. We characterize in a combinatorial fashion sample points leading to a nonexistent MLE, and nonestimability of the probability parameters under a nonexistent MLE. We formulate conditions that guarantee that the MLE exists with probability tending to one as the number of nodes increases., Comment: Published in at http://dx.doi.org/10.1214/12-AOS1078 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We study maximum likelihood estimation in log-linear models under conditional Poisson sampling schemes. We derive necessary and sufficient conditions for existence of the maximum likelihood estimator (MLE) of the model parameters and investigate estimability of the natural and mean-value parameters under a nonexistent MLE. Our conditions focus on the role of sampling zeros in the observed table. We situate our results within the framework of extended exponential families, and we exploit the geometric properties of log-linear models. We propose algorithms for extended maximum likelihood estimation that improve and correct the existing algorithms for log-linear model analysis., Comment: Published in at http://dx.doi.org/10.1214/12-AOS986 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

The deployment of improvised explosive devices (IEDs) along major roadways has been a favoured strategy of insurgents in recent war zones, both for the ability to cause damage to targets along roadways at minimal cost, but also as a means of controlling the flow of traffic and causing additional expense to opposing forces. Among other related approaches (which we discuss), the adversarial problem has an analogue in the Canadian Traveller Problem, wherein a stretch of road is blocked with some independent probability, and the state of the road is only discovered once the traveller reaches one of the intersections that bound this stretch of road. We discuss the implementation of ideas from social network analysis, namely the notion of "betweenness centrality", and how this can be adapted to the notion of deployment of IEDs with the aid of Generalized Linear Models (GLMs): namely, how we can model the probability of an IED deployment in terms of the increased effort due to Canadian betweenness, how we can include expert judgement on the probability of a deployment, and how we can extend the approach to estimation and updating over several time steps., Comment: 25 pages, 3 figures

Introduction to papers on the modeling and analysis of network data---II, Comment: Published in at http://dx.doi.org/10.1214/10-AOAS365 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Introduction to papers on the modeling and analysis of network data, Comment: Published in at http://dx.doi.org/10.1214/10-AOAS346 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Holland and Leinhardt (1981) proposed a directed random graph model, the p1 model, to describe dyadic interactions in a social network. In previous work (Petrovic et al., 2010), we studied the algebraic properties of the p1 model and showed that it is a toric model specified by a multi-homogeneous ideal. We conducted an extensive study of the Markov bases for p1 that incorporate explicitly the constraint arising from multi-homogeneity. Here we consider the properties of the corresponding toric variety and relate them to the conditions for the existence of the maximum likelihood and extended maximum likelihood estimators or the model parameters. Our results are directly relevant to the estimation and conditional goodness-of-fit testing problems in p1 models.

We present a new similarity measure tailored to posts in an online forum. Our measure takes into account all the available information about user interest and interaction --- the content of posts, the threads in the forum, and the author of the posts. We use this post similarity to build a similarity between users, based on principal coordinate analysis. This allows easy visualization of the user activity as well. Similarity between users has numerous applications, such as clustering or classification. We show that including the author of a post in the post similarity has a smoothing effect on principal coordinate projections. We demonstrate our method on real data drawn from an internal corporate forum, and compare our results to those given by a standard document classification method. We conclude our method gives a more detailed picture of both the local and global network structure., Comment: 8 Pages, 7 Figures, Short form appears in Proc. of ICWSM 2010

Networks are ubiquitous in science and have become a focal point for discussion in everyday life. Formal statistical models for the analysis of network data have emerged as a major topic of interest in diverse areas of study, and most of these involve a form of graphical representation. Probability models on graphs date back to 1959. Along with empirical studies in social psychology and sociology from the 1960s, these early works generated an active network community and a substantial literature in the 1970s. This effort moved into the statistical literature in the late 1970s and 1980s, and the past decade has seen a burgeoning network literature in statistical physics and computer science. The growth of the World Wide Web and the emergence of online networking communities such as Facebook, MySpace, and LinkedIn, and a host of more specialized professional network communities has intensified interest in the study of networks and network data. Our goal in this review is to provide the reader with an entry point to this burgeoning literature. We begin with an overview of the historical development of statistical network modeling and then we introduce a number of examples that have been studied in the network literature. Our subsequent discussion focuses on a number of prominent static and dynamic network models and their interconnections. We emphasize formal model descriptions, and pay special attention to the interpretation of parameters and their estimation. We end with a description of some open problems and challenges for machine learning and statistics., Comment: 96 pages, 14 figures, 333 references

The p_1 model is a directed random graph model used to describe dyadic interactions in a social network in terms of effects due to differential attraction (popularity) and expansiveness, as well as an additional effect due to reciprocation. In this article we carry out an algebraic statistics analysis of this model. We show that the p_1 model is a toric model specified by a multi-homogeneous ideal. We conduct an extensive study of the Markov bases for p_1 models that incorporate explicitly the constraint arising from multi-homogeneity. Our results are directly relevant to the estimation and conditional goodness-of-fit testing problems in p_1 models., Comment: 22 pages. 4 figures depicting relevant Markov moves. One section removed from previous version.

There has been an explosion of interest in statistical models for analyzing network data, and considerable interest in the class of exponential random graph (ERG) models, especially in connection with difficulties in computing maximum likelihood estimates. The issues associated with these difficulties relate to the broader structure of discrete exponential families. This paper re-examines the issues in two parts. First we consider the closure of $k$-dimensional exponential families of distribution with discrete base measure and polyhedral convex support $\mathrm{P}$. We show that the normal fan of $\mathrm{P}$ is a geometric object that plays a fundamental role in deriving the statistical and geometric properties of the corresponding extended exponential families. We discuss its relevance to maximum likelihood estimation, both from a theoretical and computational standpoint. Second, we apply our results to the analysis of ERG models. In particular, by means of a detailed example, we provide some characterization of the properties of ERG models, and, in particular, of certain behaviors of ERG models known as degeneracy.

Large contingency tables arise in many contexts but especially in the collection of survey and census data by government statistical agencies. Because the vast majority of the variables in this context have a large number of categories, agencies and users need a systematic way of constructing tables which are summaries of such contingency tables. We propose such an approach in this paper by finding members of a class of restricted log-linear models which maximize the likelihood of the data and use this to find a parsimonious means of representing the table. In contrast with more standard approaches for model search in hierarchical log-linear models (HLLM), our procedure systematically reduces the number of categories of the variables. Through a series of examples, we illustrate the extent to which it can preserve the interaction structure found with HLLMs and be used as a data simplification procedure prior to HLL modeling. A feature of the procedure is that it can easily be applied to many tables with millions of cells, providing a new way of summarizing large data sets in many disciplines. The focus is on information and description rather than statistical testing. The procedure may treat each variable in the table in different ways, preserving full detail, treating it as fully nominal, or preserving ordinality., Comment: Published in at http://dx.doi.org/10.1214/08-AOAS175 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Howard Raiffa earned his bachelor's degree in mathematics, his master's degree in statistics and his Ph.D. in mathematics at the University of Michigan. Since 1957, Raiffa has been a member of the faculty at Harvard University, where he is now the Frank P. Ramsey Chair in Managerial Economics (Emeritus) in the Graduate School of Business Administration and the Kennedy School of Government. A pioneer in the creation of the field known as decision analysis, his research interests span statistical decision theory, game theory, behavioral decision theory, risk analysis and negotiation analysis. Raiffa has supervised more than 90 doctoral dissertations and written 11 books. His new book is Negotiation Analysis: The Science and Art of Collaborative Decision Making. Another book, Smart Choices, co-authored with his former doctoral students John Hammond and Ralph Keeney, was the CPR (formerly known as the Center for Public Resources) Institute for Dispute Resolution Book of the Year in 1998. Raiffa helped to create the International Institute for Applied Systems Analysis and he later became its first Director, serving in that capacity from 1972 to 1975. His many honors and awards include the Distinguished Contribution Award from the Society of Risk Analysis; the Frank P. Ramsey Medal for outstanding contributions to the field of decision analysis from the Operations Research Society of America; and the Melamed Prize from the University of Chicago Business School for The Art and Science of Negotiation. He earned a Gold Medal from the International Association for Conflict Management and a Lifetime Achievement Award from the CPR Institute for Dispute Resolution. He holds honorary doctor's degrees from Carnegie Mellon University, the University of Michigan, Northwestern University, Ben Gurion University of the Negev and Harvard University. The latter was awarded in 2002., Comment: Published in at http://dx.doi.org/10.1214/088342307000000104 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)

What makes a problem suitable for statistical analysis? Are historical and religious questions addressable using statistical calculations? Such issues have long been debated in the statistical community and statisticians and others have used historical information and texts to analyze such questions as the economics of slavery, the authorship of the Federalist Papers and the question of the existence of God. But what about historical and religious attributions associated with information gathered from archeological finds? In 1980, a construction crew working in the Jerusalem neighborhood of East Talpiot stumbled upon a crypt. Archaeologists from the Israel Antiquities Authority came to the scene and found 10 limestone burial boxes, known as ossuaries, in the crypt. Six of these had inscriptions. The remains found in the ossuaries were reburied, as required by Jewish religious tradition, and the ossuaries were catalogued and stored in a warehouse. The inscriptions on the ossuaries were catalogued and published by Rahmani (1994) and by Kloner (1996) but there reports did not receive widespread public attention. Fast forward to March 2007, when a television ``docudrama'' aired on The Discovery Channel entitled ``The Lost Tomb of Jesus'' touched off a public and religious controversy--one only need think about the title to see why there might be a controversy! The program, and a simultaneously published book [Jacobovici and Pellegrino (2007)], described the ``rediscovery'' of the East Talpiot archeological find and they presented interpretations of the ossuary inscriptions from a number of perspectives. Among these was a statistical calculation attributed to the statistician Andrey Feuerverger: ``that the odds that all six names would appear together in one tomb are 1 in 600, calculated conservatively--or possibly even as much as one in one million.'', Comment: Published in at http://dx.doi.org/10.1214/08-AOAS162 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Data on functional disability are of widespread policy interest in the United States, especially with respect to planning for Medicare and Social Security for a growing population of elderly adults. We consider an extract of functional disability data from the National Long Term Care Survey (NLTCS) and attempt to develop disability profiles using variations of the Grade of Membership (GoM) model. We first describe GoM as an individual-level mixture model that allows individuals to have partial membership in several mixture components simultaneously. We then prove the equivalence between individual-level and population-level mixture models, and use this property to develop a Markov Chain Monte Carlo algorithm for Bayesian estimation of the model. We use our approach to analyze functional disability data from the NLTCS., Comment: Published in at http://dx.doi.org/10.1214/07-AOAS126 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Forensic science is usually taken to mean the application of a broad spectrum of scientific tools to answer questions of interest to the legal system. Despite such popular television series as CSI: Crime Scene Investigation and its spinoffs--CSI: Miami and CSI: New York--on which the forensic scientists use the latest high-tech scientific tools to identify the perpetrator of a crime and always in under an hour, forensic science is under assault, in the public media, popular magazines [Talbot (2007), Toobin (2007)] and in the scientific literature [Kennedy (2003), Saks and Koehler (2005)]. Ironically, this growing controversy over forensic science has occurred precisely at the time that DNA evidence has become the ``gold standard'' in the courts, leading to the overturning of hundreds of convictions many of which were based on clearly less credible forensic evidence, including eyewitness testimony [Berger (2006)]., Comment: Published in at http://dx.doi.org/10.1214/07-AOAS140 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Learning latent expression themes that best express complex patterns in a sample is a central problem in data mining and scientific research. For example, in computational biology we seek a set of salient gene expression themes that explain a biological process, extracting them from a large pool of gene expression profiles. In this paper, we introduce probabilistic models to learn such latent themes in an unsupervised fashion. Our models capture contagion, i.e., dependence among multiple occurrences of the same feature, using a hierarchical Bayesian scheme. Contagion is a convenient analytical formalism to characterize semantic themes underlying observed feature patterns, such as biological context. We present model variants tailored to different properties of biological data, and we outline a general variational inference scheme for approximate posterior inference. We validate our methods on both simulated data and realistic high-throughput gene expression profiles via SAGE. Our results show improved predictions of gene functions over existing methods based on stronger independence assumptions, and demonstrate feasibility of a promising hierarchical Bayesian formalism for soft clustering and latent aspects analysis., Comment: 22 pages, 4 figures

Discussion of ``The William Kruskal Legacy: 1919--2005'' by Stephen E. Fienberg, Stephen M. Stigler and Judith M. Tanur [arXiv:0710.5063], Comment: Published in at http://dx.doi.org/10.1214/088342306000000376 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)

William Kruskal (Bill) was a distinguished statistician who spent virtually his entire professional career at the University of Chicago, and who had a lasting impact on the Institute of Mathematical Statistics and on the field of statistics more broadly, as well as on many who came in contact with him. Bill passed away last April following an extended illness, and on May 19, 2005, the University of Chicago held a memorial service at which several of Bill's colleagues and collaborators spoke along with members of his family and other friends. This biography and the accompanying commentaries derive in part from brief presentations on that occasion, along with recollections and input from several others. Bill was known personally to most of an older generation of statisticians as an editor and as an intellectual and professional leader. In 1994, Statistical Science published an interview by Sandy Zabell (Vol. 9, 285--303) in which Bill looked back on selected events in his professional life. One of the purposes of the present biography and accompanying commentaries is to reintroduce him to old friends and to introduce him for the first time to new generations of statisticians who never had an opportunity to interact with him and to fall under his influence., Comment: This paper discussed in: [arXiv:0710.5072], [arXiv:0710.5074], [arXiv:0710.5077], [arXiv:0710.5079], [arXiv:0710.5081], [arXiv:0710.5084] and [arXiv:0710.5085]. Published in at http://dx.doi.org/10.1214/088342306000000420 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Statistical models with latent structure have a history going back to the 1950s and have seen widespread use in the social sciences and, more recently, in computational biology and in machine learning. Here we study the basic latent class model proposed originally by the sociologist Paul F. Lazarfeld for categorical variables, and we explain its geometric structure. We draw parallels between the statistical and geometric properties of latent class models and we illustrate geometrically the causes of many problems associated with maximum likelihood estimation and related statistical inference. In particular, we focus on issues of non-identifiability and determination of the model dimension, of maximization of the likelihood function and on the effect of symmetric data. We illustrate these phenomena with a variety of synthetic and real-life tables, of different dimension and complexity. Much of the motivation for this work stems from the "100 Swiss Francs" problem, which we introduce and describe in detail.

Mobile robots require basic information to navigate through an environment: they need to know where they are (localization) and they need to know where they are going. For the latter, robots need a map of the environment. Using sensors of a variety of forms, robots gather information as they move through an environment in order to build a map. In this paper we present a novel sampling algorithm to solving the simultaneous mapping and localization (SLAM) problem in indoor environments. We approach the problem from a Bayesian statistics perspective. The data correspond to a set of range finder and odometer measurements, obtained at discrete time instants. We focus on the estimation of the posterior distribution over the space of possible maps given the data. By exploiting different factorizations of this distribution, we derive three sampling algorithms based on importance sampling. We illustrate the results of our approach by testing the algorithms with two real data sets obtained through robot navigation inside office buildings at Carnegie Mellon University and the Pontificia Universidad Catolica de Chile., Comment: Published at http://dx.doi.org/10.1214/07-AOAS115 in the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)

In this paper, we consider the statistical analysis of a protein interaction network. We propose a Bayesian model that uses a hierarchy of probabilistic assumptions about the way proteins interact with one another in order to: (i) identify the number of non-observable functional modules; (ii) estimate the degree of membership of proteins to modules; and (iii) estimate typical interaction patterns among the functional modules themselves. Our model describes large amount of (relational) data using a relatively small set of parameters that we can reliably estimate with an efficient inference algorithm. We apply our methodology to data on protein-to-protein interactions in saccharomyces cerevisiae to reveal proteins' diverse functional roles. The case study provides the basis for an overview of which scientific questions can be addressed using our methods, and for a discussion of technical issues., Comment: 22 pages, 6 figures, 2 tables

Observations consisting of measurements on relationships for pairs of objects arise in many settings, such as protein interaction and gene regulatory networks, collections of author-recipient email, and social networks. Analyzing such data with probabilisic models can be delicate because the simple exchangeability assumptions underlying many boilerplate models no longer hold. In this paper, we describe a latent variable model of such data called the mixed membership stochastic blockmodel. This model extends blockmodels for relational data to ones which capture mixed membership latent relational structure, thus providing an object-specific low-dimensional representation. We develop a general variational inference algorithm for fast approximate posterior inference. We explore applications to social and protein interaction networks., Comment: 46 pages, 14 figures, 3 tables

Comment on Complex Causal Questions Require Careful Model Formulation: Discussion of Rubin on Experiments with ``Censoring'' Due to Death [math.ST/0612783], Comment: Published at http://dx.doi.org/10.1214/088342306000000295 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)

The growing expanse of e-commerce and the widespread availability of online databases raise many fears regarding loss of privacy and many statistical challenges. Even with encryption and other nominal forms of protection for individual databases, we still need to protect against the violation of privacy through linkages across multiple databases. These issues parallel those that have arisen and received some attention in the context of homeland security. Following the events of September 11, 2001, there has been heightened attention in the United States and elsewhere to the use of multiple government and private databases for the identification of possible perpetrators of future attacks, as well as an unprecedented expansion of federal government data mining activities, many involving databases containing personal information. We present an overview of some proposals that have surfaced for the search of multiple databases which supposedly do not compromise possible pledges of confidentiality to the individuals whose data are included. We also explore their link to the related literature on privacy-preserving data mining. In particular, we focus on the matching problem across databases and the concept of ``selective revelation'' and their confidentiality implications., Comment: Published at http://dx.doi.org/10.1214/088342306000000240 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We have measured the orbital light curve of dwarf nova OY Carinae on 8 separate occasions between 1997 September and 2005 December. The measurements were made in white light using CCD photometers on the Mt Canopus 1 m telescope. The time of eclipse in 2005 December was 168 +- 5 s earlier than that predicted by the Wood et al.(1989) ephemeris. Using the times of eclipse from our measurements and the compilation of published measurements by Pratt et al (1999) we find that the observational data are inconsistent with a constant period and indicate that the orbital period is decreasing by 5+-1 X 10^-12 s/s. This is too fast to be explained by gravitational radiation emission. It is possible that the change is cyclic with a period greater than about 80 years. This is much longer than typical magnetic activity cycles and may be due to the presence of a third object in the system. Preliminary estimates suggest that this is a brown dwarf with mass about 0.016 Msun and orbital radius >= 17 AU., Comment: 4 pages 2 figures. MNRAS submitted Final proofread version. Discussion modified with figure showing fits and residuals to models, statistical significance of fits added and minor typographical edits

We provide a polyhedral description of the conditions for the existence of the maximum likelihood estimate (MLE) for a hierarchical log-linear model. The MLE exists if and only if the observed margins lie in the relative interior of the marginal cone. Using this description, we give an algorithm for determining if the MLE exists. If the tree width is bounded, the algorithm runs in polynomial time. We also perform a computational study of the case of three random variables under the no three-factor effect model., Comment: 15 pages

Superfield expansions over four-dimensional graded spacetime $(x^\mu,\theta^\nu)$, with Minkowski coordinates $x$ extended by vector Grassmann variables $\theta$, are investigated. By appropriate identification of the physical Lorentz algebra in the even and odd parts of the superfield, a typology of `schizofields' containing both integer and half-integer spin fields is established. For two of these types, identified as `gauge potential'-like and `field strength'-like schizofields, an $sl(2/1,{\mathbb C})_{\mathbb R}$ supersymmetry at the component field level is demonstrated. Prospects for a schizofield calculus, and application of these types of fields to the particle spectrum, are adumbrated., Comment: 8 pages, LaTeX