Statistics seminars 2011-2012

Volume estimators based on Cavalieri’s principle are widely used in the bio-
sciences. For example in neuroscience, where volumetric measurements of brain
structures are of interest, systematic samples of serial sections are obtained by
magnetic resonance imaging or by a physical cutting procedure. The volume v is
then estimated by ˆv, which is the sum over the areas of the structure of interest in
the section planes multiplied by the width of the sections, t > 0.
Assessing the precision of such volume estimates is a question of great practical
importance, but statistically a challenging task due to the strong spatial dependence
of the data and typically small sample sizes. In this talk, an overview of classical
and new approaches to this problem will be presented. A special focus will be given
to some recent advances on distribution estimators and confidence intervals for ˆv;
see Hall and Ziegel (2011).

References
Hall, P. and Ziegel, J. (2011). Distribution estimators and confidence intervals for
stereological volumes. Biometrika 98, 417–431.

Susko: The data generated by large scale sequencing projects is complex, high-dimensional, multivariate discrete data. In studies of evolutionary biology, the parameter space of evolutionary trees is an unusual additional complication from a statistical perspective. In this talk I will briefly introduce the general approaches to utilizing sequence data in phylogenetic inference. A particular issue of interest in phylogenetic inference is assessments of uncertainty about the true tree or structures that might be present in it. The primary way in which uncertainty is assessed in practice is through bootstrap support (BP) for splits, large values indicating strong support for the split. A difficulty with this measure, however, has been deciding how large is large enough. We discuss the interpretation of BP and ways of adjusting it so that it has an interpretation similar to a p-value. A related issue, having to do with the behaviour of methods when data are generated from a star tree, gives rise to an interesting example in which, due to the unusual statistical nature,Bayesian and maximum likelihood methods give strikingly different results, even asymptotically.

Labbe: Recently, expression quantitative loci (eQTL) mapping studies, where expression levels of thousands of genes are viewed as quantitative traits, have been used to provide greater insight into the biology of gene regulation. Current data analysis and interpretation of eQTL studies involve the use of multiple methods and applications, the output of which is often fragmented. In this talk, we present an integrated hierarchical Bayesian model that jointly models all genes and SNPs to detect eQTLs.
We propose a model (named iBMQ) that is speci cally designed to handle a large number G of gene expressions, a large number S of regressors (genetic markers) and a small number n of individuals in what we call a "large G, large S, small n" paradigm. This method incorporates genotypic and gene expression data into a single model while 1) specifically coping with the high dimensionality of eQTL data (large number of genes), 2) borrowing strength from all gene expression data for the mapping procedures, and 3) controlling the number of false positives to a desirable level.

Streaming data is ubiquitous in a wide range of areas from engineering and information technology, finance, and commerce, to atmospheric physics, and earth sciences. The online approximation of properties of data streams is of great interest, but this approximation process is hindered by the sheer size of the data and the speed at which it is generated. Data stream algorithms typically allow only one pass over the data, and maintain sub-linear representations of the data from which target properties can be inferred with high efficiency.

In this talk we consider the online approximation of two important characterizations of data streams: cardinality and empirical Shannon entropy. We assume that the number of distinct elements observed in the stream is prohibitively large, so that the vector of cumulative
quantities cannot be stored on main computer memory for fast and efficient access. We focus on two techniques that use pseudo-random variates to form low-dimensional data sketches (using hashing and random projections), and derive estimators of the cardinality and empirical entropy. We discuss various properties of our estimators such as relative asymptotic efficiency, recursive computability, and error and complexity bounds. Finally, we present results on simulated data and seismic measurements from a volcano.

References:
Peter Clifford and Ioana A. Cosma (2011) “A statistical analysis of probabilistic counting algorithms” (to appear in the Scandinavian Journal of Statistics, preprint on arXiv:0801.3552).

Peter Clifford and Ioana A. Cosma (2009)“A simple sketching algorithm for entropy estimation” (in preparation, preprint on arXiv:0908.3961).

Dupuis: Environmental processes are often non-stationary since climate patterns cause systematic seasonal effects and long-term climate changes cause trends.  The usual limit models are not applicable for non-stationary processes, but models from standard extreme value theory can be used along with statistical modeling to provide useful inference. Traditional approaches include letting model parameters be a function of covariates or using time-varying thresholds. These approaches are inadequate for the study of heat waves however and we show how a recent pre-processing approach by Eastoe and Tawn (2009) can be used in conjunction with an innovative change-point analysis to model daily maximum temperature.  The model is then fitted to data from four U.S. cities and used to estimate the recurrence probabilities of runs over seasonally high temperatures.  We show that the probability of long and intense heat waves has increased considerably over 50 years.

Davis: The extremogram is a flexible quantitative tool that measures various types of extremal dependence in a stationary time series.  In many respects, the extremogram can be viewed as an extreme-value analogue of the autocorrelation function (ACF) for a time series.  Under mixing conditions, the asymptotic normality of the empirical extremogram was derived in Davis and Mikosch (2009).  Unfortunately, the limiting variance is a difficult quantity to estimate.  Instead we employ the stationary bootstrap to the empirical extremogram and establish that this resampling  procedure provides an asymptotically correct approximation to the central limit theorem.  This in turn can be used for constructing credible confidence bounds for the sample extremogram. The use of the stationary bootstrap for the extremogram is illustrated in a variety of real and simulated data sets. The cross-extremogram measures cross-sectional extremal dependence in multivariate time series. A measure of this dependence, especially left tail dependence, is of great importance in the calculation of portfolio risk.  We find that after devolatilizing  the marginal series, extremal dependence still remains, which suggests that the extremal dependence is not due solely to the heteroskedasticity in the stock returns process. However, for the univariate series, the filtering removes all extremal dependence.  Following Geman and Chang (2010), a return time extremogram which measures the waiting time between rare or extreme events in univariate and bivariate stationary time series is calculated. The return time extremogram suggests the existence of extremal clustering in the return times of extreme events for financial assets. The stationary bootstrap can again provide an asymptotically correct approximation to the central limit theorem and can be used for constructing credible confidence bounds for this return time extremogram.  (This is joint work with Thomas Mikosch and Ivor Cribben.)

This talk considers goodness of fit diagnostics for time-series data from processes approximately modeled by systems of nonlinear ordinary differential equations. In particular, we seek to determine three nested causes of lack of fit: (i) unmodeled stochastic forcing, (ii) mis-specified functional forms and (iii) mis-specified state variables. Testing lack of fit in differential equations is challenging since the model is expressed in terms of rates of change of the measured variables. Here, lack of fit is represented on the model scale via time-varying parameters. We develop tests for each of the three cases above through bootstrap and permutation methods.

A motivating example is presented from laboratory-based ecology in which algae are grown on nitrogen-rich medium and rotifers are introduced as a predator. The resulting data exhibit dynamics that do not correspond to those generated by classical ecological models. A hypothesized explanation is that more than one algal species are present in the chemostat. We assess the statistical evidence for this claim and show that while models incorporating multiple algal species provide better agreement with the data, their existence cannot be demonstrated without strong model assumptions. We conclude with an examination of the use of control theory to design inputs into dynamic systems to improve parameter estimation and power to detect missing components.

Du: Linear mixed-effects (LME) models are frequently used for modeling longitudinal data. One complicating factor in the analysis of such data is that samples are sometimes obtained from a population with significant underlying heterogeneity, which would be hard to capture by a single LME model. Such problems may be addressed by a finite mixture of linear mixed-effects (FMLME) models, which segments the population into subpopulations and models each subpopulation by a distinct LME model. Often in the initial stage of a study, a large number of predictors are introduced. However, their associations to the response variable vary from one component to another of the FMLME model. To enhance predictability and to obtain a parsimonious model, it is of great practical interest to identify the important effects, both fixed and random, in the model. Traditional variable selection techniques such as stepwise deletion and subset selection are computationally expensive as the number of covariates and components in the mixture model increases. In this talk, we introduce a penalized likelihood approach and propose a nested EM algorithm for efficient numerical computations. Our estimators are shown to possess desirable properties such as consistency, sparsity and asymptotic normality. We illustrate the performance of our method through simulations and a systemic sclerosis data example.

Harel: Multi-item, self-reported questionnaires are frequently used to measure aspects of health-related quality of life. Due to the latent nature of the constructs underlying these instruments, Item Response Theory models are often used to relate the observed item scores to the latent trait. However, there are no well-established guidelines for how to compare two such questionnaires. In this talk I will explore graphical methods for the comparison of multi-item self-reported questionnaires by using Partial Credit Models. This will be illustrated with the comparison of two fatigue questionnaires in patients with Systemic Sclerosis.

Ye Ting Du is an M.Sc. student in our department. He works with Abbas Khalili and Johanna Neslehova.

Daphna Harel is a Ph.D. candidate in our department. She works with Russell Steele.

Stute: The Poisson Process constitutes a well-known model for describing random events over time.  It has many applications in marketing research, insurance mathematics and finance.  Though it has been studied for decades not much is known how to check (in a non-asymptotic way) the validity of the Poisson Process.  In this talk we present the principal component decomposition of the Poisson Process which enables us to derive finite sample properties of associated goodness-of-fit tests.  In the first step we show that the Fourier-transforms of the components contain Bessel and Struve functions.  Inversion leads to densities which are modified arc sin distributions.

Blath: In this talk we consider properties of the so-called 'symbiotic branching model' describing the spatial evolution of two populations which can only reproduce if they are both present at the same location at the same time. We will put particular emphasis on the long-term dynamics of this population model. To this end, we consider a 'critical curve' separating the asymptotic behaviour of the moments of the symbiotic branching process into two qualitatively different regimes. From this result, various properties can be derived. For example, we improve a result of Etheridge and Fleischmann on the speed of the propagation of the area in which both species are simultaneously present.

McGillivray: In statistical applications of hidden Markov models (HMMs), one may have no knowledge of the number of hidden states (or order) of the model needed to be able to accurately represent the underlying process of the data. The problem of estimating the number of hidden states of the HMM is thus brought to the forefront. In this talk, we present a penalized quasi-likelihood approach for order estimation in HMMs which makes use of the fact that the marginal distribution of the observations from a HMM is a finite mixture model. The method starts with a HMM with a large number of states and obtains a model of lower order by clustering and combining similar states of the model through two penalty functions. We assess the performance of the new method via extensive simulation studies for Normal and Poisson HMMs.

Best: Statisticians are often faced with budget concerns when conducting studies. The collection of some covariates, such as genetic data, is very expensive. Other covariates, such as detailed histories, might be difficult or time-consuming to measure. This helped bring about the invention of the nested case-control study, and its more generalized version, risk-set sampled survival analysis. The literature has a good discussion of the properties of risk-set sampling in standard right-censored survival data. My interest is in extending the methods of risk-set sampling to left-truncated survival data. Left-truncated survival data arise in prevalent longitudinal studies. Since prevalent studies are easier and cheaper to conduct than incident studies, this extension is extremely practical and relevant. I will introduce the partial likelihood in this scenario.

In functional data analysis, as in its multivariate counterpart, estimates of the bivariate covariance kernel σ(s,t )and its inverse are useful for many things, and we need the inverse of a covariance matrix or kernel especially often.  However, the dimensionality of functional observations often exceeds the sample size available to estimate σ(s,t, and then the analogue S of the multivariate sample estimate is singular and non-invertible.   Even when this is not the case, the high dimensionality S often implies unacceptable sample variability and loss of degrees of freedom for model fitting.   The common practice of employing low-dimensional principal component approximations to σ(s,t) to achieve invertibility also raises serious issues.

This talk describes a functional estimate of σ(s,t) and its inverse defined by an expansion in terms of finite element basis functions.   This strategy permits the user to control the resolution of the estimate, its smoothness, and the time lag over which covariance may be nonzero.   It turns out that the matrix resulting from evaluating σ(s,t) at a discrete set of time points is almost never singular, and therefore enables the estimation of S and S-1 as a seamless single problem.  These estimates have many applications to classical statistical problems, such as discrete but unequally spaced time and spatial series, as well as to functional and longitudinal data analysis.

Jamshidian: Test of homogeneity of covariances (or homoscedasticity) among several groups has many applications in statistical analysis.  In the context of incomplete data analysis, tests of homoscedasticity among groups of cases with identical missing data patterns have been proposed to test whether data are missing completely at random (MCAR).  The proposed tests of MCAR often require large sample sizes n and/or large group sample sizes ni, and they usually fail when applied to non-normal data.  Hawkins (1981) proposed a test of multivariate normality and homoscedasticity that is an exact test for complete data when ni are small.  In this talk we present a modification of the Hawkins test for complete data to improve its performance, and extends its application to test of homoscedasticity and MCAR when data are multivariate normal and incomplete.  Moreover, we will show that the statistic used in the Hawkins test in conjunction with a nonparametric k-sample test can be used to obtain a nonparametric test of MCAR that works well for both normal and non-normal data.  It will be explained how a combination of the proposed normal-theory Hawkins test and the nonparametric test can be employed to test for homoscedasticity, MCAR, and multivariate normality.  We will present simulation studies that indicate the newly proposed tests generally outperform their existing competitors in terms of Type I error rejection rates.  Also, a power study of the proposed tests indicates good power.  The newly proposed methods use appropriate methods of imputations to impute missing data.  As such, multiple imputation is employed to assess the performance of our tests in light of imputation variability.  Moreover, examples will be presented where multiple imputation enables one to identify a group or groups whose covariance matrices differ from the majority of other groups.  Finally, an R-package that implements these new tests, called MissMech, will be briefly presented.

Chipman: In computer experiments, statistical models are commonly used as surrogates for slow-running codes.  In this talk, the usually ubiquitous Gaussian process models are nowhere to be seen, however.  Instead, an adaptive nonparametric regression model (BART) is used to deal with nonstationarities in the response surface.  By providing both point estimates and uncertainty bounds for prediction, BART provdes a basis for sequential design criteria to find optima with few function evaluations.  Similar ideas will also be illustrated in other active learning problems, such as identification of active compounds in drug discovery.

Li: The problem of selecting the most useful features from a great many (eg, thousands) of candidates arises in many areas of modern sciences. An interesting problem from genomic research is that, from thousands of genes that are active (expressed) in certain tissue cells, we want to find the genes that can be used to separate tissues of different classes (eg. cancer and normal). In this paper, we report a Bayesian logistic regression method based on heavytailed priors with moderately small degree freedom (such as 1) and small scale (such as 0.01), and using Gibbs sampling to do the computation. We show that it can distinctively separate a couple of useful features from a large number of useless ones, and discriminate many redundant correlated features. We also show that this method is very stable to the choice of scale. We apply our method to a microarray data set related to prostate cancer, and identify only 3 genes out of 6033 candidates that can separate cancer and normal tissues very well in leave-one-out cross-validation.

Rao: We derive the best predictive estimator (BPE) of the fixed parameters for a linear mixed model.  This leads to a new prediction procedure called observed best prediction (OBP), which is different from the empirical best linear unbiased prediction (EBLUP).  We show that BPE is more reasonable than the traditional estimators derived from estimation considerations, such as maximum likelihood (ML) and restricted maximum likelihood (REML), if the main interest is the prediction of the mixed effect.  We show how the OBP can significantly outperform the EBLUP in terms of mean squared prediction error (MSPE) if the underlying model is misspecified.  On the other hand, when the underlying model is correctly specified, the overall predictive performance of the OBP can be very similar to the EBLUP.  The well known Fay-Herriot small area model is used as an illustration of the methodology.  In addition, simulations and analysis of a data set on graft failure rates from kidney transplant operations will be used to show empirical performance. This is joint work with Jiming Jiang of UC-Davis and Thuan Nguyen of Oregon Health and Science University.