Dr Elena Kulinskaya, Statistical Advisory Service, Imperial College
Meta analysis on the right scale

This talk is about an approach to meta analysis and to statistical evidence developed jointly with Stephan Morgenthaler and Robert Staudte, and now written up in our book 'Meta Analysis: a guide to calibrating and combining statistical evidence'  to be published by Wiley very soon. The traditional ways of measuring evidence, in particular with p-values, are neither intuitive nor useful when it comes to making comparisons between experimental results, or when combining them. We measure evidence for an alternative hypothesis, not evidence against a null. To do this, we have in a sense adopted standardized scores for the calibration scale. Evidence for us is simply a transformation of a test statistic S to another one (called evidence T=T(S)) whose distribution is close to normal with variance 1, and whose mean grows from 0 with the parameter as it moves away from the null. Variance stabilization is used to arrive on this scale. For meta analysis the results from different studies are transformed to a common calibration scale, where it is simpler to combine and interpret them. 

Dr Richard Samworth, Statistical Laboratory, Cambridge
Computing the maximum likelihood estimator of a multidimensional log-concave density

We show that if $X_1,...,X_n$ are a random sample from a log-concave density $f$ in $\mathbb{R}^d$, then with probability one there exists a unique maximum likelihood estimator $\hat{f}_n$ of $f$.  The use of this estimator is attractive because, unlike kernel density estimation, the estimator is fully automatic, with no smoothing parameters to choose. The existence proof is non-constructive, however, and in practice we require an iterative algorithm that converges to the estimator.  By reformulating the problem as one of non-differentiable convex optimisation, we are able to exhibit such an algorithm.  We will also show how the method can be combined with the EM algorithm to fit finite mixtures of log-concave densities.  The talk will be illustrated with pictures from the R package LogConcDEAD.

This is joint work with Madeleine Cule (Cambridge), Bobby Gramacy (Cambridge) and Michael Stewart (University of Sydney).

Dr Robert Gramacy, Statistical Laboratory Cambridge
Importance Tempering
Simulated tempering (ST) is an established Markov Chain Monte Carlo (MCMC) methodology for sampling from a multimodal density pi(theta).  The technique involves introducing an auxiliary variable k taking values in a finite subset of [0,1] and indexing a set of tempered distributions, say pi_k(theta) = pi(theta)^k.  Small values of k encourage better mixing, but samples from pi are only obtained when the joint chain for (theta,k) reaches k=1. However, the entire chain can be used to estimate expectations under pi of functions of interest, provided that importance sampling (IS) weights are calculated.  Unfortunately this method, which we call importance tempering (IT),  has tended not work well in practice.  This is partly because the most immediately obvious implementation is naive and can lead to high variance estimators.  We derive a new optimal method for combining multiple IS estimators and prove that this optimal combination has a highly desirable property related to the notion of effective sample size.  The methodology is applied in two modelling scenarios requiring reversible-jump MCMC, where the naive approach to IT fails: model averaging in treed models, and model selection for mark--recapture data. 

Professor Simon Wood, University of Bath
Generalized Additive Smooth Modelling
Generalized Additive Models are GLMs in which the linear predictor is made up, partly, of a sum of smooth functions of predictor variables. I will talk about the penalized regression spline approach to GAM, as implemented, for example, in R package mgcv. In particular I will focus on two interesting aspects: low rank representation of smooth functions of several covariates and stable computation of both model coefficients and smoothing parameters. More than half the slides will have pictures.
 

Terry Speed, University of California, Berkeley
Alternative Splicing in Tumors: Detection and Interpretation
In this talk I will discuss using the Affymetrix GeneChip Human Exon and Human Gene 1.0 ST arrays for the detection of genes spliced differently in some tumors in comparison with others. I plan to begin by introducing the arrays and the expression data they produce. Next I will outline the way in which we use such data in our attempts to identify exon-tumor combinations exhibiting splicing patterns different from the majority. This will be illustrated by examples from publicly available tissue and mixture data. Then I will briefly discuss some of the additional issues which arise when we seek to enumerate such alternative splicing patterns on a genome-wide scale. Finally, I will exhibit some of the results we have found applying these methods to glioblastoma tissue samples collected as part of The Cancer Genome Atlas (TCGA) project. (This is joint work with ELizabeth Purdom, Mark Robinson, Ken Simpson, and members of the Berkeley Cancer Genome Center.)

Alexey Koloydenko & Juri Lember (Joint Talk), University of Nottingham
Adjusted Viterbi Training for Hidden Markov Models
The Expectation Maximisation (EM) procedure is a principal tool for parameter estimation in hidden Markov models (HMMs). However, in applications EM is sometimes replaced by Viterbi training, or extraction, (VT). VT is computationally less intensive and more stable, and has more of an intuitive appeal, but VT estimation is biased and does not satisfy the following fixed point property: Hypothetically, given an infinitely large sample and initialized to the true parameters, VT will generally move away from the initial values. We propose adjusted Viterbi training (VA), a new method to restore the fixed point property and thus alleviate the overall imprecision of the VT estimators, while preserving the computational advantages of the baseline VT algorithm. Simulations show that VA indeed improves estimation precision appreciably in both the special case of mixture models and more general HMMs.

We will discuss the main idea of the adjusted Viterbi training. This will also touch on tools developed specifically to analyze asymptotic behaviour of maximum a posteriori (MAP) hidden paths, also known as Viterbi alignments. Our VA correction is analytic and relies on infinite Viterbi alignments and associated limiting probability distributions. While explicit in the special case of mixture models, these limiting measures are not obvious to exist for more general HMMs. We will conclude by presenting a result that under certain mild conditions, general (discrete time) HMMs do possess the limiting distributions required for the construction of VA. 

Prof Antony Pettitt, Lancaster University
Statistical inference for assessing infection control measures for the transmission of pathogens in hospitals
Patients can acquire infections from pathogen sources within hospitals and certain pathogens appear to be found mainly in hospitals.  Methicillin-resistant Staphylococcus Aureus (MRSA) is an example of a hospital acquired pathogen that continues to be of particular concern to patients and hospital management.  Patients infected with MRSA can develop severe infections which lead to increased patient morbidity and costs for the hospital.  Pathogen transmission to a patient can occur via health-care workers that do not regularly perform hand hygiene.  Infection control measures that can be considered include isolation for colonised patients and improved hand hygiene for health-care workers.
The talk develops statistical methods and models in order to assess the effectiveness of the two control measures (i) isolation and (ii) improved hand hygiene.  For isolation, data from a prospective study carried out in a London hospital is considered and statistical models based on detailed patient data are used to determine the effectiveness of isolation.  The approach is Bayesian. For hand hygiene it is not possible, for ethical and practical reasons, to carry out a prospective study to investigate various levels of hand hygiene.  Instead hand hygiene effects are investigated by simulation using parameter values estimated from data on health-care worker hand hygiene and weekly colonisation incidence collected from a hospital ward in Brisbane.  The approach uses profile likelihoods.Both approaches involve transmission parameters where there is little information available and contrasting compromises have to be made. Conclusions about the effectiveness of the two infection control measures will be discussed. The talk involves collaborative work with Marie Forrester, Emma McBryde, Chris Drovandi, Ben Cooper, Gavin Gibson, Sean McElwain.

Alastair Young, Imperial College London
Objective Bayes and Conditional Inference
In Bayesian parametric inference, in the absence of subjective prior information about the parameter of interest, it is natural to consider use of an objective prior which leads to posterior probability quantiles which have, at least to some higher order approximation in terms of the sample size, the correct frequentist interpretation. Such priors are termed probability matching priors. In many circumstances, however, the appropriate frequentist inference is a conditional one. The key contexts involve inference in multi-parameter exponential families, where conditioning eliminates the nuisance parameter, and models which admit ancillary statistics, where conditioning on the ancillary is indicated by the conditionality principle of inference. In this talk, we consider conditions on the prior under which posterior quantiles have, to high order, the correct conditional frequentist interpretation. The key motivation for the work is that the conceptually simple objective Bayes route may provide accurate approximation to more complicated frequentist procedures. We focus on the exponential family context, where it turns out that the condition for higher order conditional frequentist accuracy reduces to a condition on the model, not the prior: when the condition is satisfied, as it is in many key situations, any first order probability matching prior (in the unconditional sense) automatically yields higher order conditional probability matching. We provide numerical illustrations, discuss the relationship between the objective Bayes inference and the parametric bootstrap, as well as giving a brief account appropriate to the ancillary statistic context, where conditional frequentist probability matching is more difficult. [This is joint work with Tom DiCiccio, Cornell].

Thomas Richardson, University of Washington
Estimation of the Relative Risk and Risk Difference

I will first review well-known differences between odds ratios, relative risks and risk differences. These results motivate the development of methods, analogous to logistic regression, for estimating the latter two quantities. I will then describe simple parametrizations that facilitate maximum-likelihood estimation of the relative risk and risk-difference. Further, these parametrizations allow for doubly-robust g-estimation of the relative risk and risk difference. (Joint work with James Robins, Harvard School of Public Health).

Geoff McLachlan, University of Queensland
On Simple Mixture Models in High-Dimensional Testing in the Detection of Differentially Expressed Genes

An important problem in microarray experiments is the detection of genes that are differentially expresse in a given number of classes. As there are usually thousands of genes to be considered simultaneously, one encounters high-dimensional testing problems. We provide a straightforward and easily implemented method for estimating the posterior probability that an individual gene is null (not differentially expressed).  The problem can be expressed in a two-component mixture framework.  Current methods of implementing this approach either have some limitations due to the minimal assumptions made or with the computationally intensive nature of more specific assumptions. By converting to a z-score the value of the test statistic used to test the significance of each gene, we propose a simple two-component normal mixture that models adequately the distribution of this score. The approach provides an estimate of the local false discovery rate (FDR) for each gene, which is taken to be the posterior probability that the gene is null.  Genes with the local FDR less than a specified threshold C are taken to be differentially expressed.  For a given C, this approach also provides estimates of the implied overall errors such as the (global) FDR and the false negative/positive rates.

Thomas Nichols, GlaxoSmithKline Clinical Imaging Centre
Cluster mass inference - a new random field theory method for neuroimaging
Functional and structural brain imaging analyses typically fit univariate models at each volume element, or voxel, producing images of T-statistics. These statistics images must be threshold or assessed insome way to identify significant regions while controlling some specified false positive rate. A widely used method for assessing signficance is the cluster size test, where an arbitary threshold is applied to the statistic image and contiguous suprathreshold voxels are grouped into clusters. The size of a cluster forms a test statistic on the null hypothesis of no activation anywhere within the cluster, and P-values are found with Random Field Theory (RFT). Various authors have reported on improved sensitivity with a modified version of the cluster size test, the cluster mass test. Cluster mass is the integral of suprathreshold intensities within the cluster, and captures information about both the extent and magnitude of the effect. However, all previous work has relied on permutation inference as no distributional results have been available. I will show recent work on deriving a null distribution for cluster mass using RFT. Using the approximate parabolic shape about local maxima and distributional results for the cluster extent and peak curvature, we produce a joint distribution for mass and peak height which is marginalized to produce a P-value for the mass statistic. We show results on simulated and real data which demonstrate the tests validity and power. 

Ian Dryden, University of Nottingham
Non-Euclidean statistics for covariance matrices, with applications to diffusion tensor imaging
The statistical analysis of covariance matrices occurs in many important applications, e.g. in diffusion tensor imaging or longitudinal data analysis. Methodology is discussed for estimating covariance matrices which takes into account the non-Euclidean nature of the space of positive semi-definite symmetric matrices. We make connections with the use of Procrustes methods in shape analysis, and comparisons are made with other estimation techniques, including using the matrix logarithm, Riemannian metric, matrix square root and Cholesky decomposition. Our main application will be diffusion tensor imaging which is used to study the fibre structure in the white matter of the brain. Diffusion weighted MR images involve applying a set of gradients in a design of directions, and the recorded data are related to the Fourier transforms of the displacement of water molecules at each voxel. Under a multivariate Gaussian diffusion model a single tensor (3 x 3 covariance matrix) is fitted at each voxel. We discuss the statistical analysis of diffusion tensors, including construction of means, spatial interpolation, geoedsics, and principal components analysis. This is joint work with Alexey Koloydenko and Diwei Zhou.

Key Names: Pearson; Fisher; Neyman and Pearson; Kullback and Leibler; Cochran, Mantel, and Heanszel; Goodman.

Co-authors: John A. D. Aston, Jiun W. Liou, and Michelle Liou.

3rd JOINT WARWICK-OXFORD STATISTICS SEMINAR

2:30 – 5:00 pm  at The Mary Ogilvie Lecture Theatre,

St. Anne’s College, University of Oxford

2:30 p.m.

Speaker 1:  Julian Besag (University of Bath, University of Washington, Seattle)

Title:  Continuum limits of Gaussian Markov random fields: resolving the conflict with geostatistics

Abstract:  For more than 30 years, Markov random fields (equivalently, graphical models with undirected edges) have been used with some success to account for spatial variation in data. Applications include agricultural crop trials, geographical epidemiology, medical imaging, remote sensing, astronomy, and microarrays. Almost all of the examples involve (hidden) Gaussian MRF formulations.

MRFs refer to fixed regular or irregular discrete lattices or arrays and questions arise regarding inconsistencies between MRFs specified at differing scales, especially for regional data. Ideally, one would often prefer an underlying continuum formulation, as in geostatistics, which can then be integrated to the regions of interest. However, limiting continuum versions of MRFs, as lattice spacing decreases, proved elusive until recently.

This talk briefly presents some motivating examples and shows that limiting processes indeed exist but are defined on arbitrary regions of the plane rather than pointwise. Especially common is the conformally invariant de Wijs process, which coincidentally was used originally by mining engineers but which became unfashionable as geostatistics developed. Convergence is generally very fast. The de Wijs process is also shown to be a natural extension of Brownian motion to the plane. Other processes, including the thin-plate spline, can be derived as limits of MRFs. The talk closes by briefly discussing data analysis.

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3:30 to 4.00 - Tea, Coffee and biscuits in foyer outside lecture theatre

4:00 p.m.

Speaker 2:   Susan Lewis (University of Southampton, UK)

Title:  Screening experiments

Abstract:  Discovery and development in science and industry often involves investigation of many features or factors that could potentially affect the performance of a product or process. In factor screening, designed experiments are used to identify efficiently the few features that influence key properties of the system under study. A brief overview of this broad area will be presented. This will be followed by discussion of a variety of methods with particular emphasis on industrial screening. Ideas will be motivated and illustrated through examples, including a case study from the automotive industry.

Prof Ming-Yen Cheng UCL
Generalized Multiparameter Likelihood Models
Multiparameter likelihood models (MLM) with multiple covariates have a wide range of applications. However, they encounter the ``curse of dimensionality" problem when the dimension of the covariates is large. We develop a generalized multiparameter likelihood model that copes with multiple covariates and adapts to dynamic structural changes well. And it includes some popular models, such as the partially linear and varying coefficient models, as special cases. When the model is fixed, a simple and effective two-step method is developed to estimate both the parametric and the nonparametric components.  The proposed estimator of the parametric component has the root-n convergence rate, and the estimator of the nonparametric component enjoys an adaptivity property. A data-driven procedure is suggested to select the bandwidths involved. Also proposed is a new initial estimator in profile likelihood estimation of the parametric part to ensure stability of the approach in general settings. We further develop an automatic procedure to identify constant parameters in the underlying model. A simulation study and an application to the  infant mortality data of China are given to demonstrate performance of the proposed methods.

James Cussens, University of York

 Model Selection using weighted MAX-SAT Solvers

This talk concerns encoding problems of statistical model selection in such a way that "weighted MAX-SAT solvers" can be used to search for the   'best' model. In this approach each model is (implicitly) encoded as a joint instantiation of n binary variables. Each of these binary variables encodes the truth/falsity of a logical proposition and weighted logical formulae are used to represent the model selection problem. Once encoded in this way we can tap into years of research and use any of the state-of-the-art solvers to conduct the search. In the talk I will show how to use this approach when the model class is that of Bayesian networks, and also for clustering. I will briefly touch on related methods which permit the calculation of marginal probabilities in discrete distributions.

Prof Sylvia Fruehwirth-Schnatter, Johannes Kepler University, Austria

Bayesian Variable Selection Problems for State Space and other Latent Variable Models

Latent Variable models are widely used in applied statistics and econometrics to deal with data where the underlying processes change either over time or between units.  Whereas estimation of these models is well understood, model selection problems are rarely studies, because such an issue usually leads to a non-regular testing problem.

Bayesian statistics offers in principle a framework for model selection even for non-regular problems, as is shortly discussed in the first part of the talk.  The practical application of the Bayesian approach, however, proves to be challenging and numerical technique like marginal likelihoods, RJMCMC or the variable selection approach have to be used.

The main contribution of this talk is to demonstrate that the Bayesian variable selection approach is useful far beyond the common problem of selecting covariates in a classical regression model and may be extended to deal model selection problems in various latent variable models.  First, it is extended to testing for the presence of unobserved heterogeneity in random effects models.  Second, dynamic regression models are considered, where one has to choose between fixed and random coefficients.  Finally, the variable selection approach is extended to state space models, where testing problems like discriminating between models with a stochastic trend, a deterministic trend and a model without trend arise.

Case studies from marketing, economics and finance will be considered for illustration.

Stein estimators and SURE shrinkage estimation for Gaussian processes using the Malliavin calculus

In this talk we present a construction of Stein type estimators for the drift of Gaussian processes using the Malliavin integration by parts formula. Then we construct an estimation and de-noising procedure for an input signal perturbed by a continuous-time Gaussian noise using the local and occupation times of Gaussian processes. This talk is based on joint works with Nicolas Privault.

Dr Laura Sangalli, Politecnico di Milano

Title : Efficient estimation of curves in more than one dimension by free-knot regression splines, with applications to the analysis of 3D cerebral vascular geometries.

Abstract : We deal with the problem of efficiently estimating a 3D curve and its derivatives, starting from a discrete and noisy observation of the curve.  We develop a regression technique based on free-knot splines, ie. regression splines where the number and position of knots are not fixed in advance but chosen in a way to minimize a penalized sum of squared errors criterion.  We thoroughly compare this technique to a classical regression method, local polynomial smoothing, via simulation studies and application to the analysis of inner carotid artery centerlines (AneuRisk Project dataset).  We show that 3D free-knot regression splines yield more accurate and efficient estimates.

 Joint work with Piercesare Secchi and Simone Vantini.