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Professor Matt Wand

Biography

Professor Matt Wand is a Distinguished Professor of Statistics at University of Technology, Sydney.

His latest research outputs are available on the website: http://matt-wand.utsacademics.info

He has held faculty appointments at Harvard University, Rice University, Texas A&M University, University of New South Wales and University of Wollongong. In 2008 Professor Wand became an elected Fellow of the Australian Academy of Science. He also has been awarded the two Australian Academy of Science honorific awards for statistical research: the Moran Medal in 1997 for outstanding research by scientists under the age of 40 and the Hannan Medal in 2013 for career research in statistical science. In 2013 he was awarded the University of Technology Sydney, Chancellor's Medal for Exceptional Research. He received the 2013 Pitman Medal from the Statistical Society of Australia in recognition of outstanding achievement in, and contribution to, the discipline of Statistics. Professor Wand is an elected fellow of the American Statistical Association and the Institute of Mathematical Statistics.

Professor Wand has co-authored two books and more than 100 papers in statistics journals. He has six packages in the R language on the Comprehensive R Archive Network.

In 2002 Professor Wand was ranked 23 among highly cited authors in mathematics and statistics for the period 1991–2001. He is also a member of the ‘ISI Highly Cited Researchers’ list. Since 2000 Professor Wand has been principal investigator on seven major grants. A recent one, an Australian Research Council Discovery Project, is titled ‘Semiparametric Regression for Streaming Data’ and will run for the years 2015–2017. Another is the `Centre of Excellence for Mathematical and Statistical Frontiers of Big Data, Big Models, New Insights' and is running during 2014-2020.

For more information visit his personal website matt-wand.utsacademics.info.

Professional

Matt serves as an associate editor for the Statistics journal: Australian and New Zealand Journal of Statistics.

He has previously served as an associate editor for the Journal of the American Statistical Association, Biometrika and Statistica Sinica.

He also participates in committee work within the Australian Academy of Science.

Image of Matt Wand
Distinguished Professor, School of Mathematical and Physical Sciences
B Mathematics (Hons Class 1), Ph D
 
Phone
+61 2 9514 2240

Research Interests

Professor Wand is chiefly interested in the development of statistical methodology for finding useful structure in large multivariate data sets.

Currently, Matt’s specific interests include: variational approximate methods, statistical methods for streaming data, generalised linear mixed models, semiparametric regression, spatial statistics, multivariate density estimation and feature significance.

He is also very interested in Statistical Computing and contributes to the field's main software repository — the ‘Comprehensive R Archive Network’.

Recent research by Wand and co-authors on real-time semiparametric regression is described on the Real-time Semiparametric Regression website (opens an external site).

Can supervise: Yes
Matt Wand is currently supervising: Cathy (Yuen Yi) Lee, PhD candidate; Andy (Sang Il) Kim, PhD candidate; Tui Nolan Masters degree candidate.

35393 Seminar (Statistics)

Books

Ruppert, D., Wand, M. & Carroll, R.J. 2003, Semiparametric Regression, 1, Cambridge University Press, New York.
Wand, M. & Jones, M.C. 1995, Kernel Smoothing, First, Chapman and Hall, London.

Conferences

Neville, S.E. & Wand, M.P. 2011, 'Generalised Extreme Value geoadditive model analysis via variational Bayes', Procedia Environmental Sciences, pp. 8-13.
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We devise a variational Bayes algorithm for fast approximate inference in Bayesian Generalized Extreme Value additive model analysis. Such models are useful for flexibly assessing the impact of continuous predictor variables on sample extremes. The new methodology allows large Bayesian models to be fitted and assessed without the significant computing costs of Monte Carlo methods. © 2010 Published by Elsevier Ltd. Ltd.

Journal articles

Dubossarsky, E., Friedman, J.H., Ormerod, J.T. & Wand, M.P. 2016, 'Wavelet-based gradient boosting', Statistics and Computing, vol. 26, no. 1-2, pp. 93-105.
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© 2014, Springer Science+Business Media New York. A new data science tool named wavelet-based gradient boosting is proposed and tested. The approach is special case of componentwise linear least squares gradient boosting, and involves wavelet functions of the original predictors. Wavelet-based gradient boosting takes advantages of the approximate $$\ell _1$$?1 penalization induced by gradient boosting to give appropriate penalized additive fits. The method is readily implemented in R and produces parsimonious and interpretable regression fits and classifiers.
Lee, C.Y.Y. & Wand, M.P. 2016, 'Variational methods for fitting complex Bayesian mixed effects models to health data', Statistics in Medicine, vol. 35, no. 2, pp. 165-188.
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© 2016 John Wiley & Sons, Ltd. We consider approximate inference methods for Bayesian inference to longitudinal and multilevel data within the context of health science studies. The complexity of these grouped data often necessitates the use of sophisticated statistical models. However, the large size of these data can pose significant challenges for model fitting in terms of computational speed and memory storage. Our methodology is motivated by a study that examines trends in cesarean section rates in the largest state of Australia, New South Wales, between 1994 and 2010. We propose a group-specific curve model that encapsulates the complex nonlinear features of the overall and hospital-specific trends in cesarean section rates while taking into account hospital variability over time. We use penalized spline-based smooth functions that represent trends and implement a fully mean field variational Bayes approach to model fitting. Our mean field variational Bayes algorithms allow a fast (up to the order of thousands) and streamlined analytical approximate inference for complex mixed effects models, with minor degradation in accuracy compared with the standard Markov chain Monte Carlo methods.
Menictas, M. & Wand, M.P. 2015, 'Variational Inference for Heteroscedastic Semiparametric Regression', Australian and New Zealand Journal of Statistics, vol. 57, no. 1, pp. 119-138.
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© 2015 Australian Statistical Publishing Association Inc. We develop fast mean field variational methodology for Bayesian heteroscedastic semiparametric regression, in which both the mean and variance are smooth, but otherwise arbitrary, functions of the predictors. Our resulting algorithms are purely algebraic, devoid of numerical integration and Monte Carlo sampling. The locality property of mean field variational Bayes implies that the methodology also applies to larger models possessing variance function components. Simulation studies indicate good to excellent accuracy, and considerable time savings compared with Markov chain Monte Carlo. We also provide some illustrations from applications.
Kayal, M., Vercelloni, J., Wand, M.P. & Adjeroud, M. 2015, 'Searching for the best bet in life-strategy: A quantitative approach to individual performance and population dynamics in reef-building corals', Ecological Complexity, vol. 23, pp. 73-84.
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© 2015 Elsevier B.V. Ecological signs of Earth's biosphere forewarn an alarming trajectory towards a global mass-extinction. Assessing species performance and susceptibilities to decline is essential to comprehend and reverse this trend. Yet it is challenging, given difficulties associated with quantifying individual and population processes that are variable across time, space, and life-stages. We describe a new approach to estimating and comparing species performances that combines empirical data, a novel theoretical consideration of population dynamics, and modern statistics. Our approach allows for a more realistic continuous representation of individual performances along development stages while taking into account non-linearity, and natural variability as captured by spatio-temporally replicated observations. We illustrate its application in a coral meta-assemblage composed of populations of the three major reef-building taxa Acropora, Pocillopora, Porites. Using a unique set of highly replicated observations of individual coral dynamics under various environmental conditions, we show how taxa differ in their investment in recruitment and size-specific aptitude for growth and survival, notably through different use of clonal shrinkage, fragmentation, fission, and fusion processes. Our results reveal contrasting life-history trade-offs among taxa which, along with differing patterns of density-dependent recruitment, modulate species responses to decline. These differences in coral life history traits reflect opposing life-strategies, imply regulation at differing life-stages, and explain divergence in species trajectories. Our findings indicate a high potential for resilience in Pocillopora and Porites populations, thanks respectively to a sustained recruitment that promotes demographic elasticity through replacement of individuals, and a steady resistance to mortality which confers persistence through lingering of individuals. Resilience in Acropora appear...
Luts, J., Broderick, T. & Wand, M.P. 2014, 'Real-Time Semiparametric Regression', Journal of Computational and Graphical Statistics, vol. 23, no. 3, pp. 589-615.
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© 2014 American Statistical Association, Institute of Mathematical Statistics, and Interface Foundation of North America. We develop algorithms for performing semiparametric regression analysis in real time, with data processed as it is collected and made immediately available via modern telecommunications technologies. Our definition of semiparametric regression is quite broad and includes, as special cases, generalized linear mixed models, generalized additive models, geostatistical models, wavelet nonparametric regression models and their various combinations. Fast updating of regression fits is achieved by couching semiparametric regression into a Bayesian hierarchical model or, equivalently, graphical model framework and employing online mean field variational ideas. An Internet site attached to this article, realtime-semiparametric-regression.net, illustrates the methodology for continually arriving stock market, real estate, and airline data. Flexible real-time analyses based on increasingly ubiquitous streaming data sources stand to benefit. This article has online supplementary material.
Neville, S.E., Ormerod, J.T. & Wand, M.P. 2014, 'Mean field variational bayes for continuous sparse signal shrinkage: Pitfalls and remedies', Electronic Journal of Statistics, vol. 8, pp. 1113-1151.
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© 2014, Institute of Mathematical Statistics. All rights received. We investigate mean field variational approximate Bayesian inference for models that use continuous distributions, Horseshoe, Negative-Exponential-Gamma and Generalized Double Pareto, for sparse signal shrinkage. Our principal finding is that the most natural, and simplest, mean field variational Bayes algorithm can perform quite poorly due to posterior dependence among auxiliary variables. More sophisticated algorithms, based on special functions, are shown to be superior. Continued fraction approximations via Lentz's Algorithm are developed to make the algorithms practical.
Wand, M.P. 2014, 'Fully simplified multivariate normal updates in non-conjugate variational message passing', Journal of Machine Learning Research, vol. 15, pp. 1351-1369.
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Fully simplified expressions for Multivariate Normal updates in non-conjugate variational message passing approximate inference schemes are obtained. The simplicity of these expressions means that the updates can be achieved very eficiently. Since the Multivariate Normal family is the most common for approximating the joint posterior density function of a continuous parameter vector, these fully simplified updates are of great practical benefit. © 2014 Matt P. Wand.
Gloag, E.S., Turnbull, L., Huang, A., Vallotton, P., Wang, H., Nolan, L.M., Mililli, L., Hunt, C., Lu, J., Osvath, S.R., Monahan, L.G., Cavaliere, R., Charles, I.G., Wand, M.P., Gee, M.L., Prabhakar, R. & Whitchurch, C.B. 2013, 'Self-organization of bacterial biofilms is facilitated by extracellular DNA', Proceedings of the National Academy of Sciences of the United States of America, vol. 110, no. 28, pp. 11541-11546.
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Twitching motility-mediated biofilm expansion is a complex, multicellular behavior that enables the active colonization of surfaces by many species of bacteria. In this study we have explored the emergence of intricate network patterns of interconnected trails that form in actively expanding biofilms of Pseudomonas aeruginosa. We have used high-resolution, phase-contrast time-lapse microscopy and developed sophisticated computer vision algorithms to track and analyze individual cell movements during expansion of P. aeruginosa biofilms. We have also used atomic force microscopy to examine the topography of the substrate underneath the expanding biofilm. Our analyses reveal that at the leading edge of the biofilm, highly coherent groups of bacteria migrate across the surface of the semisolid media and in doing so create furrows along which following cells preferentially migrate. This leads to the emergence of a network of trails that guide mass transit toward the leading edges of the biofilm. We have also determined that extracellular DNA (eDNA) facilitates efficient traffic flow throughout the furrow network by maintaining coherent cell alignments, thereby avoiding traffic jams and ensuring an efficient supply of cells to the migrating front. Our analyses reveal that eDNA also coordinates the movements of cells in the leading edge vanguard rafts and is required for the assembly of cells into the "bulldozer" aggregates that forge the interconnecting furrows. Our observations have revealed that large-scale self-organization of cells in actively expanding biofilms of P. aeruginosa occurs through construction of an intricate network of furrows that is facilitated by eDNA.
Huang, A. & Wand, M.P. 2013, 'Simple marginally noninformative prior distributions for covariance matrices', Bayesian Analysis, vol. 8, no. 2, pp. 439-452.
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A family of prior distributions for covariance matrices is studied. Members of the family possess the attractive property of all standard deviation and correlation parameters being marginally noninformative for particular hyper-parameter choices. Moreover, the family is quite simple and, for approximate Bayesian inference techniques such as Markov chain Monte Carlo and mean eld variational Bayes, has tractability on par with the Inverse-Wishart conjugate fam-ily of prior distributions. A simulation study shows that the new prior distributions can lead to more accurate sparse covariance matrix estimation. © 2013 International Society for Bayesian Analysis.
Menictas, M. & Wand, M.P. 2013, 'Variational inference for marginal longitudinal semiparametric regression', Stat, vol. 2, no. 1, pp. 61-71.
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© 2013 John Wiley & Sons Ltd. We derive a variational inference procedure for approximate Bayesian inference in marginal longitudinal semiparametric regression. Fitting and inference is much faster than existing Markov chain Monte Carlo approaches. Numerical studies indicate that the new methodology is very accurate for the class of models under consideration.
Pham, T.H., Ormerod, J.T. & Wand, M.P. 2013, 'Mean field variational Bayesian inference for nonparametric regression with measurement error', Computational Statistics and Data Analysis, vol. 68, pp. 375-387.
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A fast mean field variational Bayes (MFVB) approach to nonparametric regression when the predictors are subject to classical measurement error is investigated. It is shown that the use of such technology to the measurement error setting achieves reasonable accuracy. In tandem with the methodological development, a customized Markov chain Monte Carlo method is developed to facilitate the evaluation of accuracy of the MFVB method. © 2013 Published by Elsevier B.V. All rights reserved.
Ormerod, J.T. & Wand, M.P. 2012, 'Gaussian variational approximate inference for generalized linear mixed models', Journal of Computational and Graphical Statistics, vol. 21, no. 1, pp. 2-17.
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Variational approximation methods have become a mainstay of contemporary machine learning methodology, but currently have little presence in statistics. We devise an effective variational approximation strategy for fitting generalized linear mixed models (GLMMs) appropriate for grouped data. It involves Gaussian approximation to the distributions of random effects vectors, conditional on the responses. We show that Gaussian variational approximation is a relatively simple and natural alternative to Laplace approximation for fast, non-Monte Carlo, GLMM analysis. Numerical studies show Gaussian variational approximation to be very accurate in grouped data GLMM contexts. Finally, we point to some recent theory on consistency of Gaussian variational approximation in this context. Supplemental materials are available online. © 2012 American Statistical Association, Institute of Mathematical Statistics, and Interface Foundation of North America.
Wand, M.P. & Ormerod, J.T. 2012, 'Continued fraction enhancement of Bayesian computing', Stat, vol. 1, no. 1, pp. 31-41.
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© 2012 John Wiley & Sons, Ltd. The agéd number theoretic concept of continued fractions can enhance certain Bayesian computations. The crux of this claim is due to continued fraction representations of numerically challenging special function ratios that arise in Bayesian computing. Continued fraction approximation via Lentz's Algorithm often leads to efficient and stable computation of such quantities.
Ormerod, J.T. & Wand, M.P. 2012, 'Comment', Technometrics, vol. 54, no. 3, pp. 233-236.
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Hall, P., Ormerod, J.T. & Wand, M. 2011, 'Theory of Gaussian variational approximation for a Poisson mixed model', Statistica Sinica, vol. 21, no. 1, Special Issue, pp. 369-389.
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Likelihood-based inference for the parameters of generalized linear mixed models is hindered by the presence of intractable integrals. Gaussian variational approximation provides a fast and effective means of approximate inference. We provide some theory for this type of approximation for a simple Poisson mixed model. In particular, we establish consistency at rate m(-1/2) + n(-1), where in is the number of groups and n is the number of repeated measurements.
Chacón, J.E., Duong, T. & Wand, M.P. 2011, 'Asymptotics for general multivariate kernel density derivative estimators', Statistica Sinica, vol. 21, no. 2, pp. 807-840.
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We investigate kernel estimators of multivariate density derivative functions using general (or unconstrained) bandwidth matrix selectors. These density derivative estimators have been relatively less well researched than their density estimator analogues. A major obstacle for progress has been the intractability of the matrix analysis when treating higher order multivariate derivatives. With an alternative vectorization of these higher order derivatives, mathematical intractabilities are surmounted in an elegant and unified framework. The finite sample and asymptotic analysis of squared errors for density estimators are generalized to density derivative estimators. Moreover, we are able to exhibit a closed form expression for a normal scale bandwidth matrix for density derivative estimators. These normal scale bandwidths are employed in a numerical study to demonstrate the gain in performance of unconstrained selectors over their constrained counterparts.
Goldsmith, J., Wand, M.P. & Crainiceanu, C. 2011, 'Functional regression via variational bayes', Electronic Journal of Statistics, vol. 5, pp. 572-602.
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We introduce variational Bayes methods for fast approximate inference in functional regression analysis. Both the standard cross-sectional and the increasingly common longitudinal settings are treated. The method- ology allows Bayesian functional regression analyses to be conducted with- out the computational overhead of Monte Carlo methods. Confidence in- tervals of the model parameters are obtained both using the approximate variational approach and nonparametric resampling of clusters. The latter approach is possible because our variational Bayes functional regression ap- proach is computationally efficient. A simulation study indicates that varia- tional Bayes is highly accurate in estimating the parameters of interest and in approximating the Markov chain Monte Carlo-sampled joint posterior distribution of the model parameters. The methods apply generally, but are motivated by a longitudinal neuroimaging study of multiple sclerosis patients. Code used in simulations is made available as a web-supplement.
Faes, C., Ormerod, J.T. & Wand, M.P. 2011, 'Variational Bayesian inference for parametric and nonparametric regression with missing data', Journal of the American Statistical Association, vol. 106, no. 495, pp. 959-971.
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Bayesian hierarchical models are attractive structures for conducting regression analyses when the data are subject to missingness. However, the requisite probability calculus is challenging andMonte Carlo methods typically are employed.We develop an alternative approach based on deterministic variational Bayes approximations. Both parametric and nonparametric regression are considered. Attention is restricted to the more challenging case of missing predictor data.We demonstrate that variational Bayes can achieve good accuracy, but with considerably less computational overhead. The main ramification is fast approximate Bayesian inference in parametric and nonparametric regression models with missing data. Supplemental materials accompany the online version of this article. © 2011 American Statistical Association.
Wang, S.S.J. & Wand, M.P. 2011, 'Using infer.NET for statistical analyses', American Statistician, vol. 65, no. 2, pp. 115-126.
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We demonstrate and critique the new Bayesian inference package Infer.NET in terms of its capacity for statistical analyses. Infer.NET differs from the well-known BUGS Bayesian inference packages in that its main engine is the variational Bayes family of deterministic approximation algorithms rather than Markov chain Monte Carlo. The underlying rationale is that such deterministic algorithms can handle bigger problems due to their increased speed, despite some loss of accuracy. We find that Infer.NET is a well-designed computational framework and offers significant speed advantages over BUGS. Nevertheless, the current release is limited in terms of the breadth of models it can handle, and its inference is sometimes inaccurate. Supplemental materials accompany the online version of this article. © 2011 American Statistical Association.
Wand, M.P., Ormerod, J.T., Padoan, S.A. & Frührwirth, R. 2011, 'Mean field variational bayes for elaborate distributions', Bayesian Analysis, vol. 6, no. 4, pp. 847-900.
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We develop strategies for mean field variational Bayes approximate inference for Bayesian hierarchical models containing elaborate distributions. We loosely define elaborate distributions to be those having more complicated forms compared with common distributions such as those in the Normal and Gamma families. Examples are Asymmetric Laplace, Skew Normal and Generalized Extreme Value distributions. Such models suffer from the difficulty that the parameter updates do not admit closed form solutions. We circumvent this problem through a combination of (a) specially tailored auxiliary variables, (b) univariate quadrature schemes and (c) finite mixture approximations of troublesome density functions. An accuracy assessment is conducted and the new methodology is illustrated in an application. © 2011 International Society for Bayesian Analysis.
Wand, M.P. & Ormerod, J.T. 2011, 'Penalized wavelets: Embedding wavelets into semiparametric regression', Electronic Journal of Statistics, vol. 5, pp. 1654-1717.
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We introduce the concept of penalized wavelets to facilitate seamless embedding of wavelets into semiparametric regression models. In particular, we show that penalized wavelets are analogous to penalized splines; the latter being the established approach to function estimation in semiparametric regression. They differ only in the type of penalization that is appropriate. This fact is not borne out by the existing wavelet literature, where the regression modelling and fitting issues are overshadowed by computational issues such as efficiency gains afforded by the Discrete Wavelet Transform and partially obscured by a tendency to work in the wavelet coefficient space. With penalized wavelet structure in place, we then show that fitting and inference can be achieved via the same general approaches used for penalized splines: penalized least squares, maximum likelihood and best prediction within a frequentist mixed model framework, and Markov chain Monte Carlo and mean field variational Bayes within a Bayesian framework. Penalized wavelets are also shown have a close relationship with wide data ("p ? ? n") regression and benefit from ongoing research on that topic.
Neville, S.E., Palmer, M.J. & Wand, M.P. 2011, 'Generalized extreme value additive model analysis via mean field variational bayes', Australian and New Zealand Journal of Statistics, vol. 53, no. 3, pp. 305-330.
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We develop Mean Field Variational Bayes methodology for fast approximate inference in Bayesian Generalized Extreme Value additive model analysis. Such models are useful for flexibly assessing the impact of continuous predictor variables on sample extremes. The new methodology allows large Bayesian models to be fitted and assessed without the significant computing costs of Markov Chain Monte Carlo methods. We illustrate our new methodology with maximum rainfall data from the Sydney, Australia, hinterland. Comparisons are made between the Mean Field Variational Bayes and Markov Chain Monte Carlo approaches. © 2012 Australian Statistical Publishing Association Inc..
Hall, P., Pham, T., Wand, M.P. & Wang, S.S.J. 2011, 'Asymptotic normality and valid inference for Gaussian variational approximation', Annals of Statistics, vol. 39, no. 5, pp. 2502-2532.
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We derive the precise asymptotic distributional behavior of Gaussian variational approximate estimators of the parameters in a single-predictor Poisson mixed model. These results are the deepest yet obtained concerning the statistical properties of a variational approximation method. Moreover, they give rise to asymptotically valid statistical inference. A simulation study demonstrates that Gaussian variational approximate confidence intervals possess good to excellent coverage properties, and have a similar precision to their exact likelihood counterparts. © Institute of Mathematical Statistics, 2011.
Hall, P., Ormerod, J.T. & Wand, M.P. 2011, 'Theory of Gaussian variational approximation for a Poisson mixed model', Statistica Sinica, vol. 21, no. 1, pp. 369-389.
Likelihood-based inference for the parameters of generalized linear mixed models is hindered by the presence of intractable integrals. Gaussian variational approximation provides a fast and effective means of approximate inference. We provide some theory for this type of approximation for a simple Poisson mixed model. In particular, we establish consistency at rate m -1/2 +n-1, where m is the number of groups and n is the number of repeated measurements.
Samworth, R.J. & Wand, M.P. 2010, 'Asymptotics and optimal bandwidth selection for highest density region estimation', Annals of Statistics, vol. 38, no. 3, pp. 1767-1792.
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We study kernel estimation of highest-density regions (HDR). Our main contributions are two-fold. First, we derive a uniform-in-bandwidth asymptotic approximation to a risk that is appropriate for HDR estimation. This approximation is then used to derive a bandwidth selection rule for HDR estimation possessing attractive asymptotic properties.We also present the results of numerical studies that illustrate the benefits of our theory and methodology. © 2010 Institute of Mathematical Statistics.
Ormerod, J.T. & Wand, M.P. 2010, 'Explaining variational approximations', American Statistician, vol. 64, no. 2, pp. 140-153.
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Variational approximations facilitate approximate inference for the parameters in complex statistical models and provide fast, deterministic alternatives to Monte Carlo methods. However, much of the contemporary literature on variational approximations is in Computer Science rather than Statistics, and uses terminology, notation, and examples from the former field. In this article we explain variational approximation in statistical terms. In particular, we illustrate the ideas of variational approximation using examples that are familiar to statisticians. © 2010 American Statistical Association.
Al Kadiri, M., Carroll, R.J. & Wand, M.P. 2010, 'Marginal longitudinal semiparametric regression via penalized splines', Statistics and Probability Letters, vol. 80, no. 15-16, pp. 1242-1252.
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We study the marginal longitudinal nonparametric regression problem and some of its semiparametric extensions. We point out that, while several elaborate proposals for efficient estimation have been proposed, a relative simple and straightforward one, based on penalized splines, has not. After describing our approach, we then explain how Gibbs sampling and the BUGS software can be used to achieve quick and effective implementation. Illustrations are provided for nonparametric regression and additive models. © 2010 Elsevier B.V.
Marley, J.K. & Wand, M.P. 2010, 'Non-standard semiparametric regression via BRugs', Journal of Statistical Software, vol. 37, no. 5, pp. 1-30.
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We provide several illustrations of Bayesian semiparametric regression analyses in the BRugs package. BRugs facilitates use of the BUGS inference engine from the R computing environment and allows analyses to be managed using scripts. The examples are chosen to represent an array of non-standard situations, for which mixed model software is not viable. The situations include: the response variable being outside of the one-parameter exponential family, data subject to missingness, data subject to measurement error and parameters entering the model via an index.
Kauermann, G., Ormerod, J.T. & Wand, M.P. 2010, 'Parsimonious classification via generalized linear mixed models', Journal of Classification, vol. 27, no. 1, pp. 89-110.
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We devise a classification algorithm based on generalized linear mixed model (GLMM) technology. The algorithm incorporates spline smoothing, additive model-type structures and model selection. For reasons of speed we employ the Laplace approximation, rather than Monte Carlo methods. Tests on real and simulated data show the algorithm to have good classification performance. Moreover, the resulting classifiers are generally interpretable and parsimonious. © 2010 Springer Science+Business Media, LLC.
Naumann, U., Luta, G. & Wand, M.P. 2010, 'The curvHDR method for gating flow cytometry samples', BMC Bioinformatics, vol. 11.
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Background: High-throughput flow cytometry experiments produce hundreds of large multivariate samples of cellular characteristics. These samples require specialized processing to obtain clinically meaningful measurements. A major component of this processing is a form of cell subsetting known as gating. Manual gating is time-consuming and subjective. Good automatic and semi-automatic gating algorithms are very beneficial to high-throughput flow cytometry.Results: We develop a statistical procedure, named curvHDR, for automatic and semi-automatic gating. The method combines the notions of significant high negative curvature regions and highest density regions and has the ability to adapt well to human-perceived gates. The underlying principles apply to dimension of arbitrary size, although we focus on dimensions up to three. Accompanying software, compatible with contemporary flow cytometry infor-matics, is developed.Conclusion: The method is seen to adapt well to nuances in the data and, to a reasonable extent, match human perception of useful gates. It offers big savings in human labour when processing high-throughput flow cytometry data whilst retaining a good degree of efficacy. © 2010 Naumann et al; licensee BioMed Central Ltd.
Wand, M.P. & Ormerod, J.T. 2010, 'On semiparametric regression with o'sullivan penalised splines', Australian and New Zealand Journal of Statistics, vol. 52, no. 2, pp. 239-239.
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Pearce, N.D. & Wand, M.P. 2009, 'Explicit connections between longitudinal data analysis and kernel machines', Electronic Journal of Statistics, vol. 3, no. 0, pp. 797-823.
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Naumann, U. & Wand, M.P. 2009, 'Automation in high-content flow cytometry screening', Cytometry Part A, vol. 75, no. 9, pp. 789-797.
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High-content flow cytometric screening (FC-HCS) is a 21st Century technology that combines robotic fluid handling, flow cytometric instrumentation, and bioinformatics software, so that relatively large numbers of flow cytometric samples can be processed and analysed in a short period of time. We revisit a recent application of FC-HCS to the problem of cellular signature definition for acute graft-versus-host-disease. Our focus is on automation of the data processing steps using recent advances in statistical methodology. We demonstrate that effective results, on par with those obtained via manual processing, can be achieved using our automatic techniques. Such automation of FC-HCS has the potential to drastically improve diagnosis and biomarker identification. © 2009 International Society for Advancement of Cytometry.
Duong, T., Koch, I. & Wand, M.P. 2009, 'Highest density difference region estimation with application to flow cytometric data', Biometrical Journal, vol. 51, no. 3, pp. 504-521.
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Motivated by the needs of scientists using flow cytometry, we study the problem of estimating the region where two multivariate samples differ in density. We call this problem highest density difference region estimation and recognise it as a two-sample analogue of highest density region or excess set estimation. Flow cytometry samples are typically in the order of 10 000 and 100 000 and with dimension ranging from about 3 to 20. The industry standard for the problem being studied is called Frequency Difference Gating, due to Roederer and Hardy (2001). After couching the problem in a formal statistical framework we devise an alternative estimator that draws upon recent statistical developments such as patient rule induction methods. Improved performance is illustrated in simulations. While motivated by flow cytometry, the methodology is suitable for general multivariate random samples where density difference regions are of interest. © 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.
Staudenmayer, J., Lake, E.E. & Wand, M.P. 2009, 'Robustness for general design mixed models using the t-distribution', Statistical Modelling, vol. 9, no. 3, pp. 235-255.
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The t-distribution allows the incorporation of outlier robustness into statistical models while retaining the elegance of likelihood-based inference. In this paper, we develop and implement a linear mixed model for the general design of the linear mixed model using the univariate t-distribution. This general design allows a considerably richer class of models to be fit than is possible with existing methods. Included in this class are semi-parametric regression and smoothing and spatial models. © 2009 SAGE Publications.
Ruppert, D., Wand, M.P. & Carroll, R.J. 2009, 'Semiparametric regression during 2003-2007.', Electron J Stat, vol. 3, pp. 1193-1256.
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Semiparametric regression is a fusion between parametric regression and nonparametric regression that integrates low-rank penalized splines, mixed model and hierarchical Bayesian methodology - thus allowing more streamlined handling of longitudinal and spatial correlation. We review progress in the field over the five-year period between 2003 and 2007. We find semiparametric regression to be a vibrant field with substantial involvement and activity, continual enhancement and widespread application.
Wand, M.P. 2009, 'Semiparametric regression and graphical models', Australian and New Zealand Journal of Statistics, vol. 51, no. 1, pp. 9-41.
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Semiparametric regression models that use spline basis functions with penalization have graphical model representations. This link is more powerful than previously established mixed model representations of semiparametric regression, as a larger class of models can be accommodated. Complications such as missingness and measurement error are more naturally handled within the graphical model architecture. Directed acyclic graphs, also known as Bayesian networks, play a prominent role. Graphical model-based Bayesian 'inference engines', such as bugs and vibes, facilitate fitting and inference. Underlying these are Markov chain Monte Carlo schemes and recent developments in variational approximation theory and methodology. © 2009 Australian Statistical Publishing Association Inc.
Gottardo, R., Brinkman, R.R., Luta, G. & Wand, M.P. 2009, 'Recent bioinformatics advances in the analysis of high throughput flow cytometry data.', Adv Bioinformatics, p. 461763.
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Duong, T., Cowling, A., Koch, I. & Wand, M.P. 2008, 'Feature significance for multivariate kernel density estimation', Computational Statistics and Data Analysis, vol. 52, no. 9, pp. 4225-4242.
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Multivariate kernel density estimation provides information about structure in data. Feature significance is a technique for deciding whether features-such as local extrema-are statistically significant. This paper proposes a framework for feature significance in d-dimensional data which combines kernel density derivative estimators and hypothesis tests for modal regions. For the gradient and curvature estimators distributional properties are given, and pointwise test statistics are derived. The hypothesis tests extend the two-dimensional feature significance ideas of Godtliebsen et al. [Godtliebsen, F., Marron, J.S., Chaudhuri, P., 2002. Significance in scale space for bivariate density estimation. Journal of Computational and Graphical Statistics 11, 1-21]. The theoretical framework is complemented by novel visualization for three-dimensional data. Applications to real data sets show that tests based on the kernel curvature estimators perform well in identifying modal regions. These results can be enhanced by corresponding tests with kernel gradient estimators. © 2008 Elsevier Ltd. All rights reserved.
Fan, Y., Leslie, D.S. & Wand, M.P. 2008, 'Generalised linear mixed model analysis via sequential Monte Carlo sampling', Electronic Journal of Statistics, vol. 2, no. 0, pp. 916-938.
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Padoan, S.A. & Wand, M.P. 2008, 'Mixed model-based additive models for sample extremes', Statistics and Probability Letters, vol. 78, no. 17, pp. 2850-2858.
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We consider additive models fitting and inference when the response variable is a sample extreme. Non-linear covariate effects are handled using the mixed model representation of penalised splines. A fitting algorithm based on likelihood approximations is derived. The efficacy of the resulting methodology is demonstrated via application to simulated and real data. © 2008 Elsevier B.V. All rights reserved.
Kuo, F.Y., Dunsmuir, W.T.M., Sloan, I.H., Wand, M.P. & Womersley, R.S. 2008, 'Quasi-Monte Carlo for highly structured generalised response models', Methodology and Computing in Applied Probability, vol. 10, no. 2, pp. 239-275.
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Highly structured generalised response models, such as generalised linear mixed models and generalised linear models for time series regression, have become an indispensable vehicle for data analysis and inference in many areas of application. However, their use in practice is hindered by high-dimensional intractable integrals. Quasi-Monte Carlo (QMC) is a dynamic research area in the general problem of high-dimensional numerical integration, although its potential for statistical applications is yet to be fully explored. We survey recent research in QMC, particularly lattice rules, and report on its application to highly structured generalised response models. New challenges for QMC are identified and new methodologies are developed. QMC methods are seen to provide significant improvements compared with ordinary Monte Carlo methods. © Springer Science+Business Media, LLC 2007.
Smith, A.D.A.C. & Wand, M.P. 2008, 'Streamlined variance calculations for semiparametric mixed models', Statistics in Medicine, vol. 27, no. 3, pp. 435-448.
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Semiparametric mixed model analysis benefits from variability estimates such as standard errors of effect estimates and variability bars to accompany curve estimates. We show how the underlying variance calculations can be done extremely efficiently compared with the direct nave approach. These streamlined calculations are linear in the number of subjects, representing a two orders of magnitude improvement. Copyright © 2007 John Wiley & Sons, Ltd.
Ormerod, J.T., Wand, M.P. & Koch, I. 2008, 'Penalised spline support vector classifiers: Computational issues', Computational Statistics, vol. 23, no. 4, pp. 623-641.
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We study computational issues for support vector classification with penalised spline kernels. We show that, compared with traditional kernels, computational times can be drastically reduced in large problems making such problems feasible for sample sizes as large as ~106. The optimisation technology known as interior point methods plays a central role. Penalised spline kernels are also shown to allow simple incorporation of low-dimensional structure such as additivity. This can aid both interpretability and performance. © 2008 Springer-Verlag.
Wand, M.P. & Ormerod, J.T. 2008, 'On semiparametric regression with O'Sullivan penalized splines', Australian and New Zealand Journal of Statistics, vol. 50, no. 2, pp. 179-198.
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An exposition on the use of O'Sullivan penalized splines in contemporary semiparametric regression, including mixed model and Bayesian formulations, is presented. O'Sullivan penalized splines are similar to P-splines, but have the advantage of being a direct generalization of smoothing splines. Exact expressions for the O'Sullivan penalty matrix are obtained. Comparisons between the two types of splines reveal that O'Sullivan penalized splines more closely mimic the natural boundary behaviour of smoothing splines. Implementation in modern computing environments such as Matlab, r and bugs is discussed. © 2008 Australian Statistical Publishing Association Inc.
Ganguli, B. & Wand, M.P. 2007, 'Feature significance in generalized additive models', Statistics and Computing, vol. 17, no. 2, pp. 179-192.
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This paper develops inference for the significance of features such as peaks and valleys observed in additive modeling through an extension of the SiZer-type methodology of Chaudhuri and Marron (1999) and Godtliebsen et al. (2002, 2004) to the case where the outcome is discrete. We consider the problem of determining the significance of features such as peaks or valleys in observed covariate effects both for the case of additive modeling where the main predictor of interest is univariate as well as the problem of studying the significance of features such as peaks, inclines, ridges and valleys when the main predictor of interest is geographical location. We work with low rank radial spline smoothers to allow to the handling of sparse designs and large sample sizes. Reducing the problem to a Generalised Linear Mixed Model (GLMM) framework enables derivation of simulation-based critical value approximations and guards against the problem of multiple inferences over a range of predictor values. Such a reduction also allows for easy adjustment for confounders including those which have an unknown or complex effect on the outcome. A simulation study indicates that our method has satisfactory power. Finally, we illustrate our methodology on several data sets. © Springer Science+Business Media, LLC 2007.
Wand, M.P. 2007, 'Fisher information for generalised linear mixed models', Journal of Multivariate Analysis, vol. 98, no. 7, pp. 1412-1416.
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The Fisher information for the canonical link exponential family generalised linear mixed model is derived. The contribution from the fixed effects parameters is shown to have a particularly simple form. © 2007 Elsevier Inc. All rights reserved.
Oakes, S.R., Robertson, F.G., Kench, J.G., Gardiner-Garden, M., Wand, M.P., Green, J.E. & Ormandy, C.J. 2007, 'Loss of mammary epithelial prolactin receptor delays tumor formation by reducing cell proliferation in low-grade preinvasive lesions', Oncogene, vol. 26, no. 4, pp. 543-553.
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Top quartile serum prolactin levels confer a twofold increase in the relative risk of developing breast cancer. Prolactin exerts this effect at an ill defined point in the carcinogenic process, via mechanisms involving direct action via prolactin receptors within mammary epithelium and/or indirect action through regulation of other hormones such as estrogen and progesterone. We have addressed these questions by examining mammary carcinogenesis in transplants of mouse mammary epithelium expressing the SV40T oncogene, with or without the prolactin receptor, using host animals with a normal endocrine system. In prolactin receptor knockout transplants the area of neoplasia was significantly smaller (7 versus 17%; P<0.001 at 22 weeks and 7 versus 14%; P=0.009 at 32 weeks). Low-grade neoplastic lesions displayed reduced BrdU incorporation rate (11.3 versus 17% P=0.003) but no change in apoptosis rate. Tumor latency increased (289 days versus 236 days, P<0.001). Tumor frequency, growth rate, morphology, cell proliferation and apoptosis were not altered. Thus, prolactin acts directly on the mammary epithelial cells to increase cell proliferation in preinvasive lesions, resulting in more neoplasia and acceleration of the transition to invasive carcinoma. Targeting of mammary prolactin signaling thus provides a strategy to prevent the early progression of neoplasia to invasive carcinoma. &copy; 2007 Nature Publishing Group All rights reserved.
Werneck, G.L., Costa, C.H.N., Walker, A.M., David, J.R., Wand, M. & Maguire, J.H. 2007, 'Multilevel modelling of the incidence of visceral leishmaniasis in Teresina, Brazil', Epidemiology and Infection, vol. 135, no. 2, pp. 195-201.
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Epidemics of visceral leishmaniasis (VL) in major Brazilian cities are new phenomena since 1980. As determinants of transmission in urban settings probably operate at different geographic scales, and information is not available for each scale, a multilevel approach was used to examine the effect of canine infection and environmental and socio-economic factors on the spatial variability of incidence rates of VL in the city of Teresina. Details on an outbreak of greater than 1200 cases of VL in Teresina during 1993-1996 were available at two hierarchical levels: census tracts (socio-economic characteristics, incidence rates of human VL) and districts, which encompass census tracts (prevalence of canine infection). Remotely sensed data obtained by satellite generated environmental information at both levels. Data from census tracts and districts were analysed simultaneously by multilevel modelling. Poor socio-economic conditions and increased vegetation were associated with a high incidence of human VL. Increasing prevalence of canine infection also predicted a high incidence of human VL, as did high prevalence of canine infection before and during the epidemic. Poor socio-economic conditions had an amplifying effect on the association between canine infection and the incidence of human VL. Focusing interventions on areas with characteristics identified by multilevel analysis could be a cost-effective strategy for controlling VL. Because risk factors for infectious diseases operate simultaneously at several levels and ecological data usually are available at different geographical scales, multilevel modelling is a valuable tool for epidemiological investigation of disease transmission. &copy; 2007 Cambridge University Press.
Ganguli, B. & Wand, M.P. 2006, 'Additive models for geo-referenced failure time data', Statistics in Medicine, vol. 25, no. 14, pp. 2469-2482.
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Asthma researchers have found some evidence that geographical variations in susceptibility to asthma could reflect the effect of community level factors such as exposure to violence. Our methodology was motivated by a study of age at onset of asthma among children of inner-city neighbourhoods in East Boston. Cox's proportional hazards model was not appropriate since there was not enough information about the nature of geographical variations so as to impose a parametric relationship. In addition, some of the known risk factors were believed to have non-linear log-hazard ratios. We extend the geoadditive models of Kamman and Wand to the case where the outcome measure is a possibly censored time to event. We reduce the problem to one of fitting a Poisson mixed model by using Poisson approximations in conjunction with a mixed model formulation of generalized additive modelling. Our method allows for low-rank additive modelling, provides likelihood-based estimation of all parameters including the amount of smoothing and can be implemented using standard software. We illustrate our method on the East Boston data. Copyright &copy; 2005 John Wiley & Sons, Ltd.
Zhao, Y., Staudenmayer, J., Coull, B.A. & Wand, M.P. 2006, 'General design Bayesian generalized linear mixed models', Statistical Science, vol. 21, no. 1, pp. 35-51.
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Linear mixed models are able to handle an extraordinary range of complications in regression-type analyses. Their most common use is to account for within-subject correlation in longitudinal data analysis. They are also the standard vehicle for smoothing spatial count data. However, when treated in full generality, mixed models can also handle spline-type smoothing and closely approximate kriging. This allows for nonparametric regression models (e.g., additive models and varying coefficient models) to be handled within the mixed model framework. The key is to allow the random effects design matrix to have general structure; hence our label general design. For continuous response data, particularly when Gaussianity of the response is reasonably assumed, computation is now quite mature and supported by the R, SAS and S-PLUS packages. Such is not the case for binary and count responses, where generalized linear mixed models (GLMMs) are required, but are hindered by the presence of intractable multivariate integrals. Software known to us supports special cases of the GLMM (e.g., PROC NLMIXED in SAS or glmmML in R) or relies on the sometimes crude Laplace-type approximation of integrals (e.g., the SAS macro glimmix or glmmPQL in R). This paper describes the fitting of general design generalized linear mixed models. A Bayesian approach is taken and Markov chain Monte Carlo (MCMC) is used for estimation and inference. In this generalized setting, MCMC requires sampling from nonstandard distributions. In this article, we demonstrate that the MCMC package WinBUGS facilitates sound fitting of general design Bayesian generalized linear mixed models in practice. &copy; Institute of Mathematical Statistics, 2006.
Wand, M. 2006, 'Support vector machine classification', Parabola, vol. 42, no. 2, pp. 21-37.
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Pearce, N.D. & Wand, M.P. 2006, 'Penalized splines and reproducing kernel methods', American Statistician, vol. 60, no. 3, pp. 233-240.
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Two data analytic research areas - penalized splines and reproducing kernel methods - have become very vibrant since the mid-1990s. This article shows how the former can be embedded in the latter via theory for reproducing kernel Hilbert spaces. This connection facilitates cross-fertilization between the two bodies of research. In particular, connections between support vector machines and penalized splines are established. These allow for significant reductions in computational complexity, and easier incorporation of special structure such as additivity. &copy; 2006 American Statistical Association.
Salganik, M.P., Hardie, D.L., Swart, B., Dandie, G.W., Zola, H., Shaw, S., Shapiro, H., Tinckam, K., Milford, E.L. & Wand, M.P. 2005, 'Detecting antibodies with similar reactivity patterns in the HLDA8 blind panel of flow cytometry data', Journal of Immunological Methods, vol. 305, no. 1, pp. 67-74.
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The blind panel collected for the 8th Human Leucocyte Differentiation Antigens Workshop (HLDA8; http://www.hlda8.org) included 49 antibodies of known CD specificities and 76 antibodies of unknown specificity. We have identified groups of antibodies showing similar patterns of reactivity that need to be investigated by biochemical methods to evaluate whether the antibodies within these groups are reacting with the same molecule. Our approach to data analysis was based on the work of Salganik et al. (in press) [Salganik, M.P., Milford E.L., Hardie D.L., Shaw, S., Wand, M.P., in press. Classifying antibodies using flow cytometry data: class prediction and class discovery. Biometrical Journal]. &copy; 2005 Elsevier B.V. All rights reserved.
Salganik, M.P., Milford, E.L., Hardie, D.L., Shaw, S. & Wand, M.P. 2005, 'Classifying antibodies using flow cytometry data: Class prediction and class discovery', Biometrical Journal, vol. 47, no. 5, pp. 740-754.
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Classifying monoclonal antibodies, based on the similarity of their binding to the proteins (antigens) on the surface of blood cells, is essential for progress in immunology, hematology and clinical medicine. The collaborative efforts of researchers from many countries have led to the classification of thousands of antibodies into 247 clusters of differentiation (CD). Classification is based on flow cytometry and biochemical data. In preliminary classifications of antibodies based on flow cytometry data, the object requiring classification (an antibody) is described by a set of random samples from unknown densities of fluorescence intensity. An individual sample is collected in the experiment, where a population of cells of a certain type is stained by the identical fluorescently marked replicates of the antibody of interest. Samples are collected for multiple cell types. The classification problems of interest include identifying new CDs (class discovery or unsupervised learning) and assigning new antibodies to the known CD clusters (class prediction or supervised learning). These problems have attracted limited attention from statisticians. We recommend a novel approach to the classification process in which a computer algorithm suggests to the analyst the subset of the "most appropriate" classifications of an antibody in class prediction problems or the "most similar" pairs/ groups of antibodies in class discovery problems. The suggested algorithm speeds up the analysis of a flow cytometry data by a factor 10-20. This allows the analyst to focus on the interpretation of the automatically suggested preliminary classification solutions and on planning the subsequent biochemical experiments. &copy; 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.
Crainiceanu, C., Ruppert, D., Claeskens, G. & Wand, M.P. 2005, 'Exact likelihood ratio tests for penalised splines', Biometrika, vol. 92, no. 1, pp. 91-103.
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Penalised-spline-based additive models allow a simple mixed model representation where the variance components control departures from linear models. The smoothing parameter is the ratio of the random-coefficient and error variances and tests for linear regression reduce to tests for zero random-coefficient variances. We propose exact likelihood and restricted likelihood ratio tests for testing polynomial regression versus a general alternative modelled by penalised splines. Their spectral decompositions are used as the basis of fast simulation algorithms. We derive the asymptotic local power properties of the tests under weak conditions. In particular we characterise the local alternatives that are detected with asymptotic probability one. Confidence intervals for the smoothing parameter are obtained by inverting the tests for a fixed smoothing parameter versus a general alternative. We discuss F and R tests and show that ignoring the variability in the smoothing parameter estimator can have a dramatic effect on their null distributions. The powers of several known tests are investigated and a small set of tests with good power properties is identified. The restricted likelihood ratio test is among the best in terms of power. &copy; 2005 Biometrika Trust.
Swart, B., Salganik, M.P., Wand, M.P., Tinckam, K., Milford, E.L., Drbal, K., Angelisova, P., Horejsi, V., MacArdle, P., Bailey, S., Hollemweguer, E., Hodge, G., Nairn, J., Millard, D., Dagdeviren, A., Dandie, G.W. & Zola, H. 2005, 'The HLDA8 blind panel: Findings and conclusions', Journal of Immunological Methods, vol. 305, no. 1, pp. 75-83.
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There were over 600 antibodies submitted to HLDA8, with many of unknown specificity. Of these, 101 antibodies were selected for a blind panel study that also included 5 negative controls and 27 positive controls of known CD specificity making a total of 133 antibodies in the final panel. Of the 101 unknowns, 31 antibodies were identified during the course of this blind panel study as being specific for known molecules and included some specific for MHC class II antigens, CD45 isoforms and the Dombrock antigen. Several antibody pairs among those in the blind panel were found to have very similar staining patterns and were therefore compared by immunohistochemical and/or Western blot analyses for identity. &copy; 2005 Elsevier B.V. All rights reserved.
Ganguli, B., Staudenmayer, J. & Wand, M.P. 2005, 'Additive models with predictors subject to measurement error', Australian and New Zealand Journal of Statistics, vol. 47, no. 2, pp. 193-202.
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This paper develops a likelihood-based method for fitting additive models in the presence of measurement error. It formulates the additive model using the linear mixed model representation of penalized splines. In the presence of a structural measurement error model, the resulting likelihood involves intractable integrals, and a Monte Carlo expectation maximization strategy is developed for obtaining estimates. The method's performance is illustrated with a simulation study. &copy; 2005 Australian Statistical Publishing Association Inc.
Crainiceanu, C.M., Ruppert, D. & Wand, M.P. 2005, 'Bayesian analysis for penalized spline regression using WinBUGS', Journal of Statistical Software, vol. 14.
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Penalized splines can be viewed as BLUPs in a mixed model framework, which allows the use of mixed model software for smoothing. Thus, software originally developed for Bayesian analysis of mixed models can be used for penalized spline regression. Bayesian inference for nonparametric models enjoys the flexibility of nonparametric models and the exact inference provided by the Bayesian inferential machinery. This paper provides a simple, yet comprehensive, set of programs for the implementation of nonparametric Bayesian analysis in Win BUGS. Good mixing properties of the MCMC chains are obtained by using low-rank thin-plate splines, while simulation times per iteration are reduced employing WinBUGS specific computational tricks.
Durbán, M., Harezlak, J., Wand, M.P. & Carroll, R.J. 2005, 'Simple fitting of subject-specific curves for longitudinal data', Statistics in Medicine, vol. 24, no. 8, pp. 1153-1167.
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We present a simple semiparametric model for fitting subject-specific curves for longitudinal data. Individual curves are modelled as penalized splines with random coefficients. This model has a mixed model representation, and it is easily implemented in standard statistical software. We conduct an analysis of the long-term effect of radiation therapy on the height of children suffering from acute lymphoblastic leukaemia using penalized splines in the framework of semiparametric mixed effects models. The analysis revealed significant differences between therapies and showed that the growth rate of girls in the study cannot be fully explained by the group-average curve and that individual curves are necessary to reflect the individual response to treatment. We also show how to implement these models in S-PLUS and R in the appendix. Copyright &copy; 2004 John Wiley & Sons, Ltd.
Wright, R.J., Finn, P., Contreras, J.P., Cohen, S., Wright, R.O., Staudenmayer, J., Wand, M., Perkins, D., Weiss, S.T. & Gold, D.R. 2004, 'Chronic caregiver stress and IgE expression, allergen-induced proliferation, and cytokine profiles in a birth cohort predisposed to atopy', Journal of Allergy and Clinical Immunology, vol. 113, no. 6, pp. 1051-1057.
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Background Psychologic stress modifies immune function and cytokine production. Objective We examined relationships between caregiver stress on the following markers of early childhood immune response: (1) IgE expression (n=215); (2) mitogen-induced and allergen-specific (Dermatophagoides farinae [Der f 1] and cockroach [Bla g 2]) proliferative response (n=114); and (3) subsequent cytokine expression (INF-?, TNF-?, IL-10, and IL-13) in a prospective birth cohort predisposed to atopy. Methods Caregiver stress was measured at 2-month intervals for the first 2 years of life and yearly thereafter by using the Perceived Stress Scale. A subsequent blood sample obtained from the children (median age, 2.1 years; range, 18-32 months) was analyzed for total serum IgE level and allergen-induced proliferation quantified as the stimulation index (SI; mean thymidine incorporation of the stimulated sample divided by that of the unstimulated sample). The relationship between stress and the proliferative response (SI >3 vs SI ?3), and total IgE level (?100 IU/mL vs >100 IU/mL) was examined by using logistic regression. The relationship between cytokine levels and stress was analyzed by using linear regression. Results In adjusted analyses higher caregiver stress in the first 6 months after birth was associated with a Der f 1 SI of greater than 3 (odds ratio [OR], 1.5; 95% CI, 1.0-2.3) and nominally associated with a Bla g 2 SI of greater than 3 (OR, 1.13; 95% CI, 0.7-1.8). Higher stress between ages 6 and 18 months was associated with a high total IgE level (OR, 2.03; 95% CI, 1.1-3.6). Higher stress was significantly associated with increased production of TNF-?, with a suggested trend between higher stress and reduced INF-? production. Conclusion Increased stress in early childhood was associated with an atopic immune profile in these children predisposed to atopy-asthma.
Myatt, T.A., Johnston, S.L., Zuo, Z., Wand, M., Kebadze, T., Rudnick, S. & Milton, D.K. 2004, 'Detection of airborne rhinovirus and its relation to outdoor air supply in office environments', American Journal of Respiratory and Critical Care Medicine, vol. 169, no. 11, pp. 1187-1190.
Rhinoviruses are major causes of morbidity in patients with respiratory diseases; however, their modes of transmission are controversial. We investigated detection of airborne rhinovirus in office environments by polymerase chain reaction technology and related detection to outdoor air supply rates. We sampled air from 9 A.M. to 5 P.M. each workday, with each sample run for 1 work week. We directly extracted RNA from the filters for nested reverse transcriptase-polymerase chain reaction analysis of rhinovirus. Nasal lavage samples from building occupants with upper respiratory infections were also collected. Indoor carbon dioxide (CO2) concentrations were recorded every 10 minutes as a surrogate for outdoor air supply. To increase the range of CO2 concentrations, we adjusted the outdoor air supply rates every 3 months. Generalized additive models demonstrated an association between the probability of detecting airborne rhinovirus and a weekly average CO2 concentration greater than approximately 100 ppm, after controlling for covariates. In addition, one rhinovirus from a nasal lavage contained an identical nucleic acid sequence similar to that in the building air collected during the same week. These results suggest that occupants in buildings with low outdoor air supply may have an increased risk of exposure to infectious droplet nuclei emanating from a fellow building occupant.
Ganguli, B. & Wand, M.P. 2004, 'Feature significance in geostatistics', Journal of Computational and Graphical Statistics, vol. 13, no. 4, pp. 954-973.
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Geographically referenced data are routinely smoothed using kriging or spline methodology. Features in the resulting surface such as peaks, inclines, ridges, and valleys are often of interest. This article develops inference for the significance of such features through extension of methodology for univariate features known as SiZer. We work with low rank radial spline smoothers. These allow the handling of sparse designs, large sample sizes, and simulation-based critical value approximation. We illustrate the methodology on two geostatistical datasets.
French, J.L. & Wand, M.P. 2004, 'Generalized additive models for cancer mapping with incomplete covariates', Biostatistics, vol. 5, no. 2, pp. 177-191.
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Maps depicting cancer incidence rates have become useful tools in public health research, giving valuable information about the spatial variation in rates of disease. Typically, these maps are generated using count data aggregated over areas such as counties or census blocks. However, with the proliferation of geographic information systems and related databases, it is becoming easier to obtain exact spatial locations for the cancer cases and suitable control subjects. The use of such point data allows us to adjust for individual-level covariates, such as age and smoking status, when estimating the spatial variation in disease risk. Unfortunately, such covariate information is often subject to missingness. We propose a method for mapping cancer risk when covariates are not completely observed. We model these data using a logistic generalized additive model. Estimates of the linear and non-linear effects are obtained using a mixed effects model representation. We develop an EM algorithm to account for missing data and the random effects. Since the expectation step involves an intractable integral, we estimate the E-step with a Laplace approximation. This framework provides a general method for handling missing covariate values when fitting generalized additive models. We illustrate our method through an analysis of cancer incidence data from Cape Cod, Massachusetts. These analyses demonstrate that standard complete-case methods can yield biased estimates of the spatial variation of cancer risk. &copy; Oxford University Press 2004; all rights reserved.
Ngo, L. & Wand, M.P. 2004, 'Smoothing with mixed model software', Journal of Statistical Software, vol. 9, pp. 1-54.
Smoothing methods that use basis functions with penalization can be formulated as fits in a mixed model framework. One of the major benefits is that software for mixed model analysis can be used for smoothing. We illustrate this for several smoothing models such as additive and varying coefficient models for both S-PLUS and SAS software. Code for each of the illustrations is available on the Internet.
Salganik, M.P., Wand, M.P. & Lange, N. 2004, 'Comparison of feature significance quantile approximations', Australian and New Zealand Journal of Statistics, vol. 46, no. 4, pp. 569-581.
Curve estimates and surface estimates often contain features such as inclines, bumps or ridges which may signify an underlying structural mechanism. However, spurious features are also a common occurrence and it is important to identify those features that are statistically significant. A method has been developed recently for recognising feature significance based on the derivatives of the function estimate. It requires simultaneous confidence intervals and tests, which in turn require quantiles for the maximal deviation statistics. This paper reviews and compares various approximations to these quantiles. Applying upcrossing-probability theory to this problem yields better quantile approximations than the use of an independent blocks method.
Kammann, E.E. & Wand, M.P. 2003, 'Geoadditive models', Journal of the Royal Statistical Society. Series C: Applied Statistics, vol. 52, no. 1, pp. 1-18.
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A study into geographical variability of reproductive health outcomes (e.g. birth weight) in Upper Cape Cod, Massachusetts, USA, benefits from geostatistical mapping or kriging. However, also observed are some continuous covariates (e.g. maternal age) that exhibit pronounced non-linear relationships with the response variable. To account for such effects properly we merge kriging with additive models to obtain what we call geoadditive models. The merging becomes effortless by expressing both as linear mixed models. The resulting mixed model representation for the geoadditive model allows for fitting and diagnosis using standard methodology and software.
Wand, M.P. 2003, 'Smoothing and mixed models', Computational Statistics, vol. 18, no. 2, pp. 223-249.
Smoothing methods that use basis functions with penalisation can be formulated as maximum likelihood estimators and best predictors in a mixed model framework. Such connections are at least a quarter of a century old but, perhaps with the advent of mixed model software, have led to a paradigm shift in the field of smoothing. The reason is that most, perhaps all, models involving smoothing can be expressed as a mixed model and hence enjoy the benefit of the growing body of methodology and software for general mixed model analysis. The handling of other complications such as clustering, missing data and measurement error is generally quite straightforward with mixed model representations of smoothing.
Kim, J.Y., Hauser, R., Wand, M.P., Herrick, R.F., Houk, R.S., Aeschliman, D.B., Woodin, M.A. & Christiani, D.C. 2003, 'Association of Expired Nitric Oxide with Urinary Metal Concentrations in Boilermakers Exposed to Residual Oil Fly Ash', American Journal of Industrial Medicine, vol. 44, no. 5, pp. 458-466.
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Background: Exposure to metal-containing particulate matter has been associated with adverse pulmonary responses. Metals in particulate matter are soluble, hence are readily recovered in urine of exposed individuals. This study investigated the association between urinary metal concentrations and the fractional concentration of expired nitric oxide (FENO) in boilermakers (N = 32) exposed to residual oil fly ash (ROFA). Methods: Subjects were monitored at a boiler overhaul site located in the New England area, USA. FENO and urine samples were collected pre- and post-workshift for 5 consecutive workdays. Metals investigated included vanadium (V), chromium (Cr), manganese (Mn), nickel (Ni), copper (Cu), and lead (Pb). Results: The median FENO was 7.5 ppb (95% CI: 7.4-8.0), and the median creatinine-adjusted urinary metal concentrations (?g/g creatinine) were: vanadium, 1.37; chromium, 0.48; manganese, 0.30; nickel, 1.52; copper, 3.70; and lead, 2.32. Linear mixed-effects models indicated significant inverse exposure-response relationships between log FENO and the log-transformed urinary concentrations of vanadium, manganese, nickel, copper, and lead at several lag times, after adjusting for smoking status. Conclusions: Urine samples may be utilized as a biomarker of occupational metal exposure. The inverse association between FENO and urinary metal concentrations suggests that exposure to metals in particulate matter may have an adverse effect on respiratory health. &copy; 2003 Wiley-Liss, Inc.
Kim, J.Y., Hauser, R., Wand, M.P., Herrick, R.F., Amarasiriwardena, C.J. & Christiani, D.C. 2003, 'The association of expired nitric oxide with occupational particulate metal exposure', Environmental Research, vol. 93, no. 2, pp. 158-166.
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Toxicologic studies have shown that soluble transition metals in residual oil fly ash (ROFA) can induce pulmonary injury. In this study, we investigated the association between the fractional concentration of expired nitric oxide (FENO) and exposure to metal constituents of particulate matter with an aerodynamic mass median diameter?2.5?m (PM2.5) in boilermakers exposed to ROFA and metal fume. Metals investigated included vanadium, chromium, manganese, nickel, copper, and lead. Subjects were monitored for 5 consecutive days during boiler repair overhauls in 1999 (n=20) and 2000 (n=14). In 1999, we found a significant inverse association between log-transformed FENO and PM2.5 metal concentrations. LogFENO changed by -0.03 (95% CI: -0.04, -0.01), -0.56 (95% CI: -0.88, -0.24), -0.09 (95% CI: -0.16, -0.02), and -0.04 (95% CI: -0.07, -0.02) per ?g/m3 of PM2.5 vanadium, chromium, manganese, and nickel, respectively. In 2000, no significant associations were observed, most likely due to exposure misclassification resulting from the use of respirators. The inverse association between PM2.5 metal exposure and F ENO in subjects with limited respirator usage suggests that soluble transition metals might be partially responsible for the adverse pulmonary responses seen in workers exposed to ROFA. &copy; 2003 Elsevier Science (USA). All rights reserved.
Hauser, R., Rice, T.M., Krishha Murthy, G.G., Wand, M., Lewis, D., Bledsoe, T. & Paulauskis, J. 2003, 'The upper airway response to pollen is enhanced by exposure to combustion particulates: A pilot human experimental challenge study', Environmental Health Perspectives, vol. 111, no. 5, pp. 676-680.
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Kim, J.Y., Wand, M., Hauser, R., Mukherjee, S., Herrick, R.F. & Christiani, D.C. 2003, 'Association of expired nitric oxide with occupational particulate exposure', Environmental Health Perspectives, vol. 111, no. 4, pp. 472-477.
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Kim, J.Y., Wand, M.P., Hauser, R., Mukherjee, S., Herrick, R.F. & Christiani, D.C. 2003, 'Association of expired nitric oxide with occupational particulate exposure', Environmental Health Perspectives, vol. 111, no. 5, pp. 676-680.
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Particulate air pollution has been associated with adverse respiratory health effects. This study assessed the utility of expired nitric oxide to detect acute airway responses to metal-containing fine particulates. Using a repeated-measures study design, we investigated the association between the fractional concentration of expired nitric oxide (FENO) and exposure to particulate matter with an aerodynamic mass median diameter of ? 2.5 ?m (PM2.5) in boilermakers exposed to residual oil fly ash and metal fumes. Subjects were monitored for 5 days during boiler repair overhauls in 1999 (n = 20 or 2000 (n = 14). The Wilcoxon median baseline FENO was 10.6 ppb [95% confidence interval (CI): 9.1, 12.7] in 1999 and 7.4 ppb (95% CI: 6.7, 8.0) in 2000. The Wilcoxon median PM2.5 8-hr time-weighted average was 0.56 mg/m3 (95% CI: 0.37, 0.93) in 1999 and 0.86 mg/m3 (95% CI: 0.65, 1.07) in 2000. FENO levels during the work week were significantly lower than baseline FENO in 1999 (p < 0.001). A significant inverse exposure-response relationship between log-transformed FENO and the previous workday's PM2.5 concentration was found in 1999, after adjusting for smoking status, age, and sampling year. With each 1 mg/m3 incremental increase in PM2.5 exposure, log FENO decreased by 0.24 (95% CI: -0.38, -0.10) in 1999. The lack of an exposure-response relationship between PM2.5 exposure and FENO in 2000 could be attributable to exposure misclassification resulting from the use of respirators. In conclusion, occupational exposure to metal-containing fine particulates was associated with significant decreases in FENO in a survey of workers with limited respirator usage.
Hauser, R., Rice, T.M., Murthy, G.G.K., Wand, M.P., Lewis, D., Bledsoe, T. & Paulauskis, J. 2003, 'The upper airway response to pollen is enhanced by exposure to combustion particulates: A pilot human experimental challenge study', Environmental Health Perspectives, vol. 111, no. 4, pp. 472-477.
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Although human experimental studies have shown that gaseous pollutants enhance the inflammatory response to allergens, human data on whether combustion particulates enhance the inflammatory response to allergen are limited. Therefore, we conducted a human experimental study to investigate whether combustion particulates enhance the inflammatory response to aeroallergens. "Enhancement" refers to a greater-than-additive response when combustion particulates are delivered with allergen, compared with the responses when particulates and allergen are delivered alone. Eight subjects, five atopic and three nonatopic, participated in three randomized exposure-challenge sessions at least 2 weeks apart (i.e., clean air followed by allergen, particles followed by no allergen, or particles followed by allergen). Each session consisted of nasal exposure to combustion particles (target concentration of 1.0 mg/m3) or clean air for 1 hr, followed 3 hr later by challenge with whole pollen grains or placebo. Nasal lavage was performed immediately before particle or clean air exposure, immediately after exposure, and 4, 18 and 42 hr after pollen challenge. Cell counts, differentials, and measurement of cytokines were performed on each nasal lavage. In atopic but not in nonatopic subjects, when allergen was preceded by particulates, there was a significant enhancement immediately after pollen challenge in nasal lavage leukocytes and neutrophils (29.7 103 cells/mL and 25.4 103 cells/mL, respectively). This represents a 143% and 130% enhancement, respectively. The enhanced response for interleukin-4 was 3.23 pg/mL (p = 0.06), a 395% enhancement. In atopic subjects there was evidence of an enhanced response when particulates, as compared to clean air, preceded the allergen challenge.
Cai, T., Hyndman, R.J. & Wand, M. 2002, 'Mixed model-based hazard estimation', Journal of Computational and Graphical Statistics, vol. 11, no. 4, pp. 784-798.
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Aerts, M., Claeskens, G. & Wand, M.P. 2002, 'Some theory for penalized spline generalized additive models', Journal of Statistical Planning and Inference, vol. 103, no. 1-2, pp. 455-470.
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Generalized additive models have become one of the most widely used modern statistical tools. Traditionally, they are fit through scatterplot smoothing and the backfitting algorithm. However, a more recent development is the direct fitting through the use of low-rank smoothers (Hastie, J. Roy. Statist. Soc. Ser. B 58 (1996) 379). A particularly attractive example of this is through use of penalized splines (Marx and Eilers, Comput. Statist. Data Anal. 28 (1998) 193). Such an approach has a number of advantages, particularly regarding computation. In this paper, we exploit the explicitness of penalized spline additive models to derive some useful and revealing theoretical approximations. &copy; 2002 Elsevier Science B.V. All rights reserved.
Wand, M.P. 2002, 'Vector differential calculus in statistics', American Statistician, vol. 56, no. 1, pp. 55-62.
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Many statistical operations benefit from differential calculus. Examples include optimization of likelihood functions and calculation of information matrices. For multiparameter models differential calculus suited to vector argument functions is usually the most efficient means of performing the required calculations. We present a primer on vector differential calculus and demonstrate its application to statistics through several worked examples.
Betensky, R.A., Lindsey, J.C., Ryan, L.M. & Wand, M.P. 2002, 'A local likelihood proportional hazards model for interval censored data', Statistics in Medicine, vol. 21, no. 2, pp. 263-275.
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We discuss the use of local likelihood methods to fit proportional hazards regression models to right and interval censored data. The assumed model allows for an arbitrary, smoothed baseline hazard on which a vector of covariates operates in a proportional manner, and thus produces an interpretable baseline hazard function along with estimates of global covariate effects. For estimation, we extend the modified EM algorithm suggested by Betensky, Lindsey, Ryan and Wand. We illustrate the method with data on times to deterioration of breast cosmeses and HIV-1 infection rates among haemophiliacs. Copyright &copy; 2002 John Wiley & Sons, Ltd.
Werneck, G.L., Costa, C.H.N., Walker, A.M., David, J.R., Wand, M. & Maguire, J.H. 2002, 'The urban spread of visceral leishmaniasis: Clues from spatial analysis', Epidemiology, vol. 13, no. 3, pp. 364-367.
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Background. The pattern of spread of visceral leishmaniasis in Brazilian cities is poorly understood. Methods. We used geographic information systems and spatial statistics to evaluate the distribution of 1061 cases of visceral leishmaniasis in Teresina, Brazil, in 1993 through 1996. Results. A locally weighted (LOESS) regression model, which was fit as a smoothed function of spatial coordinates, demonstrated large-scale variation, with high incidence rates in peripheral neighborhoods that bordered forest land and pastures. Moran's I indicated small-scale variation and clustering up to 300 m, roughly the flight range of the sand fly vector. Conclusions. Spatial analytical techniques can identify high-risk areas for targeting control interventions.
French, J.L., Kammann, E.E. & Wand, M.P. 2002, 'Erratum: Semiparametric nonlinear mixed-effects models and their application (Journal of the American Statistical Association (2002) 96 (1285-1288))', Journal of the American Statistical Association, vol. 97, no. 458, p. 661.
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Zeng, Q., Wand, M., Young, A.J., Rawn, J., Milford, E.L., Mentzer, S.J. & Greenes, R.A. 2002, 'Matching of flow-cytometry histograms using information theory in feature space.', Proceedings / AMIA ... Annual Symposium. AMIA Symposium, pp. 929-933.
Flow cytometry is a widely available technique for analyzing cell-surface protein expression. Data obtained from flow cytometry is frequently used to produce fluorescence intensity histograms. Comparison of histograms can be useful in the identification of unknown molecules and in the analysis of protein expression. In this study, we examined the combination of a new smoothing technique called SiZer with information theory to measure the difference between cytometry histograms. SiZer provides cross-bandwidth smoothing and allowed analysis in feature space. The new methods were tested on a panel of monoclonal antibodies raised against proteins expressed on peripheral blood lymphocytes and compared with previous methods. The findings suggest that comparing information content of histograms in feature space is effective and efficient for identifying antibodies with similar cell-surface binding patterns.
Cai, T., Hyndman, R.J. & Wand, M.P. 2002, 'Mixed model-based hazard estimation', Journal of Computational and Graphical Statistics, vol. 11, no. 4, pp. 784-798.
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This article proposes a new method for estimation of the hazard function from a set of censored failure time data, with a view to extending the general approach to more complicated models. The approach is based on a mixed model representation of penalized spline hazard estimators. One payoff is the automation of the smoothing parameter choice through restricted maximum likelihood. Another is the option to use standard mixed model software for automatic hazard estimation.
Kim, E.Y., Zeng, Q., Rawn, J., Wand, M., Young, A.J., Milford, E., Mentzer, S.J. & Greenes, R.A. 2002, 'Using a neural network with flow cytometry histograms to recognize cell surface protein binding patterns.', Proceedings / AMIA ... Annual Symposium. AMIA Symposium, pp. 380-384.
Flow cytometric systems are being used increasingly in all branches of biological science including medicine. To develop analytic tools for identifying unknown molecules such as the antibodies that recognize different structure in the identical antigens, we explored use of a neural network in flow cytometry data comparison. Peak locations were extracted from flow cytometry histograms and we used the Marquardt backpropagation neural networks to recognize identical or similar binding patterns between antibodies and antigens based on the peak locations. The neural network showed 93.8% to 99.6% correct classification rates for identical or similar molecules. This suggests that the neural network technique can be useful in flow cytometry histogram data analysis.
Mammen, E., Marron, J.S., Turlach, B.A. & Wand, M.P. 2001, 'A General Projection Framework for Constrained Smoothing', Statistical Science, vol. 16, no. 3, pp. 232-248.
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There are a wide array of smoothing methods available for finding structure in data. A general framework is developed which shows that many of these can be viewed as a projection of the data, with respect to appropriate norms. The underlying vector space is an unusually large product space, which allows inclusion of a wide range of smoothers in our setup (including many methods not typically considered to be projections). We give several applications of this simple geometric interpretation of smoothing. A major payoff is the natural and computationally frugal incorporation of constraints. Our point of view also motivates new estimates and helps understand the finite sample and asymptotic behavior of these estimates.
Parise, H., Wand, M.P., Ruppert, D. & Ryan, L. 2001, 'Incorporation of historical controls using semiparametric mixed models', Journal of the Royal Statistical Society. Series C: Applied Statistics, vol. 50, no. 1, pp. 31-42.
The analysis of animal carcinogenicity data is complicated by various statistical issues. A topic of recent debate is how to control for the effect of the animals' body weight on the outcome of interest, the onset of tumours. We propose a method which incorporates historical information from the control animals in previously conducted experiments. We allow non-linearity in the effects of body weight by modelling the relationship nonparametrically through a penalized spline. A simple extension of the penalized spline model allows the relationship between weight and onset of the tumour to vary from one experiment to another.
Coull, B.A., Schwartz, J. & Wand, M.P. 2001, 'Respiratory health and air pollution: additive mixed model analyses.', Biostatistics, vol. 2, no. 3, pp. 337-349.
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We conduct a reanalysis of data from the Utah Valley respiratory health/air pollution study of Pope and co-workers (Pope et al., 1991) using additive mixed models. A relatively recent statistical development (e.g. Wang, 1998; Verbyla et al., 1999; Lin and Zhang, 1999), the methods allow for smooth functional relationships, subject-specific effects and time series error structure. All three of these are apparent in the Utah Valley data.
Coull, B.A., Ruppert, D. & Wand, M.P. 2001, 'Simple incorporation of interactions into additive models', Biometrics, vol. 57, no. 2, pp. 539-545.
Often, the functional form of covariate effects in an additive model varies across groups defined by levels of a categorical variable. This structure represents a factor-by-curve interaction. This article presents penalized spline models that incorporate factor-by-curve interactions into additive models. A mixed model formulation for penalized splines allows for straightforward model fitting and smoothing parameter selection. We illustrate the proposed model by applying it to pollen ragweed data in which seasonal trends vary by year.
Zeng, Q., Young, A.J., Boxwala, A., Rawn, J., Long, W., Wand, M., Salganik, M., Milford, E.L., Mentzer, S.J. & Greenes, R.A. 2001, 'Molecular identification using flow cytometry histograms and information theory.', Proceedings / AMIA ... Annual Symposium. AMIA Symposium, pp. 776-780.
Flow cytometry is a common technique for quantitatively measuring the expression of individual molecules on cells. The molecular expression is represented by a frequency histogram of fluorescence intensity. For flow cytometry to be used as a knowledge discovery tool to identify unknown molecules, histogram comparison is a major limitation. Many traditional comparison methods do not provide adequate assessment of histogram similarity and molecular relatedness. We have explored a new approach applying information theory to histogram comparison, and tested it with histograms from 14 antibodies over 3 cell types. The information theory approach was able to improve over traditional methods by recognizing various non-random correlations between histograms in addition to similarity and providing a quantitative assessment of similarity beyond hypothesis testing of identity.
Moore, P.E., Laporte, J.D., Abraham, J.H., Schwartzman, I.N., Yandava, C.N., Silverman, E.S., Drazen, J.M., Wand, M.P., Panettieri, R.A. & Shore, S.A. 2000, 'Polymorphism of the ?2-adrenergic receptor gene and desensitization in human airway smooth muscle', American Journal of Respiratory and Critical Care Medicine, vol. 162, no. 6, pp. 2117-2124.
We examined the influence of two common polymorphic forms of the ?2-adrenergic receptor (?2AR): the Gly16 and Glu27 alleles, on acute and long-term ?2AR desensitization in human airway smooth muscle (HASM) cells. In cells from 15 individuals, considered without respect to genotype, pretreatment with Isoproterenol (ISO) at 10-7 M for 1 h or 24 h caused approximately 25% and 64% decreases in the ability of subsequent ISO (10-6 M) stimulation to reduce HASM cell stiffness as measured by magnetic twisting cytometry. Similar results were obtained with ISO-induced cyclic adenosine monophosphate (cAMP) as the outcome indicator. Data were then stratified post hoc by genotype. Cells containing at least one Glu27 allele (equivalent to presence of the Gly16Glu27 haplotype) showed significantly greater acute desensitization than did cells with no Glu27 allele, whether ISO-induced cell stiffness (34% versus 19%, p < 0.03) or cAMP formation (58% versus 11%, p < 0.02) was measured. Likewise, cells with any Glu27 allele showed greater long-term desensitization of cell stiffness and cAMP formation responses than did cells without the Glu27 allele. The distribution of genotypes limited direct conclusions about the influence of the Gly16 allele. However, presence of the Gly16Gln27 haplotype was associated with less acute and long-term desensitization of ISO-induced cAMP formation than was seen in cells without the Gly16Gln27 haplotype (14% versus 47%, p < 0.09 for short-term desensitization; 32% versus 84%, p < 0.01 for long-term desensitization), suggesting that the influence of Glu27 is not through its association with Gly16. The Glu27 allele was in strong linkage disequilibrium with the Arg19 allele, a polymorphic form of the ?2AR upstream peptide of the 5?-leader cistron of the ?2AR, and this polymorphism in the ?2AR 5?-flanking region may explain the effects of the Glu27 allele. Cells with any Arg19 allele showed significantly greater acute and long-term desensitization of ISO...
Wand, M.P. 2000, 'A comparison of regression spline smoothing procedures', Computational Statistics, vol. 15, no. 4, pp. 443-462.
Regression spline smoothing involves modelling a regression function as a piecewise polynomial with a high number of pieces relative to the sample size. Because the number of possible models is so large, efficient strategies for choosing among them are required. In this paper we review approaches to this problem and compare them through a simulation study. For simplicity and conciseness we restrict attention to the univariate smoothing setting with Gaussian noise and the truncated polynomial regression spline basis.
Zanobetti, A., Wand, M.P., Schwartz, J. & Ryan, L.M. 2000, 'Generalized additive distributed lag models: quantifying mortality displacement.', Biostatistics, vol. 1, no. 3, pp. 279-292.
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There are a number of applied settings where a response is measured repeatedly over time, and the impact of a stimulus at one time is distributed over several subsequent response measures. In the motivating application the stimulus is an air pollutant such as airborne particulate matter and the response is mortality. However, several other variables (e.g. daily temperature) impact the response in a possibly non-linear fashion. To quantify the effect of the stimulus in the presence of covariate data we combine two established regression techniques: generalized additive models and distributed lag models. Generalized additive models extend multiple linear regression by allowing for continuous covariates to be modeled as smooth, but otherwise unspecified, functions. Distributed lag models aim to relate the outcome variable to lagged values of a time-dependent predictor in a parsimonious fashion. The resultant, which we call generalized additive distributed lag models, are seen to effectively quantify the so-called 'mortality displacement effect' in environmental epidemiology, as illustrated through air pollution/mortality data from Milan, Italy.
Wechsler, M.E., Grasemann, H., Deykin, A., Silverman, E.K., Yandava, C.N., Israel, E., Wand, M. & Drazen, J.M. 2000, 'Exhaled nitric oxide in patients with asthma: Association with NOS1 genotype', American Journal of Respiratory and Critical Care Medicine, vol. 162, no. 6, pp. 2043-2047.
An increased concentration of nitric oxide (NO) in exhaled air (FeNO) is now recognized as a critical component of the asthmatic phenotype. When we identified patients with asthma on the basis of a standard case definition alone, we found that they were remarkably heterogeneous with respect to their FeNO. However, when we included genotype at a prominent asthma candidate gene (i.e., NOS1) in the case definition, and determined the number of AAT repeats in intron 20, we identified a remarkably homogenous cohort of patients with respect to FeNO. Both mean FeNO (p = 0.00008) and variability around the mean (p = 0.000002) were significantly lower in asthmatic individuals with a high number (? 12) of AAT repeats at this locus than in those with fewer repeats. These data provide a biologically tenable link between genotype at a candidate gene in a region of linkage, NOS1, and an important component of the asthmatic phenotype, FeNO. We show that addition of NOS1 genotype to the case definition of asthma allows the identification of a uniform cohort of patients, with respect to FeNO, that would have been indistinguishable by other physiologic criteria. Out isolation of this homogenous cohort of patients ties together the well-established associations among asthma, increased concentrations of NO in the exhaled air of asthmatic individuals, and variations of trinucleotide repeat sequences as identified in several neurologic conditions.
Thurston, S.W., Wand, M.P. & Wiencke, J.K. 2000, 'Negative binomial additive models', Biometrics, vol. 56, no. 1, pp. 139-144.
The generalized additive model is extended to handle negative binomial responses. The extension is complicated by the fact that the negative binomial distribution has two parameters and is not in the exponential family. The methodology is applied to data involving DNA adduct counts and smoking variables among ex-smokers with lung cancer. A more detailed investigation is made of the parametric relationship between the number of adducts and years since quitting while retaining a smooth relationship between adducts and the other covariates.
Wand, M.P. 1999, 'A central limit theorem for local polynomial backfitting estimators', Journal of Multivariate Analysis, vol. 70, no. 1, pp. 57-65.
Additive models based on backfitting estimators are among the most important recent contributions to modern statistical modelling. However, the statistical properties of backfitting estimators have received relatively little attention. Recently, J.-D. Opsomer and D. Ruppert (1997, Ann. Statist. 25, 186-211; 1998, J. Amer. Statist. Assoc. 93, 605-619) and J.-D. Opsomer (1997, preprint 96-12, Department of statistics, Iowa State University) derived their mean squared error properties in the case of local polynomial smoothers. In this paper the asymptotic distributional behaviour of backfitting estimators is investigated. &copy; 1999 Academic Press.
Opsomer, J.D., Ruppert, D., Wand, M.P., Holst, U. & Hössjer, O. 1999, 'Kriging with nonparametric variance function estimation', Biometrics, vol. 55, no. 3, pp. 704-710.
A method for fitting regression models to data that exhibit spatial correlation and heteroskedasticity is proposed. It is well known that ignoring a nonconstant variance does not bias least-squares estimates of regression parameters, thus, data analysts are easily lead to the false belief that moderate heteroskedasticity can generally be ignored. Unfortunately, ignoring nonconstant variance when fitting variograms can seriously bias estimated correlation functions. By modeling heteroskedasticity and standardizing by estimated standard deviations, our approach eliminates this bias in the correlations. A combination of parametric and nonparametric regression techniques is used to iteratively estimate the various components of the model. The approach is demonstrated on a large data set of predicted nitrogen runoff from agricultural lands in the Midwest and Northern Plains regions of the U.S.A. For this data set, the model comprises three main components. (1) the mean function, which includes farming practice variables, local soil and climate characteristics, and the nitrogen application treatment, is assumed to be linear in the parameters and is fitted by generalized least squares; (2) the variance function, which contains a local and a spatial component whose shapes are left unspecified, is estimated by local linear regression; and (3) the spatial correlation function is estimated by fitting a parametric variogram model to the standardized residuals, with the standardization adjusting the variogram for the presence of heteroskedasticity. The fitting of these three components is iterated until convergence. The model provides an improved fit to the data compared with a previous model that ignored the heteroskedasticity and the spatial correlation.
Wand, M.P. 1999, 'On the optimal amount of smoothing in penalised spline regression', Biometrika, vol. 86, no. 4, pp. 936-940.
The optimal amount of smoothing in penalised spline regression is investigated. In particular, a simple closed form approximation to the optimal smoothing parameter is derived. Comparisons with its exact counterpart show it to be a useful starting point for measuring the optimal amount of smoothing in penalised spline regression. It also lends itself to the development of quick and simple rules for automatic smoothing parameter selection. &copy; 1999 Biometrika Trust.
Gijbels, I., Pope, A. & Wand, M.P. 1999, 'Understanding exponential smoothing via kernel regression', Journal of the Royal Statistical Society. Series B: Statistical Methodology, vol. 61, no. 1, pp. 39-50.
Exponential smoothing is the most common model-free means of forecasting a future realization of a time series. It requires the specification of a smoothing factor which is usually chosen from the data to minimize the average squared residual of previous one-step-ahead forecasts. In this paper we show that exponential smoothing can be put into a nonparametric regression framework and gain some interesting insights into its performance through this interpretation. We also use theoretical developments from the kernel regression field to derive, for the first time, asymptotic properties of exponential smoothing forecasters.
Betensky, R.A., Lindsey, J.C., Ryan, L.M. & Wand, M.P. 1999, 'Local EM estimation of the hazard function for interval-censored data', Biometrics, vol. 55, no. 1, pp. 238-245.
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We propose a smooth hazard estimator for interval-censored survival data using the method of local likelihood. The model is fit using a local EM algorithm. The estimator is more descriptive than traditional empirical estimates in regions of concentrated information and takes on a parametric flavor in regions of sparse information. We derive two different standard error estimates for the smooth curve, one based on asymptotic theory and the other on the bootstrap. We illustrate the local EM method for times to breast cosmesis deterioration (Finkelstein, 1986, Biometrics 42, 845-854) and for times to HIV-1 infection for individuals with hemophilia (Kroner et al., 1994, Journal of AIDS 7, 279-286). Our hazard estimates for each of these data sets show interesting structures that would not be found using a standard parametric hazard model or empirical survivorship estimates.
Augustyns, I. & Wand, M.P. 1998, 'Bandwidth selection for local polynomial smoothing of multinomial data', Computational Statistics, vol. 13, no. 4, pp. 447-461.
We develop a rule for choosing bandwidths for local polynomial smoothing of ordered multinomial data. Our method is a variant of the double smoothing idea and is particularly geared towards good performance near the boundaries of the data, through the use of exact risk expressions.
Wand, M.P. 1998, 'Finite sample performance of deconvolving density estimators', Statistics and Probability Letters, vol. 37, no. 2, pp. 131-139.
Recent studies have shown that the asymptotic performance of nonparametric curve estimators in the presence of measurement error will often be very much inferior to that when the observations are error-free. For example, deconvolution of Gaussian measurement error worsens the usual algebraic convergence rates of kernel estimators to very slow logarithmic rates. However, the slow convergence rates mean that very large sample sizes may be required for the asymptotics to take effect, so the finite sample properties of the estimator may not be very well described by the asymptotics. In this article finite sample calculations are performed for the important cases of Gaussian and Laplacian measurement error which provide insight into the feasibility of deconvolving density estimators for practical sample sizes. Our results indicate that for lower levels of measurement error deconvolving density estimators can perform well for reasonable sample sizes. &copy; 1998 Elsevier Science B.V. All rights reserved.
Hyndman, R.J. & Wand, M.P. 1997, 'NONPARAMETRIC AUTOCOVARIANCE FUNCTION ESTIMATION', Australian Journal of Statistics, vol. 39, no. 3, pp. 313-324.
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Wand, M.P. 1997, 'Data-Based Choice of Histogram Bin Width', American Statistician, vol. 51, no. 1, pp. 59-64.
The most important parameter of a histogram is the bin width because it controls the tradeoff between presenting a picture with too much detail ("undersmoothing") or too little detail ("oversmoothing") with respect to the true distribution. Despite this importance there has been surprisingly little research into estimation of the "optimal"' bin width. Default bin widths in most common statistical packages are, at least for large samples, quite far from the optimal bin width. Rules proposed by, for example, Scott lead to better large sample performance of the histogram, but are not consistent themselves. In this paper we extend the bin width rules of Scott to those that achieve root-n rates of convergence to the L2-optimal bin width, thereby providing firm scientific justification for their use. Moreover, the proposed rules are simple, easy and fast to compute, and perform well in simulations.
Wand, M.P. & Gutierrez, R.G. 1997, 'Exact risk approaches to smoothing parameter selection', Journal of Nonparametric Statistics, vol. 8, no. 4, pp. 337-354.
The past decade has seen the development of a large number of second-generational smoothing parameter selectors as a response to the high degree of variability of cross-validatory methods. However, most of these rules rely on asymptotic approximations which make them subject to adverse performance when the approximations are poor. They are also difficult to extend to those settings where asymptotics is difficult. We aim to alleviate each of these problems by developing rules based on exact expressions for the risk.
Carroll, R.J., Fan, J., Gijbels, I. & Wand, M.P. 1997, 'Generalized partially linear single-index models', Journal of the American Statistical Association, vol. 92, no. 438, pp. 477-489.
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The typical generalized linear model for a regression of a response Y on predictors (X, Z) has conditional mean function based on a linear combination of (X, Z). We generalize these models to have a nonparametric component, replacing the linear combination ?T 0X + ?T 0Z by ?0(?T 0X) + ?T 0Z, where ?0() is an unknown function. We call these generalized partially linear single-index models (GPLSIM). The models include the "single-index" models, which have ?0 = 0. Using local linear methods, we propose estimates of the unknown parameters (?0, ?0) and the unknown function ?0 () and obtain their asymptotic distributions. Examples illustrate the models and the proposed estimation methodology.
Ruppert, D., Wand, M.P., Holst, U. & Hössjer, O. 1997, 'Local polynomial variance-function estimation', Technometrics, vol. 39, no. 3, pp. 262-273.
The conditional variance function in a heteroscedastic, nonparametric regression model is estimated by linear smoothing of squared residuals. Attention is focused on local polynomial smoothers. Both the mean and variance functions are assumed to be smooth, but neither is assumed to be in a parametric family. The biasing effect of preliminary estimation of the mean is studied, and a degrees-of-freedom correction of bias is proposed. The corrected method is shown to be adaptive in the sense that the variance function can be estimated with the same asymptotic mean and variance as if the mean function were known. A proposal is made for using standard bandwidth selectors for estimating both the mean and variance functions. The proposal is illustrated with data from the LIDAR method of measuring atmospheric pollutants and from turbulence-model computations.
Turlach, B.A. & Wand, M.P. 1996, 'Fast computation of auxiliary quantities in local polynomial regression', Journal of Computational and Graphical Statistics, vol. 5, no. 4, pp. 337-350.
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We investigate the extension of binning methodology to fast computation of several auxiliary quantities that arise in local polynomial smoothing. Examples include degrees of freedom measures, cross-validation functions, variance estimates, and exact measures of error. It is shown that the computational effort required for such approximations is of the same order of magnitude as that required for a binned local polynomial smooth.
Hall, P. & Wand, M.P. 1996, 'On the accuracy of binned kernel density estimators', Journal of Multivariate Analysis, vol. 56, no. 2, pp. 165-184.
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The accuracy of the binned kernel density estimator is studied for general binning rules. We derive mean squared error results for the closeness of this estimator to both the true density and the unbinned kernel estimator. The binning rule and smoothness of the kernel function are shown to influence the accuracy of the binned kernel estimators. Our results are used to compare commonly used binning rules, and to determine the minimum grid size required to obtain a given level of accuracy. &copy; 1996 Academic Press, Inc.
González-Manteiga, W., Sánchez-Sellero, C. & Wand, M.P. 1996, 'Accuracy of binned kernel functional approximations', Computational Statistics and Data Analysis, vol. 22, no. 1, pp. 1-16.
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Virtually all common bandwidth selection algorithms are based on a certain type of kernel functional estimator. Such estimators can be computationally very expensive, so in practice they are often replaced by fast binned approximations. This is especially worthwhile when the bandwidth selection method involves iteration. Results for the accuracy of these approximations are derived and then used to provide an understanding of the number of binning grid points required to achieve a given level of accuracy. Our results apply to both univariate and multivariate settings. Multivariate contexts are of particular interest since the cost due to having a higher number of grid points can be quite significant.
Herrmann, E., Wand, M., Engel, J. & Gasser, T. 1995, 'A bandwidth selector for bivariate kernel regression', Journal of The Royal Statistical Society Series B-methodological, vol. 57, no. 1, pp. 171-180.
Ruppert, D., Sheather, S.J. & Wand, M. 1995, 'An effective bandwidth selector for local least squares regression', Journal of the American Statistical Association, vol. 90, no. 432, pp. 1257-1270.
Local least squares kernel regression provides an appealing solution to the nonparametric regression, or "scatterplot smoothing," problem, as demonstrated by Fan, for example. The practical implementation of any scatterplot smoother is greatly enhanced by the availability of a reliable rule for automatic selection of the smoothing parameter. In this article we apply the ideas of plug-in bandwidth selection to develop strategies for choosing the smoothing parameter of local linear squares kernel estimators. Our results are applicable to odd-degree local polynomial fits and can be extended to other settings, such as derivative estimation and multiple nonparametric regression. An implementation in the important case of local linear fits with univariate predictors is shown to perform well in practice. A by-product of our work is the development of a class of nonparametric variance estimators, based on local least squares ideas, and plug-in rules for their implementation.
Aldershof, B., Marron, J.S., Park, B.U. & Wand, M.P. 1995, 'Facts about the gaussian probability density function', Applicable Analysis, vol. 59, no. 1-4, pp. 289-306.
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Fan, J., Heckman, N.E. & Wand, M. 1995, 'Local polynomial kernel regression for generalized linear models and quasi-likelihood functions', Journal of the American Statistical Association, vol. 90, no. 429, pp. 141-150.
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Wand, M. 1994, 'Fast computation of multivariate kernel estimators', Journal of Computational and Graphical Statistics, vol. 3, no. 4, pp. 433-445.
Ruppert, D. & Wand, M.P. 1994, 'Multivariate Locally Weighted Least Squares Regression', The Annals of Statistics, vol. 22, no. 3, pp. 1346-1370.
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Wand, M. & Jones, M.C. 1994, 'Multivariate plug-in bandwidth selection', Computational Statistics, vol. 9, pp. 97-116.
Wand, M. & Jones, M.C. 1993, 'Comparison of smoothing parameterizations in bivariate kernel density estimation', Journal of the American Statistical Association, vol. 88, no. 422, pp. 520-528.
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Wand, M.P. & Devroye, L. 1993, 'How easy is a given density to estimate?', Computational Statistics and Data Analysis, vol. 16, no. 3, pp. 311-323.
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In data analytic applications of density estimation one is usually interested in estimating the density over its support. However, common estimators such as the basic kernel estimator use a single smoothing parameter over the whole of the support. While this will be adequate for some densities there will be other densities that will be very difficult to estimate using this approach. The purpose of this article is to quantify how easy a particular density is to estimate using a global smoothing parameter. By considering the asymptotic expected L1 error we obtain a scale invariant functional that is useful for measuring degree of estimation difficulty. Implications for the transformation kernel density estimators, which attempt to overcome the inadequacy of the basic kernel estimator, are also discussed. &copy; 1993.
Devroye, L. & Wand, M. 1993, 'On the influence of the density on the kernel estimate', Statistics, vol. 24, pp. 215-233.
Jones, M.C. & Wand, M.P. 1992, 'Asymptotic effectiveness of some higher order kernels', Journal of Statistical Planning and Inference, vol. 31, no. 1, pp. 15-21.
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'Spline-equivalent' kernels and 'exponential power' kernels are considered as higher order kernels for use in kernel estimation of a function and its derivatives. They form two more practicable classes of alternatives to 'optimal' polynomial kernels, along with Gaussian-based ones. Both firstnamed families of kernels exhibit good theoretical performance for orders four and/or six, actually improving on the polynomial kernels for many such cases. &copy; 1992.
Marron, J.S. & Wand, M.P. 1992, 'Exact Mean Integrated Squared Error', The Annals of Statistics, vol. 20, no. 2, pp. 712-736.
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Wand, M.P. 1992, 'Finite sample performance of density estimators under moving average dependence', Statistics and Probability Letters, vol. 13, no. 2, pp. 109-115.
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We study the finite sample performance of kernel density estimators through exact mean integrated squared error formulas when the data belong to an infinite order moving average process. It is demonstrated that dependence can have a significant influence, even in situations where the asymptotic performance is unaffected. &copy; 1992.
Wand, M.P. 1992, 'Error analysis for general multtvariate kernel estimators', Journal of Nonparametric Statistics, vol. 2, no. 1, pp. 1-15.
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Ruppert, D. & Wand, M.P. 1992, 'CORRECTING FOR KURTOSIS IN DENSITY ESTIMATION', Australian Journal of Statistics, vol. 34, no. 1, pp. 19-29.
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Wand, M., Marron, J.S. & Ruppert, D. 1991, 'Transformations in density estimation', Journal of the American Statistical Association, vol. 86, no. 414, pp. 343-353.
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Carroll, R.J. & Wand, M. 1991, 'Semiparametric estimation in logistic measurement error models', Journal of The Royal Statistical Society Series B-methodological, vol. 53, no. 3, pp. 573-585.
SCOTT, D.A.V.I.D.W. 1991, 'Feasibility of multivariate density estimates', Biometrika, vol. 78, no. 1, pp. 197-205.
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Wand, M. 1990, 'On exact L1 rates of convergence in non-parametric kernel regression', Scandinavian Journal of Statistics, vol. 17, no. 3, pp. 251-256.
Hardle, W., Marron, J.S. & Wand, M. 1990, 'Bandwidth choice for density derivatives', Journal of The Royal Statistical Society Series B-methodological, vol. 52, no. 1, pp. 223-232.
Wand, M.P. & Schucany, W.R. 1990, 'Gaussian-based kernels', Canadian Journal of Statistics, vol. 18, no. 3, pp. 197-204.
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Hall, P. & Wand, M.P. 1988, 'Minimizing L1 distance in nonparametric density estimation', Journal of Multivariate Analysis, vol. 26, no. 1, pp. 59-88.
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We construct a simple algorithm, based on Newton's method, which permits asymptotic minimization of L1 distance for nonparametric density estimators. The technique is applicable to multivariate kernel estimators, multivariate histogram estimators, and smoothed histogram estimators such as frequency polygons. It has an "adaptive" or "data-driven" version. We show theoretically that both theoretical and adaptive forms of the algorithm do indeed minimize asymptotic L1 distance. Then we apply the algorithm to derive concise formulae for asymptotically optimal smoothing parameters. We also give numerical examples of applications of the adaptive algorithm. &copy; 1988.
Hall, P. & Wand, M.P. 1988, 'On nonparametric discrimination using density differences', Biometrika, vol. 75, no. 3, pp. 541-547.
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We propose a technique for nonparametric discrimination in which smoothing parameters are chosen jointly, according to a criterion based on the difference between two densities. The approach is suitable for categorical, continuous and mixed data, and uses information from both populations to determine the smoothing parameter for any one population. In the case of categorical data, optimal performance is sometimes achieved using negative smoothing parameters, a property which does not emerge if the smoothing parameters are chosen individually. &copy; 1988 Biometrika Trust.
Hall, P. & Wand, M.P. 1988, 'On the minimization of absolute distance in kernel density estimation', Statistics and Probability Letters, vol. 6, no. 5, pp. 311-314.
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We given an algorithm for determining the window-size which minimizes mean absolute distance in kernel density estimation and discuss the practical implications of our results. &copy; 1988.
Luts, J. & Wand, M.P., 'Variational inference for count response semiparametric regression'.
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Fast variational approximate algorithms are developed for Bayesian semiparametric regression when the response variable is a count, i.e. a non-negative integer. We treat both the Poisson and Negative Binomial families as models for the response variable. Our approach utilizes recently developed methodology known as non-conjugate variational message passing. For concreteness, we focus on generalized additive mixed models, although our variational approximation approach extends to a wide class of semiparametric regression models such as those containing interactions and elaborate random effect structure.
Hall, P., Pham, T., Wand, M.P. & Wang, S.S.J., 'Asymptotic normality and valid inference for Gaussian variational approximation', Annals of Statistics, vol. 39, no. 5, pp. 2502-2532.
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We derive the precise asymptotic distributional behavior of Gaussian variational approximate estimators of the parameters in a single-predictor Poisson mixed model. These results are the deepest yet obtained concerning the statistical properties of a variational approximation method. Moreover, they give rise to asymptotically valid statistical inference. A simulation study demonstrates that Gaussian variational approximate confidence intervals possess good to excellent coverage properties, and have a similar precision to their exact likelihood counterparts.