Dec 10, 11
9 am - 5 pm

Get presentation slides.

The workshop is free, but please register in advance with nancy@stat.ubc.ca

The workshop is designed to provide something of value to as wide a range of participants as possible, ranging from those interested in whether FDA might prove useful in their research, to statistical methodologists looking for research problems and interested in new techniques.

Each lecture will begin with one or more case studies, and the initial lectures will be almost entirely case studies. These aim to show the range of applications possible, show what insights might be gained from using FDA methods, and illustrate the challenges that are specific or particularly relevant to the analysis functional data. Case studies are not "how to" sessions, but rather address questions like, "Why should I consider this approach?" and "What should I watch out for?"

The first half of the first day will also be more oriented to the preliminaries of functional data analysis:

• What are functional data?
• How should they be prepared for analysis?
• How do we convert discrete noisy data to smooth functions?
• What data exploration tools are useful?
• Do the data display both phase and amplitude variation?
• What about principal components analysis and other exploratory methods?

The remainder of the first day and some of the second day will consider linear models for functional data. This is a vast topic, and includes relatively basic topics like functional versions of analysis of variance and regression analysis, as well as issues less familiar to statisticians such as how differential equations can be used to model functional data. All approaches assume that the goal is to explain variation in one or more response variables by variation in one or more input or independent variables where, naturally, at least one of the variables involved is functional.

For more information on functional data analysis, see the FDA website.

Dec 10, 11
9 am - 5 pm

Get presentation slides.

The workshop is free, but please register in advance with nancy@stat.ubc.ca

The workshop is designed to provide something of value to as wide a range of participants as possible, ranging from those interested in whether FDA might prove useful in their research, to statistical methodologists looking for research problems and interested in new techniques.

Each lecture will begin with one or more case studies, and the initial lectures will be almost entirely case studies. These aim to show the range of applications possible, show what insights might be gained from using FDA methods, and illustrate the challenges that are specific or particularly relevant to the analysis functional data. Case studies are not "how to" sessions, but rather address questions like, "Why should I consider this approach?" and "What should I watch out for?"

The first half of the first day will also be more oriented to the preliminaries of functional data analysis:

• What are functional data?
• How should they be prepared for analysis?
• How do we convert discrete noisy data to smooth functions?
• What data exploration tools are useful?
• Do the data display both phase and amplitude variation?
• What about principal components analysis and other exploratory methods?

The remainder of the first day and some of the second day will consider linear models for functional data. This is a vast topic, and includes relatively basic topics like functional versions of analysis of variance and regression analysis, as well as issues less familiar to statisticians such as how differential equations can be used to model functional data. All approaches assume that the goal is to explain variation in one or more response variables by variation in one or more input or independent variables where, naturally, at least one of the variables involved is functional.

For more information on functional data analysis, see the FDA website.

Differential equations (DIFE's) can represent the underlying processes giving rise to observed functional data, and as such can offer a number of potential advantages over parametric or nonparametric basis expansion models.

• DIFE's explicitly model the behavior of derivatives, and link this behavior to the observed function itself. Consequently, they model the rate of change in the data as well as their amplitudes.
• Solutions to a linear DIFE of order m span an m-dimensional space, and consequently have the capacity to model curve-to-curve variation as well as to fit the data.
• We can build known structural features into DIFE models more easily than is usually the case for conventional functional models.
• Derivative estimates based on DIFE's are usually superior to those derived from conventional data smoothers.
• And finally a DIFE offers a wider range of ways to introduce stochastic behavior into models.

In spite of the enormous importance of DIFE models in many areas of science and engineering, existing methods for actually identifying or estimating a differential equation from noisy data remain crude, inefficient, and unable to deliver estimates of sampling error.

I will discuss a technique for going directly from the discrete and noisy data to a DIFE that is based on the work of Heckman and Ramsay (2000). Some illustrations of its performance for simulated data will be offered as well as examples from chemical engineering and for medical data on treatment regimes for lupus.

Heckman, N. and Ramsay, J. O. (2000) Penalized regression with model-based penalties. The Canadian Journal of Statistics, 28, 241-258.

Accelerated longitudinal design (ALD) sampling schemes consist of a few observations per sampling unit over a short time span. ALD data are combined across independent units to provide an estimate of overall population curve and predictions of individual patterns of change. Extending the work of Ruppert, Wand and Carroll (2003), we develop computationally efficient procedure for the longitudinal penalized regression splines (P-spline) methods under ALD sampling schemes. Major advantage of the P-spline methodology is that the models can be fit using standard mixed models software (e.g. PROC MIXED in SAS).

Extensive simulation studies indicate good performance of our method in the settings considered. We compare balanced and complete longitudinal designs to ALDs using the Berkeley Growth study data and we apply our method to the longitudinal brain volume measurements from an ongoing pediatric magnetic resonance imaging (MRI) developmental study.

This talk is based on joint work with Louise Ryan, Nicholas Lange and Jay Giedd.

The Principal Components Decomposition of a discrete data matrix is a widely used technique of Multivariate Analysis. Given some multivariate data from a population, its objective is to find some new variables that explain the most important causes of variability, while being uncorrelated between them. These new variables, called Principal Components, help to analyze the data more easily, by reducing the dimensionality, while at the same time they don't interfere with each other.

We can interpret a discrete data matrix as a group of discrete variables. We are interested in knowing if it would be possible to do the same with any continuous random variable, obtaining the equivalent to the Principal Components. The answer is affirmative, and we can find a series of random variables, called Principal Directions, with analogous properties to those of the discrete case. Actually, we can obtain them by generalizing all the mathematical operations from the discrete case, to their continuous equivalent in the general case. Thanks to this decomposition of this random variable, we find a more convenient way to represent it. It has some applications, like formulating a continuous extension of multidimensional scaling, obtaining a graphical test to distinguish between similar distributions, improving some tests of independence by relating principal components, contributing to the study of the asymptotic distribution of some statistics related to Rao's ANOQE (a generalization of ANOVA), studying some tests of goodness of fit, etc.

Cluster analysis (unsupervised learning) is a challenging problem. Its goal is to detect the separated subsets or clusters in high-dimensional data sets. Many clustering methods have been proposed. If we apply different clustering methods to the same data set, the partitions usually are different. Cluster validation methods are needed to compare the different partitions for the same data set and to check the appropriateness of a given partition for a data set, without the knowledge of the known cluster structure. We consider these based on the degree of separation among clusters.

In this talk, we propose a separation index to measure the degree of separation (or magnitude of gap) between pair of clusters. The separation index is intuitively appealing and easy to compute. The separation indexes can be used to compare partitions from different clustering methods. Based on the separation index, we propose a method to visualize the degree of separation between pair of clusters in a low-dimensional space. We also propose a sequential clustering method based on the separation index to simultaneously estimate the number of clusters and obtain partitions.

In this paper, we propose a general methodology to sample sequentially from a series of probability distributions known up to a normalizing constant and defined on a common space; i.e. in a context where one usually uses Markov chain Monte Carlo (MCMC). These probability distributions are approximated by a cloud of weighted random samples which are propagated over time using Sequential Monte Carlo methods. This methodology allows us to derive not only simple algorithms to make parallel Markov chain Monte Carlo runs interact in a principled way but also to obtain new methods for global optimization and sequential Bayesian estimation. We demonstrate the performance of these algorithms through simulation for various integration and global optimization tasks arising in the context of Bayesian inference. Precise convergence results have been established and will also be presented.

In the CS talk (to be given Thursday, 29 April) I will emphasize the applications whereas I will emphasize the methodology in the stats talk.

There has been increasing emphasis on cost comparison and cost-effectiveness analysis in health care delivery systems. However, the correct analysis of health care costs may be impeded by the following characteristics of health care costs: (1) a high proportion of patients with zero costs, (2) a highly skewed distribution for non-zero costs, and (3) heteroscedasticity (non-constant variance). Without taking these characteristics into consideration, statistical analyses of health care cost costs can lead to unreliable inferences and predictions. In this talk we first briefly review some statistical methods for one sample, two sample, and ANOVA type problems, which can adjust for these characteristics. We then introduce a new semi-parametric regression model for cost data with some zero values. The proposed approach is based on a monotone transformation of cost observations and the analysis of the data on the transformed scale. The main complication in such the approach is the retransformation bias, which arises when one transforms back results to the original scale for prediction and forecasting. To allow for the fact that transformation of the nonzero responses may not achieve normality and homoscedasticity, we fit a semi-parametric heteroscedastic regression model to transformed nonzero responses and then propose two non-parametric estimators for the mean response on the original scale.

We consider the problem of finding a bias-robust estimator of a scale parameter of a symmetric unimodal parametric distribution F when a proportion of the observations arise from an arbitrary contaminating distribution. The class of scale functionals considered are those which are location invariant, scale equivariant, Fisher consistent at F, and weakly continuous. Lower bounds for the (suitably defined) maximal asymptotic biases are derived as well as lower bounds on gross errors sensitivities (the limiting case as the proportion of contamination approaches 0). Computational verifications show that these lower bounds are surprisingly tight in some cases.

A key step in developing the minimax bias theory is to define an appropriate bias functional which is not only scale equivariant but also explicitly takes into account the unidentifiability of the scale parameter induced by the contamination model. For the case of the "p -shorth" (the shortest interval containing at least 100p% of the data), we show that under some general conditions on F, satisfied by the Cauchy distribution as well as all strongly unimodel distributions, the maximum asymptotic bias under our definition coincides with that under the usual definition used in the robustness literature. For the special case of F being the Cauchy distribution, we show that the 0.5- shorth (normalized to be consistent at F) not only attains the lower bound on gross errors sensitivity, but is the scale estimator which has minimax asymptotic bias under contamination-- and this is true for all proportions of contamination between 0 and 1/2.

My talk will contain two parts: 1) a simulation based approach to survey design with an application to the health sciences and 2) an outline of some problems related to the analysis of web surveys.

In order to better inform study design decisions when sampling patients within and across health care providers we develop a simulation-based approach for designing complex multi-stage samples. The approach explores the tradeoff between competing design goals such as precision of estimates, coverage of the target population and cost. We elicit a number of sensible candidate designs, evaluate these designs with respect to multiple sampling goals, investigate their tradeoffs, and identify the design that is the best compromise among all goals. This approach recognizes that, in the practice of sampling, precision of the estimates is not the only important goal, and that there are tradeoffs with coverage and cost that should be explicitly considered. We construct a sample frame with all phase III clinical cancer treatment trials that are conducted by cooperative oncology groups of the National Cancer Institute from October 1, 1998 through December 31, 1999. Simulation results for our study suggest sampling a different number of trials and institutions than initially considered.

Web surveys have substantially gained in popularity in recent years. Because respondents are usually allowed to self-select into web surveys, however, most web surveys completely ignore the tried and true statistical concept of a random sample. As of today, a "right" way to adjust for the selection bias resulting from self-selection has not yet emerged from the literature. Researchers in biostatistics have long used Rosenbaum and Rubin's propensity score methodology for causal inference from observational data. I will outline challenges in applying this approach to survey sampling.

In this talk I will discuss robust regression estimates for censored data. The extension of the Least Squares estimate to the case of censored responses was first proposed by Miller (1976) and later revisited by Buckley and James (1979). More recently Ritov (1990) and Lai and Ying (1994) studied M-estimates for censored responses. Unfortunately, these estimates require a monotone estimating equation and hence are only robust against low-leverage outliers. We propose an extension of high breakdown regression estimates to the case of censored responses. In particular, our approach extends the Least Median of Squares estimates [Rousseeuw, 1984], the S-estimates [Rousseeuw and Yohai, 1984], the MM-estimates [Yohai, 1987], and the tau-estimates [Yohai and Zamar, 1988]. I will illustrate our proposal with an example, discuss its properties, and present results of a Monte Carlo study that explored its finite-sample properties. If time permits I will also present an algorithm to compute these estimates.

(Joint work with Victor Yohai - University of Buenos Aires)

Ancestral graph models, introduced by Richardson and Spirtes (2002), are a new class of graphical models that generalizes both Markov random fields (underlying undirected graph) and Bayesian networks (underlying DAG = directed acyclic graph). A key feature of ancestral graphs is that they can encode all conditional independence structures which may arise from a Bayesian network/DAG model with selection and unobserved variables.

In this talk, we consider Gaussian ancestral graph models and present a new algorithm for maximum likelihood estimation. We call this new algorithm iterative conditional fitting (ICF) since in each step of the procedure, a conditional distribution is estimated, subject to constraints, while a marginal distribution is held fixed. We show that in the considered Gaussian case, ICF may be implemented by regressions on "pseudo-variables". The ICF algorithm is in duality to the well-known iterative proportional fitting algorithm, in which a marginal distribution is fitted for a fixed conditional distribution. Finally, the ICF approach seems promising for future development of methodology in the case of discrete variables.

This is joint work with Thomas Richardson, Department of Statistics, University of Washington.

Modelling autocorrelated count data is a challenging problem. Unlike the situation for continuous data, for which the multivariate normal distribution is available, for count data there is no convenient and flexible class of multivariate distributions that can capture the shape of the distribution and the autocorrelation. Parameter-driven (latent variable) models are one means of overcoming these difficulties. However, these models can be difficult to interpret and to estimate -- and to apply in the absence of information about the latent process.

In this talk, we formulate a general class of parameter-driven models for count data. Our formulation is conducive to relatively simple interpretation of the models, and to robust estimation of the regression coefficients. Furthermore, our class includes two popular models: the generalized linear mixed model (GLMM) and the hidden Markov model (HMM). We will describe, based on our specification of the class, the connection between these two models and their common properties.

We apply these ideas to the analysis of both multiple sclerosis and polio incidence data.

This work is joint with Brian Leroux of the University of Washington.

Stream health is often measured by the multimetric benthic index of biotic integrity (B-IBI). For Puget Sound Lowland (PSL), the B-IBI comprises ten metrics which quantify the well-being of benthic inhabitants of the stream. Each metric is converted to a score of 1, 3, or 5, where a higher value indicates a healthier stream with respect to the metric. Summing the metric scores yields the B-IBI. Stream health is then rated as very poor, poor, fair, good, or excellent according to the index value.

Discretization is the conventional method to standardize B-IBI metric values measured on different scales. This scoring scheme requires subjective and space/time-dependent input on the cutoff points of metric scales. In contrast, simple statistical standardization (centering and division by the standard deviation) appears to be more natural, is non-study-specific, and maps the metric space onto a continuous scale centered at 0. Our stream health index for the Puget Sound Lowland (SHIPSL) is the sum of the ten metric values standardized as such.

Bootstrap simulation results show that SHIPSL is a more efficient measure of stream health. Without sacrificing information on biotic integrity, SHIPSL reduces bias and variability of the health index, and eliminates sensitivity of the rating to slight changes in the metric scoring scheme.

Modelling autocorrelated count data is a challenging problem. Unlike the situation for continuous data, for which the multivariate normal distribution is available, for count data there is no convenient and flexible class of multivariate distributions that can capture the shape of the distribution and the autocorrelation. Parameter-driven (latent variable) models are one means of overcoming these difficulties. However, these models can be difficult to interpret and to estimate -- and to apply in the absence of information about the latent process.

In this talk, we formulate a general class of parameter-driven models for count data. Our formulation is conducive to relatively simple interpretation of the models, and to robust estimation of the regression coefficients. Furthermore, our class includes two popular models: the generalized linear mixed model (GLMM) and the hidden Markov model (HMM). We will describe, based on our specification of the class, the connection between these two models and their common properties.

We apply these ideas to the analysis of both multiple sclerosis and polio incidence data.

This work is joint with Brian Leroux of the University of Washington.

Saddlepoint theory provides highly accurate approximations for density and distribution functions using an available cumulant generating function. The density appproximation was introduced to statistics by Daniels in 1954 and given broader context in Barndorff-Nielsen and Cox in 1979. But the greater implications for statistics only emerged with the distribution function approximation by Lugannani and Rice in 1980. Research since then offers substantial implications for statistical theory. These recent results will be surveyed and some presently apparent directions indicated.

The saddlepoint is available in contexts where a cumulant generating function is accessible. But the high accuracy is available in large generality. The stand in for the cumulant generating function is an observed likelihood function provided its argument has been appropriately calibrated; the calibration involves only the locally determined form of the likelihood function near the observed data.

The availability of an accurate approximation for a probability is of interest, but the real importance comes from the related asymptotic theory that provides a widely unique decomposition of a statistical model and the determination of a p-value free of nuisance parameters. These recent statistical methods will be reviewed together with examples.