Stats
Section EventsStats
Section EventsABSTRACT: I introduce a family of prior distributions over multivariate distributions, based on the use of a `Dirichlet diffusion tree' to generate exchangeable data sets. These priors can be viewed as generalizations of Dirichlet processes and of Dirichlet process mixtures, but unlike simple mixtures, they can capture the hierarchical structure present in many distributions, by means of the latent diffusion tree underlying the data. This latent tree also provides a hierarchical clustering of the data, which, unlike ad hoc clustering methods, comes with probabilistic indications of uncertainty. The relevance of each variable to the clustering can also be determined. Although Dirichlet diffusion trees are defined in terms of a continuous-time process, posterior inference involves only finite-dimensional quantities, allowing computation to be performed by reasonably efficient Markov chain Monte Carlo methods. The methods are demonstrated on problems of modeling a two-dimensional density and of clustering gene expression data.
ABSTRACT: A three-dimensional random vector $(X_0,X_1,X_3)$ with
independent components is transformed into a bivariate vector
$(Y_1,Y_2)=(\phi_1(X_0,X_1),\phi_2(X_0,X_2))$. We want to identify the
distributions of $X_i$'s having observed $Y_i$'s. For coding functions
$\phi_i$'s falling into a semigroup scheme described in Kotlarski and
Sasvari (1992) such an identification is possible even in quite abstract
settings. A thorough review is given in Prakasa Rao's (1992) monograph.
Here we consider a new coding method - independent random choices (with
the same unknown probability): $X_0$ or $X_1$ for $Y_1$ and $X_0$ or $X_2$
for $Y_2$. It appears that in this case the full identification of the
model is possible. Also this new approach will be combined with standard
coding functions used earlier.
A somewhat related question of identification of a finite bivariate
mixtures has been treated recently in Hall and Zhou (2001) (earlier
studied also by Luboi\'nska and Niemiro (1991)).
Matias Salibian-Barrera, Carleton University, will speak on "Inference for Estimates Defined by Estimating Equations Under Weak Distribution Assumptions" at 10:30a.m. in N638 Ross.
ABSTRACT: In this talk I will discuss some inference methods based on statistics defined by estimating equations. In particular, I will considerstatistics determined by quasi-likelihood estimating equations (which are based on first and second moments of the response variable) and robust regression estimates (which also require weak distribution assumptions on the response). I will describe a bootstrap method that is asymptotically correct and significantly faster than the classical bootstrap. For quasi-likelihood estimates I will compare confidence intervals built with this method with those obtained using the "sandwich" variance estimate. Part of this work is joint work with Ruben H. Zamar, Department of Statistics, University of British Columbia.
Francois Perron, Universite de Montreal, will speak on "Improving on the Metropolis-Hastings Independent Algorithm" at 10:30a.m. in N638 Ross.
ABSTRACT: In this talk we will give an introduction to the Metropolis-Hastings Independent Algorithm (MHIA) and we will see how this algorithm can be improved. Our first approach will use a control variate based on the sample generated by the proposal. We will derive the variance of our estimator for fixed sample sizes n and show that, as n tends to infinity, the variance of our estimator is asymptotically better than the one obtained with the MHIA . Our second approach will be based on Jensen's inequality. We will use a Rao-Blackwellisation and we will exploit the lack of symmetry in the MHIA. We will find and upper bound on the improvements that we can obtain by these methods.
Steven Wang, York University, will present a talk entitled "Extract biological Information from On-line Abstracts" at 10:30a.m. in N638 Ross.
ABSTRACT: Many on-line abstracts and articles have become available to researchers. For example, MEDLINE offers access to more than 12 millions articles publsihed on medical journals. To search, dowload and read these articles can be very time consuming and even impossible. In an attempt to free researchers from this tedious research routine, we propose a simple but effective algorithm that can "read" through all the abstracts of interests and make predictions. We will demostrate this approach by using 15,000 abstracts from MEDLINE. I will also outline other bioinformatics projects that we are working on.
ABSTRACT: This talk describes an efficient sampler for the Bayesian estimation of covariance matrices using Markov chain Monte Carlo (MCMC). The methods apply to decomposable covariance selection models with a hyper inverse Wishart (HIW) prior for the covariance matrix. The conjugate properties of the HIW distribution are used to generate from reduced conditional distributions. In particular, the covariance matrix is integrated out of all conditional distributions and is not generated in the MCMC. The resulting sampler is shown to have a much faster convergence rate than existing methods. The computational complexity of one iteration of the MCMC is shown to be similar to existing methods, so the gain in convergence rate is significant. An efficient mixture estimate of the posterior mean of the inverse covariance matrix is given.
Xuming He, University of Illiniois will be giving a talk entitled "Estimation in Semiparametric Models with Unspecified Dependence Structures" at 10:30am in N638 Ross.
ABSTRACT: We consider M-estimators for partly linear models with possibly dependent observations such as those from a longitudinal study. We approximate the nonparametric function in the model by a regression spline and show that any M-estimation algorithm for the usual linear models can be used to obtain consistent estimates of the semiparametric model and valid large sample inferences on the linear components without any specification of the error distribution and the covariance structure. Included as special cases are the analysis of the conditional mean and median functions for longitudinal data and for certain spatial data. Advantages of this approach from both the theoretical and practical points of view are discussed in the talk.
ABSTRACT: The goal of the talk is to introduce some recent progress by the speaker and his students/collaborators in the area of data analysis, in order to seek feedback from the statistics group.We first give a short introduction to the connection between the global dynamics and congitive tasks such as associative memory and pattern recognition of neural networks. We then focus on the clustering problem for data sets in high dimensional feature spaces, and we describe a new neural network architecture, recently designed in collaboration with Y. Cao, and report some numerical simulations as well as comparisons with other clustering algorithms. The talk shall keep the technical details to minimal.
ABSTRACT: In genetic linkage analysis, families are examined to
find patterns of genetic marker transmissions that coincide with patterns
of disease. This analysis can identify chromosomal regions likely to
harbour genes that increase susceptibility to disease. However, for
complex diseases, it is likely that multiple genes act on different
symptoms or comorbid conditions to increase risk of disease. Although
clinical data on symptoms is normally available, it is not clear what is
the best way to use such information.
I will describe an extension to "model-free" linkage analysis methods
that uses multivariate clinical data. This method adaptively finds the
individual characteristics that are associated with the strongest evidence
for linkage, through the use of classification and regression trees
(CART). Bootstrapping can be used to stabilize cutpoint selection, and
cross-validation optimizes the size of the regression tree. The methods
will be illustrated on a data set of 68 families ascertained to have at
least two cases of asthma. Due to the adaptive nature of the algorithm,
results must be interpreted cautiously and validated in independent
data.
ABSTRACT: Consider a continuous natural exponential family on the line for which we test whether the parameter is equal to a fixed value or not. For this we can use either the two sided UMPU test or the generalized likelihood ratio test. For a given level of acceptance the results can coincide. This is what happens for a Gaussian family with known variance, for a gamma family and -although this is more difficult to check- for an inverse Gaussian family. We prove that, conversely, if the tests coincide for any level of acceptance, then the family is of one of the three types above. The proof is achieved by the cancellation of a determinant of order 6, leading to a differental equation. Computations have needed Mathematica..
ABSTRACT: Adaptive designs have often been proposed as a way
sequentially to assign more patients to better treatments, based on
outcomes of previous treatments in clinical trials.
In this talk, we discuss the optimal adaptive designs for two-arm
clinical trial based on the following optimality criterion: for a fixed
power of a test and fixed signifant level, minimizes the expected number
of treatment failures.
An optimal sequential design is proposed and is compared to some other
adaptive designs.
ABSTRACT: In teaching and thinking about least-squares regression
we tend to use the geometry of ‘subject space' where points are variables
and the axes represent subjects.
However, many important concepts for applications, including paradoxes
like Simpson's paradox, are perhaps more easily understood in ordinary
‘variable space' where points are observations and axes, variables.
The simple geometry of the ellipse seems to be the key to visualizing
regression concepts in variable space.
This talk will explore ways in the connections between data ellipsoids
and confidence ellipsoids can be used to provide clearer insights into
properties of regression that are frequently misunderstood.
Rong Zhu, University of British Columbia, will speak on "The generalized AR(1) process and its applications to non-normal time series" from 2:30p.m. to 3:30p.m. in N638 Ross.
ABSTRACT: The theory of continuous-time generalized AR(1) processes is developed for modelling non-normal time series with equally or unequally-spaced time observations, which may be count data or positive-valued data. Such a process is a Markov process represented as the sum of a dependent term (involving extended thinning operation) and an innovation term. The stationary distribution of a continuous-time generalized AR(1) process can have support on non-negative integers or positive reals; common distributions such as Poisson, negative binomial and Gamma are included. In this talk, we will introduce the continuous-time generalized AR(1) process and its properties, as well as the characterization of its stationary distribution. The modelling procedure will be illustrated by two real cases with count data and positive-valued data respectively.
Refreshments will be served in N620 Ross at 3:30p.m.
Dietrich von Rosen, Upssala University, Sweden, will give a talk entitled "Restricted Expected Multivariate Least Squares" from 10:30a.m. to 11:30a.m. in N620 Ross.
ABSTRACT: A new approach of estimating parameters in multivariate models is introduced. A fitting function will be used. The idea is to estimate parameters so that the fitting function equals or will be close to its expected value. The function will be decomposed into two parts. From one part, which will be independent of the mean parameters, the dispersion matrix is estimated: This estimator is inserted in the second part which then yields the estimators of the mean parameters. The Growth Curve model will illustrate the approach.
Augustine Wong, York University, will give a talk on "Applications of Likelihood Based Asymptotic Inference" from 10:30a.m. to 11:30a.m. in N638 Ross.
ABSTRACT: Recent third order approximate inference procedures have
evolved either from the saddlepoint method (Daniels 1954, 1987;
Barndorff-Nielsen & Cox 1979; Lugannani & Rice 1980) or from the direct
analysis and Taylor series expansion of log density functions
(Barndorff-Nielsen 1986; Fraser 1990; Fraser & Reid 1995). Tail
probabilities at a particular parameter value can be obtained by these
methods; however, it is generally restricted to the canonical parameter of
the exponential model or to the location parameter of the transformation
model. For more general models, Barndorff-Nielsen (1986) proposed a method
which depends on the existence of an ancillary statistic.
A more general approach to obtain third order approximate inference for
any scalar parameter will be presented and will emphasize on applying the
method to various areas.
ABSTRACT: The classical Wishart distributions were introduced by J.
Wishart in 1928 as distributions of the maximum likelihood (ML) estimator of
a completely unknown covariance matrix $\Sigma$ in a sample of i.i.d.
multivariate normal observations from $\mathbb{R}^I$, where $I$ is a finite
set.. Thus the Wishart distributions live on ${\rm\bf P}(I)$, the cone of
positive definite $I\times I$ matrices. One specific Wishart distribution is
denoted $W_{\lambda,\Sigma}$, where $\lambda>\frac{|I|-1}{2}$ is the {\it
shape parameter} and $\Sigma\in{\rm\bf P}(I)$ is the {\it multivariate scale
parameter}. Thus the expectation
$\mathbb{E}(W_{\lambda,\Sigma})=2\lambda\Sigma$. When the degrees of freedom
$f:=2\lambda$ is a noninteger there is no reference to a normal distribution
and the Wishart distributions have a \lq\lq life of their own\rq\rq.
Many classical and recently developed statistical models in multivariate
statistical analysis involve inference in the covariance matrix, i.e., the
parameter $\Sigma$ is restricted to a subset $\Theta\subseteq {\rm\bf
P}(I)$. Some of the important features of $\Theta$ are that it is a subcone
(or it can at least be nicely parametrized by a cone) and it is a
homogeneous space under a natural action of a Lie group. This leads directly
into a generalization of the classical Wishart distributions to the socalled
general Wishart distributions on homogeneous cones. It is of course
important to investigate which results about the classical Wishart
distributions can be generalized to this new class of distributions on
homogeneous cones.
Consider for example the following classical result: Suppose that the
random variable $S$ follows the Wishart distribution $W_{\lambda, \Sigma}$
on ${\rm\bf P}(I)$. Let $I=I_1\dot\cup I_2$ be a decomposition of $I$. Let
$\Sigma\equiv(\Sigma_{ij}|i,j=1,2)$ and $S=(S_{ij}|i,j=1,2)$ be the
corresponding decompositions of $\Sigma$ and $S$, respectively. Then
We shall discuss how this classical result extends to the general Wishart
distribution on homogeneous cones. As time permits it, several other
generalizations will be mentioned together with several questions still to
be investigated.
(i): $S_{22\cdot 1}:=S_{22}-S_{21}S_{11}^{-1}S_{12}$ and
$(S_{21},S_{11})$ are independent.
(ii): $S_{22\cdot 1}$ follows the Wishart distribution
$W_{\lambda-|I_1|/2,\Sigma_{22\cdot 1}}$ on ${\rm\bf P}(I_2)$, where
$\Sigma_{22\cdot 1}$ is defined similar to $S_{22\cdot 1}$.
(iii): The distribution of $S_{21}S_{11}^{-1}$ {\bf given} $S_{11}$ is
the normal distribution on $\mathbb{R}^{I_2\times I_1}$ with expectation
$\Sigma_{21}\Sigma_{11}^{-1}$ and $(I_2\times I_1)\times(I_2\times I_1)$
covariance matrix $\Sigma_{22\cdot 1}\otimes S^{-1}_{11}$.
(iv): $S_{11}$ follows the Wishart distribution
$W_{\lambda,\Sigma_{11}}$ on ${\rm\bf P}(I_1)$.
ABSTRACT: Recently, there has been an interest in options on volatility. For instance, Peter Carr recently studied the hedging of variance (=squared volatility) swaps (Carr, P., Replicating Variance Swaps. Risk Course on Correlation, New York, May 2001). I will describe how the distribution of the integral of squared volatility (=sum of squared returns) may be obtained, if volatility is modelled as a square-root process (Heston model). The properties of the square-root process will be reviewed, and formulas will be given for moments of all order and also for options on integrated volatility.
ABSTRACT: In this talk we extend statistical inverse problems to compact Riemannian manifolds. The approach is to use aspects of inverse spectral geometry associated with the Laplace-Beltrami operator on compact Riemannian manifolds. Although the vast majority of inverse problems usually take place in Euclidean space, certain problems can be genuinely non-Euclidean. In so doing, researchers tend to work in the spaces that are specific to the problem and it therefore seems that the time is now appropriate at unifying the theory. This can be viewed positively for both applied and pure researchers where the advantage for applied researchers is that this would provide a blueprint for tackling problems in other spaces that would fall under this broad category. For pure researchers, this would provide a formal mathematical description of the problem and therefore can be used as a foundation for broadening the theory even further.
ABSTRACT: The epidemiology of AIDS poses many challenging statistical and mathematical problems. In particular, tracking and forecasting the epidemic is complicated by very long incubation times. Individuals with HIV infections are often diagnosed before developing AIDS, and the ensuing treatment makes it difficult to model incubation times. I shall describe a new model that accounts for early detection without introducing bias. We use a Gibbs sampler Monte Carlo simulation to estimate the probabilities of diagnoses and the total number of new HIV infections each year among homosexual men in Ontario. This talk describes work in progress as part of a MITACS project on "Mathematical and Computer Modelling of Epidemics with Public Health Applications".
ABSTRACT: Bell Canada provides a full range of traditional
telephone services as well as new and emerging services such as Internet,
wireless, satellite and complex data transmission for business customers.
In today's highly competitive and quickly changing telecommunications
environment it is important to develop an in-depth understanding of
customer needs and to optimize profitability of its operations.
The Database Marketing Center of Bell Canada is developing statistical
analyses to identify optimal customer strategies and operational solutions
primarily for the marketing initiatives. It uses statistical methods to
analyze and predict customer choices for telecommunications services,
identify cost effective distribution channels, forecast staffing levels in
call centers and plan network utilization to name just a few examples. In
the presentation we will describe these and other applications of
statistics to solve business questions in Bell Canada.
ABSTRACT: The inverse Wishart distribution is the traditional conjugate prior for the covariance parameter in a multivariate $N(0,\Sigma)$ model. When it is known that the components $X_i$ of a normal random vector $X$ have certain dependence properties, the appropriate conjugate prior is the hyper inverse Wishart. We will see how, in a Bayesian framework, this distribution is used for normal models Markov with respect to a given graph.
ABSTRACT: Assessing whether or not missing data are non-ignorable is often crucial to the choice of modeling strategy for analysing longitudinal data. Current existing diagnostic tools are either based on empirical exploratory data analysis plots or logistic models for the dropout process, such as selection models (Diggle & Kenward, 1994). Most methods have been developed in connection to maximum likelihood inference, and depend on untested distribution assumptions. In this paper we study informative dropouts in longitudinal studies, in connection to quasi-likelihood or estimating equations approaches. In such a context, the missingness appears essentially to be of two types, ignorable or nonignorable, according to whether the mean zero assumption of estimating equations holds. We propose a generalised score-type test based on the quadratic inference function (Qu, Lindsay & Li, 2000) to test for nonignorable missingness. We illustrate the proposed score test by both simulations and real data examples, in which we give a thorough comparison of our method to the Wald-type test proposed by Chen and Little (1999).
ABSTRACT: The empirical likelihood method was proposed by Owen (1988, 1990) as a device for constructing confidence regions with independent observations. Historically, the first application of the concept behind empirical likelihood was suggested by Hartley and Rao (1968) for finite populations where a design-based likelihood under simple random sampling based on a multinomial distribution was used. Formal applications of the method in survey sampling was introduced by Chen and Qin (1993) and (Chen and Sitter, 1999). In this talk, we will briefly review the use of empirical likelihood method in estimating the finite population means and totals. We will then discuss some recent applications on estimating the finite population distribution function and quantiles, estimation of variance and other quadratic functions. It has been shown that the model-calibrated pseudo empirical maximum likelihood estimator (Wu and Sitter, 2001) is optimal among a class of estimators. It also provides an easy way to obtain positive or range-restricted weights for regression estimators. Some problems for future investigation will be noted.
ABSTRACT: We propose a new class of state space models for longitudinal discrete response data where the observation equation is specified in an additive form involving both deterministic and dynamic components. These models allow us to explicitly address the effects of trend, seasonal or other time-varying covariates while preserving the power of state space models in modeling dynamic pattern of data. We develop different Markov chain Monte Carlo algorithms to carry out statistical inference for models with binary and binomial responses. In a simulation experiment we investigate the mixing and convergence properties of these algorithms. In particular, we demonstrate that a joint state variable update is preferable over individual updates. In addition, different prior choices are studied. Finally, we illustrate the applicability of the proposed state space mixed models for longitudinal binomial response data in the analysis of the Tokyo rainfall data (Kitagawa 1987).
ABSTRACT: The proportional hazards regression model usually assumes that the covariate has a log-linear effect on the hazard function. In this talk, we consider a semiparametric survival model with flexible covariate effects. We suppose the baseline hazard function is parameterized, while the risk function associated with covariates is modeled in a semiparametric way. A maximum generalized profile likelihood estimator for the parameters is considered. We show that the resulting parameter estimator is root-n consistent, asymptotically normal and achieves the semiparametric efficiency bound. An estimator for the nonparametric risk function is also given and its asymptotic properties are derived. Its rate of convergence is shown to be comparable to that in a standard nonparametric regression model. The counting process theory is used to tackle the technical difficulties. An application to mouse leukemia data is presented to illustrate the proposed method. An extensive simulation study is also carried out to validate this method.
Refreshments will be served in N620 Ross at 10:00a.m.
Dr Lu is a candidate for the advertised Statistics position.
ABSTRACT: Logistic regression models are commonly used for studying
binary or proportional response variables. An important problem is to
screen a number $p$ of potential explanatory variables in order to select
a subset of them which are most related to a response variable. Several
criteria such as AIC, BIC, and stochastic complexity criterion are
available for this variable selection procedure. However, simply applying
these criteria for an exhaustive search of the best subset is
computationally infeasible, even when $p$ is moderately large (e.g. $p=20$
which implies $2^{20}$ candidate subsets available for selection).
We propose an MCMC random search procedure incorporating the above
criteria to overcome the computational difficulty. Using this procedure we
only need to search a sample of the candidate subsets in order to find the
best one. We have studied various properties of this procedure concerning
the consistency of the associated selection criteria, the convergence of
the Markov chain generated and the probability and the efficiency of
selecting the optimal model. The performance of our procedure is also
assessed by a simulation study.
ABSTRACT: The Internet has spawned a renewed interest in the
analysis of co-occurrence data. Correspondence analysis can be applied to
such data to yield useful information. A less well-known technique called
canonical correspondence analysis (CCA) is suitable when such data come
with covariates. We show that CCA is equivalent to a classification
technique known as linear discriminant analysis (LDA). Both CCA and LDA
are examples of a general feature extraction problem.
LDA as a feature extraction technique, however, is restrictive: it can
not pick up high-order features in the data. We propose a much more
general method, of which LDA is a special case. Our method does not assume
the density functions of each class to belong to any parametric family. We
then compare our method in the QDA (quadratic discriminant analysis)
setting with a competitor, known as the sliced average variance estimator
(SAVE). Our study shows that SAVE over-emphasizes second-order differences
among classes.
Our approach to feature extraction is exploratory and has applications
in dimension reduction, automatic exploratory data analysis, and data
visualization. We also investigate strategies to incorporate the
exploratory feature extraction component into formal probabilistic models.
In particular, we study the problem of reduced-rank non-parametric
discriminant analysis by combining our work in feature extraction with
projection pursuit density estimation.
Mu Zhu is a Statistics candidate.
Refreshments will be served in N620 Ross at 10:00a.m.
ABSTRACT: Some of you may have experienced back pain on
occasion. To find out
the association between lifting technique and back pain, the Ergonomics
Research Group at
Queen's University and the DuPont Company in Kingston began a project
using electromagnetic
sensors which were attached to a human subject. Each sensor recorded its
position in Cartesian
coordinates and its orientation in three consecutive Euler angles. From
several experiments for
calibration, systematic errors and considerable distortions caused
possibly by metal masses in
the test volume were detected. To correct these errors, especially in
orientations, they required
statistical analyses for modeling and experimental design for
calibration. A model for matched
pairs of orientations (Prentice, 1989) turned out to be the best model
for the orientation data
from the calibration experiments. This model was developed based on
Chang's model (1986) for
matched pairs of vectors. The parameters of both models are rotation
matrices in SO(3).
At my talk, I will first propose an optimal design for Chang's model
in general
p-dimensional space based on the minimum number of observations required
for unique
estimation of a rotation matrix parameter and selection of covariate
values for minimum volume
of confidence regions. This design could save cost and time in
practice.
Second, I will also propose an optimal design for Prentice's model in
3-dimensional space
based on the minimum number of observations for unique estimation of the
two rotation matrix
parameters. I found, however, this minimum number of observations does
not give minimum
volume of confidence regions. Recently I have proved that a design,
which gives the minimum
volume of confidence regions, is unique in the canonical form of
rotation matrix. An interesting
fact about this design is it results in four different pairs of
estimation, i.e., not unique estimation.
Thus we have a dilemma between uniqueness and level of accuracy. This
problem is the main
topic of my current research.
Finally, I will present simulation results, which confirm the
efficiency of the proposed
optimal design for Prentice model.
Refreshments will be served in N620 Ross at 10:00a.m.
Note that Hwashin Hyun Shin is a candidate for a position in Statistics.
ABSTRACT: A maximum weighted likelihood method is proposed to
combine all
the relevant data from different sources to improve the quality of
statistical inference especially
when the sample sizes are moderate or small.
In this talk, I discuss the linearly weighted maximum likelihood
estimator (MLE), a special
case of the maximum weighted likelihood estimator (MWLE). A procedure
for adaptively
choosing the weights by using cross-validation is proposed. The maximum
entropy theorem and
I-divergence geometry are shown to be closely related to the linear
MWLE. The derivation of the
weighted likelihood by using the maximum entropy principle will also be
presented. The
saddlepoint approximations to the distributions of the linear MWLE and
MWLE derived from
the estimating equations are derived for small sample sizes. Results of
simulation studies and
applications will be discussed in the presentation.
Refreshments will be served in N620 Ross at 3:30p.m.
ABSTRACT: An important tool in signal detection in brain imaging is
the distribution of the maximum of a random field on some Euclidean space,
which is taken to be the parameter space of the random field. One approach
that has proved successful in approximating the tail of this distribution
is the so-called EC or (expected) Euler characteristic approach. This
approach uses techniques from integral geometry to approximate the
distribution of the maximum of the random field and provides an
approximation of the tail in terms of certain integral invariants of S,
known in integral geometry as the intrinsic volumes of S. A fundamental
assumption of this approach was that the random field was isotropic,
i.e. invariant under the group of Euclidean rigid motions. However, when
studying random fields on manifolds such as the cortical surface, there is
no notion of isotropy because of the irregularity of the
manifold. Further, the distribution of the maximum of the random field is
invariant under diffeomorphisms of the parameter space, so that our
inference should not be based on how we visualize the data, i.e.,
flattening the cortical surface to view data in the sulci of the cortical
surface should not affect the inference about the presence of a signal or
not.
These ideas point to the fact that the relevant geometry in the study
of such random fields is one that is intrinsic to the random field
itself. In other words, the relevant geometry is derived from the
Riemannian structure that the random field induces on the manifold and not
(necessarily) from the space in which we visualize the data. In the case
of an isotropic field on a Euclidean space, the assumption of isotropy
restricts the geometry to be the geometry of the Euclidean space (modulo a
constant).
We describe the EC approach and how it can be extended smoothly to
random fields on manifolds as well as how to estimate the relevant
geometric quantities that appear in the EC approximation. We illustrate
the method on some fMRI (functional magnetic resonance imaging) data,
restricted to the cortical surface.
If time permits, we will describe a version of the classical Kinematic
Fundamental Formula from integral geometry for smooth vector-valued
Gaussian random fields. This has applications for the EC approach in a
non-Gaussian framework, i.e. for random fields built from i.i.d. Gaussian
fields.
Refreshments will be served in N620 Ross at 11:00a.m.
Jonathan Taylor has applied for our advertised position in Statistics. Please feel free to peruse his application in N522 Ross. If anyone would like to join the Hiring Committee and Mr Taylor for lunch or dinner please contact srainey@yorku.ca.
ABSTRACT: Disordered materials, such as concrete or fibreglass, can be simulated by random sets that have as their basis random sphere packings. Likelihood methods cannot be directly used in fitting these random sets, but there are a wide variety of new statistics that can be used to describe the properties and structure of the simulated materials. These statistics are unique to random set inference, and can also be applied to the study of hard-core point processes and the liquid state.
ABSTRACT: A new method based on parameterization is proposed to
construct a general goodness-of-fit test. It includes the traditional
Kolmogorov-Smirnov, Cramer-von Mises and Anderson-Darling tests, as well
as new distribution-free tests, which are robust and generally much more
powerful (sometimes several times more powerful) than the old
ones. Actually, they are the analogues of the old ones in representation
but make great improvements and innovations.
When the new tests are used to test the goodness of fit for normality,
they outperform the Shapiro-Wilk test, the most powerful test of normality
in the literature, and usually dominate the Anderson-Darling test, the
best existing EDF (empirical distribution function) test.
The new one-sample tests have been developed into general two-sample
and k-sample tests. Conventional tests are sensitive to location
difference between distributions, but are dull to detect the variation in
scale and shape. The new tests are sensitive to location, scale and
shape. In fact, if the distributions of the two sampled populations have
the same shape and variance but different locations or means, they are as
powerful as the old tests. Otherwise, they are much better with power
improvement reaching up to several times.
Professor Paul Gustafson, Department of Statistics, University of Bristish Columbia, will speak on "Two Tales of Measurement Error" from 10:30a.m. to 11:30a.m. in N638 Ross.
ABSTRACT: Many statistical techniques can yield biased estimates if
covariate measurement error is present but not accounted for in the
analysis. While this is well-known in general, there is still much to
learn about the effects of measurement error in specific situations, and
the best ways to adjust for these effects. This talk presents two
vignettes towards this end.
First, consider case-control analysis with a dichotomous covariate that
is subject to misclassification. If the misclassification probabilities
are known, then methods are available to adjust odds-ratio estimates. We
study the realistic scenario where reasonable guesses, but not exact
values, are available for these probabilities. If the analysis proceeds by
simply treating the guesses as exact, then even small discrepancies
between the guesses and the actual probabilities can seriously degrade
odds-ratio estimates. We show that this problem is mitigated by a Bayes
analysis which incorporates uncertainty about the misclassification
probabilities as prior information.
Second, we make some comparisons between the bias induced by
measurement error in continuous covariates to that induced by measurement
error in discrete covariates. Our findings relate to the common practice
of creating a dichotomous covariate by thresholding a continuous
covariate.
ABSTRACT: We propose a new statistical method: the quadratic inference function approach to analyze longitudinal data in a semiparametric framework defined by a set of mean zero estimating functions. It allows one to get an efficient estimator of regression parameter and provides a chi-squared inference function for testing. The estimating function bootstrap method is discussed to correct for anticonservative testing behavior when the asymptotic theory fails for skewed data. The quadratic inference function approach is compared to the generalized estimating equations method (Liang and Zeger, 1986), as illustrated by simulations and applications to published biomedical data.
ABSTRACT: Reliable small area statistics are needed in formulating policies and programs, allocation of government funds, market research and so forth. Traditional area-specific direct survey estimators are not suitable for this purpose because the sample size in a small area is typically too small to provide estimators with acceptable precision. It is necessary therefore to use indirect estimators by borrowing data from related small areas to increase the effective sample size and thus the precision. In this talk I will give an overview of recent model-based methods, in particular, empirical Bayes and hierarchical Bayes methods.Techniques for measuring the variability of the estimators will also be discussed. Recent applications of model-based methods will be presented.
Everyone is encouraged to attend. Statistics graduate students are expected to attend.
ABSTRACT: A conjunction is defined in the brain mapping literature as the occurrence of the same event at the same location in two or more independent 3D brain images. The images are smooth isotropic 3D random fields of test statistics, and the event occurs when the image exceeds a fixed high threshold. We give a simple approximation to the probability of a conjunction occurring anywhere in a fixed region, so that we can test for a local increase in mean of the images at the same unknown location in all images, a generalization of the split-t test. This is the corollary to a more general result on the expected Minkowski functionals of the set of points where a conjunction occurs.
Everyone is encouraged to attend. Statistics graduate students are expected to attend.
ABSTRACT: Density estimates are just smoothed versions of the
empirical measure. The main problem in density estimation thus is the
data-based choice of an estimate from a class of estimates. Particular
instances include the choice of a bandwidth in kernel estimation, the
choice of a threshold level for wavelet estimates, the choice of a
parameter in the Box-Cox transformation (when used before a kernel or
histogram estimate), and the choice of the number of basis functions in a
series estimate. We propose a general method that has the property that
for all densities, under conditions limiting the size of the complexity of
the classes of estimates, the expected L1 error is not more than about 3
times the optimal L1 error (the error corresponding to the estimate from
the class if the density were revealed to us beforehand).
This is joint work with Gabor Lugosi.
Everyone is encouraged to attend. Statistics graduated students are expected to attend.
ABSTRACT: Left truncation of survival data occurs when subjects whose survival times are shorter than either fixed or random quantities, are simply not observed. Often in epidemiologic studies, prevalent cases with a disease are identified through a cross-sectional study carried out over a short time period. These cases are then followed for a fixed time period at the end of which the subjects will either have failed or have been censored. When interest lies in estimating the survival distribution, from onset, of subjects with the disease, one must take into account that the survival times of the cases identified in such a study are left truncated or length-biased; the long survivors tend to be those cases identified at the start of the study. I shall give a brief overview of length-bias and discuss how one may estimate the "true" unbiased distribution from length-biased data. In particular, I shall propose an unconditional approach that, while requiring stronger assumptions than the traditional conditional method of the literature, produces narower confidence intervals and hence more precise inference. The problem of estimating the survival distribution of subjects identified with dementia, including Alzheimer's disease, threads its way through this talk that has a surprising conclusion.
Everyone is encouraged to attend. Statistics graduate students are expected to attend.
ABSTRACT: It is natural in a clinical trial or an epidemiologic study to investigate the effect of intervention or exposure in different subgroups of subjects. However problems of interpretation may result if such analyses are not pre-specified and issues involving multiplicity, estimation bias and type II errors are not taken into account. In this talk, I discuss strategies that may be adopted to address such problems, using recent examples from the clinical and epidemiologic literature to illustrate controversies that may arise.
Everyone is encouraged to attend. Statistics graduate students are expected to attend.
ABSTRACT: Priors for which Bayesian and frequentist inference agree (at least to some order of approximation) are called ‘matching priors', and have been proposed as candidates for noninformative priors in Bayesian inference. I will present some recent work on various aspects of the matching problem, with applications to $p$-value, confidence limits, and tolerance limits.
Everyone is encouraged to attend. Statistics graduate students are expected to attend.
Professor Nancy Reid, our next speaker, is an internationally recognised statistician. She is amongest the most well known statistians in the world. Nancy has a long list of honours including "The Committee of Presidents of Statistical Societies award" which is the statistical version of the "Fields Medal". She will deliver the "Wald Lecture" this year at IMS meeting which is the most important event of the IMS (Institute of Mathematical Statistics) meeting. To emphasize the importance of the Wald lecture, I just list some of the past Wald lecturers: Samuel Karlin, Bradly Efron, Peter Bickel, Peter Huber, L.D. Brown and Ulf Grenander. This is an impressive list of big guns in statistics. Having Nancy's name on this list is indeed an honour for the statistics community in Canada.
Duncan Murdoch, University of Western Ontario, will speak on "Perfect Sampling: Not Just for Markov Chains?" from 10:30a.m. to 11:30a.m. in N638 Ross.
ABSTRACT: Propp and Wilson's (1996) coupling from the past (CFTP) algorithm generates a sample from the limiting distribution of a Markov chain. In this talk I argue that the underlying idea of CFTP is more widely applicable, and will demonstrate attempts to apply it to two problems: approximation of limits and simulation of stochastic differential equations.
Everyone is encouraged to attend. Statistics graduate students are expected to attend.
ABSTRACT: e study the weak limit behavior of certain types of point
processes obtained by replacing the original observations by the bootstrap
sample.
The usual bootstrap fails asymptotically in cases for which there exist
a Poisson point process or a fixed point measure in the limit. A
subsampling method resolves this problem.
Bayesian bootstrap, Efron's bootstrap and other Monte Carlo methods
will be discussed and we will present several applications to our
results.
Everyone is encouraged to attend. Statistics graduate students are expected to attend.
ABSTRACT: Reliable small area statistics are needed in formulating policies and programs, allocation of government funds, market research and so forth. Traditional area-specific direct survey estimators are not suitable for this purpose because the sample size in a small area is typically too small to provide estimators with acceptable precision. It is necessary therefore to use indirect estimators by borrowing data from related small areas to increase the effective sample size and thus the precision. In this talk I will give an overview of recent model-based methods, in particular, empirical Bayes and hierarchical Bayes methods.Techniques for measuring the variability of the estimators will also be discussed. Recent applications of model-based methods will be presented.
Everyone is encouraged to attend. Statistics graduate students are expected to attend.
ABSTRACT: The relationship between statistics and research in human
genetics goes back to the work of early statisticians such as Fisher,
Haldane, Pearson, and others. Because of random aspects of the process of
meiosis that produces human gametes, transmission of genetic information
from parents to offspring is inherently probabilistic. The subject of
interest, the gene, is usually not directly observable and its location on
the genome is unknown. The development of new molecular technologies has
dramatically changed the nature and the volume of data that are available
for the study of complex human disease. It is estimated that the
completion of the Human Genome Project will reveal the location of 100,000
genes. As investigators turn their attention from simple single-gene
Mendelian disease to the "genetic dissection of complex traits" such as
diabetes and cardiovascular disease that depend on many genes, new
statistical methods are required to model heterogeneity from known and
unspecified sources.
I will review current statistical approaches to finding genes for
complex human disease and highlight some recent advances.
Everyone is encouraged to attend. Statistics graduate students are expected to attend.
ABSTRACT:We propose a data-based procedure for combining a number of individual classifiers in order to construct more effective classification rules. Under some regularity conditions, the resulting combined classifier turns out to have a misclassification error rate which is asymptotically, almost surely, lower than that of any of the individual classifiers. The procedure is also quite easy to use in practice and produces good results.
Everyone is encouraged to attend. Statistics graduate students are expected to attend.
ABSTRACT: The talk discusses the application of the characteristic aspects of graphical models to event history analysis. In graphical models the vertices of the graph are ment to represent variables and missing edges stand for some kind of conditional independences depending on the kind of graph. The conditional independence restrictions that a graph imposes on the underlying statistical model lead to considerable simplifications regarding computational efforts of various kinds (e.g. test and estimation procedures). Additionally, the graphical representation itself is helpful in visualizing the dependence structure and by this the reasoning about the underlying multivariate setting. Application of graphical models to event history analysis, however, is not straightforward if one wants to appropriately take the dynamic structure into account, especially in the continuous time situation. Therefore, I propose to replace the concept of conditional independence shown in the graph by the dynamic concept of local independence. Roughly spoken, a process Y(t) (e.g. counting process) is locally independent of another process X(t) if its intensity at t is not altered by including the information on the past of X(s), st. This concept is closely related to the ideas of non-causality discussed in the time series literature. Unlike conditional independence, local independence is not symmetric thus calling for a new semantic for the graphical representation. I will focus on the notion of separation in the corresponding local independence graphs which is the key to the reduction of complexity in multivariate settings that graphical models in general are aimed at.
J.O. Ramsay, McGill University, will speak on "Some Uses of Differential Equations in Statistics from 10:30a.m. to 11:30a.m. in N638 Ross.
ABSTRACT: How do we define functions, such as densities and
nonlinear regression models, in practice? Usually by specifying a family
of parametric curves. Or perhaps by an expansion in terms of a set of
basis functions. But a powerful and flexible alternative, familiar to
engineers and scintists, is the definition of a function by specifying a
relationship among its derivatives and itself.
Beginning with a basic introduction to differential equations, the talk
will develop a few interesting applications. DIFEs for density estimation,
for monotone function estimation, for models of binary data, and for
surfaces and volumes over complex sets will be discussed, and illustrated
by some real data analyses.
About the speaker: Jim Ramsay is the 1998 SSC (Statistical Soceity of Canada) Gold medalist. He is a professor in the Dept. of Pyschology and an adjunct professor of the Dept. of Mathematics and Statistics at McGill University.
ABSTRACT: The left-truncated logarithmic series, Poisson and negative binomial distributions are studied in detail. In particular, the distribution of the sum of n independent random variables having the logarithmic series distribution truncated on the left at the point c, called the generalised Stirling distribution of the first kind, is obtained and some of its properties are investigated. The n-fold convolution of the left truncated negative binomial distribution, called the generalised Lah distribution, is also derived. Further, it is shown that, under certain limiting conditions, the generalised Lah distribution approaches the generalised Stirling distribution of the first kind and the generalised Stirling distribution of the second kind. The seminar will survey several papers in this area.
ABSTRACT: In many researchers' opinion self-rated pain is the most
natural measure of musculoskeletal disorders. However, self-rated pain can
only be assessed from the trial subjects' own statements, and there is at
present no gold standard for how to make such measurements.
In the PRIM Study, which is a 3-year follow-up study on work-related
musculoskeletal disorders, self-rated pain is measured through
questionnaires, where the average trouble in the upper limbs during the
last 3 months is scored on a 10-point scale.
Such scale responses are usually analysed by logistic regression after
dichotomizing the scale
or by a proportional odds model (or the like) on the full scale (McCullagh
1980). The drawback to the former approach is that valuable information
about the responses is lost, whereas the drawback to the latter approach
is that the assumptions implied by the proportional odds model are rarely
fulfilled when the response can take a large number of different values.
I propose a quasi-likelihood approach with robust variance estimation
to analyse self-rated pain scored on a 10-point scale. The mean value
structure is parameterized logit linear; the variance structure resembles
a binomial variance with overdispersion. The quasi-likelihood method
requires only a minimum of assumptions, so the approach applies in general
to ordinal interval-scales. Furthermore, the method is easily extended to
a longitudinal set-up, either by use of generalized estimating equations
or by a generalized linear mixed model. The proposed method is employed on
data from the PRIM study.
Dr Ebenezer Lartey, University of Western Ontario, will speak on "Score function--essential tool for change-point analysis" from 11.30a.m to 12.30p.m. in S501 Ross.
ABSTRACT: Unified methodology for the detection of parameter
changes at unknown times adapting the Bayes-type procedure introduced by
Chernoff & Zacks (1964) for change-point analysis.
The statistic obtained in the detection of these changes is defined
in terms of the score function and are a
reversed partial sum of the score function. It is shown that these
statistics have locally best properties. The
asymptotic distributions are shown to be equivalent to those of
stochastic integrals defined by the
generalized Brownian-bridge type processes in higher dimensions.
Performance of the derived statistic based on this methodology is
compared with that of the maximum
likelihood ratio statistic introduced by Horvath (1993) for detecting
the constancy of the parameters of the
normal distribution.
Dr Lartey is a candidate for the contractually limited position in Statistics.
Jerry Brunner, University of Toronto, will give a talk entitled "Bayesian Cluster Analysis" from 10:30a.m. to 11:30a.m. in N638 Ross.
ABSTRACT: Cluster analysis is presented as a procedure for partitioning observed data into subsets that are deemed to have arisen from the same probability distribution. There are two main types of Bayesian cluster analysis, those based on a fixed-partition model and those based on a random-partition or mixture model. We observe that the most natural random-partition cluster analysis is equivalent to a fixed-partition cluster analysis with a particular choice of prior. This explains a surprising connection between parametric fixed-partition Bayesian cluster analysis and non-parametric Bayesian density estimation. The connection is generalized, and for both fixed and random-partition cluster analysis, numerical problems are overcome by methods originating in the literature on density estimation.
ABSTRACT: There has recently been a proliferation of models and methods for the analysis of clustered survival data. In many situations however, the response of interest may not be characterized by a single event time, but rather several. These may correspond to the event times of a point process or transition times of a multi-state model. In this talk I will review some of the common methods of analysis in these settings and describe some recent developments. An emphasis is given to semi-Markov models and multivariate mixing distributions. If time permits issues pertaining to selection effects will be discussed. Examples will be drawn from medical applications in oncology, infectious disease and transplantation studies.
Everyone is encouraged to attend.
ABSTRACT: We study sufficiency in terms of the extent to which the Taylor series expanison of the normed log likelihood function generated by the sampling distribution of a statistic conforms to that of the sample normed log likelihood function and give an associated self-similar property of mathematical likelihood.
Everyone is encouraged to attend.
ABSTRACT: The motto "there is nothing more practical than a good theory" applies particularly well to the field of statistics. Quoting Donald Rubin, Stat & Comp. 1993 "The special training statisticians receive in mapping real problems into formal probability models, computing inferences from data and models, and exploring the adequacy of theses inferences, is not really part of any other formal discipline, yet is critical to the quality of empirical research". But much of the research and writing on applied statistics resembles a collection of modified techniques (usually modifications that increase the complexity of existing techniques) rather than a critical examination of issues regarding how to apply theory and technique.
Some current work in the area of meta-analysis of clinical trials will be presented. The work draws heavily on work by RA Fisher and WG Cochran that has perhaps been largely ignored by many current researchers in meta-analysis. It will be used to advance the thesis that theory needs to be directly applied in applications and that with "good application" of theory the techniques tend to be simple and transparent.
Additionally--a thesis advocated by Rob Kass, that good applications are simply those that spur on further research in statistical theory--will also be discussed.
Everyone is encouraged to attend.
ABSTRACT: State space models for multivariate binomial time series are considered with the focus on the development of the Kalman filter and smoothing for state variables. A Monte Carlo approach employing the latent variable representation is proposed, which transplants classical Kalman filter and smoothing developed for Gaussian state space models to discrete models and leads to a conceptually simple and computationally convenient approach to the class of models. The method is illustrated by a simulation study and two data analysis examples.
Everyone is encouraged to attend. Statistics graduate students are expected to attend.
ABSTRACT: In this talk I shall describe the nature and current methods of biostatistics, with particular emphasis on the differences between biostatistics and statistics, and the relationships between biostatistics and its related disciplines. I shall also present important current research areas in biostatistics. Finally, I shall discuss career in biostatisttics and what does it take to make a biostatistician.
Everyone is encouraged to attend. Statistics graduate students are expected to attend.
ABSTRACT: We consider a life testing situation in which systems are subject to failure from independent cometing causes. Following a failure, immediate (stage-1) procedure are used in attempt to reach a definitive diagnostics. If these procedures fail to result in a diagnosis, this phenomenon is called masking. Stage-2 procedures, such as failure analysis or autopsy, provide definitive diagnosis for a sample of the masked cases. We show how stage-1 and stage-2 information can be combined to provide statistical inference about: (a) survival functions of the individual risks; (b) the proportions of failures associated with individual risks; and, (c) probability, for a specified masked case, that each of the masked competing risks is responsible for the failure.
Everyone is encouraged to attend.
ABSTRACT: Bahadur-Kiefer (B-K) theorems deal with the asymptotic behaviour of the error made approximating an estimator by a linear or first order representation; weak B-K theorems give the limiting distribution of this approximation error while strong B-K theorems give the set of almost sure limit points. In this talk, we will consider an approach to deriving B-K theorems based on a ``delta method'' for sequences of minimizers. This approach can easily be applied to derive B-K theorems for the sample quantiles and other estimators under both standard and non-standard conditions.
Everyone is encouraged to attend.
ABSTRACT: In selecting the best dosage choice for the estimation of the $ED_{50}$, it is natural to try to minimize the length of the confidence intervals. In this presentation the dose allocation that minimizes the length of the likelihood-based confidence intervals is presented and compared with alternative allocations that have been proposed based on the length of different types of confidence intervals, such as those based on the asymptotic variance or on Fieller's Theorem. Effective strategies to deal with the parameter dependence of these allocations are explored. A series of experiments to evaluate the effect of small doses per fraction on the radiation tolerance of the rat cervical spinal cord provide the motivation and an illustration for the proposed procedures.
Everyone is encouraged to attend. Statistics graduate students are expected to attend.
ABSTRACT: Under a semiparametric framework, estimation of parameters for mixed nonlinear models poses, in general, serious problems in ensuring consistency (not to mention asymptotic optimality) of fixed parameter estimates (both first and second order), and unbiasedness of random parameter estimates. The reason for this seems to be the practice of making the random parameters part of the nonlinear predictor in order to satisfy certain range restrictions on the conditinal means. Consequently, computation of the marginal mean and the covariance becomes intractable in general which renders usual estimation methods for fixed nonlinear models inapplicable. Note that for the usual nonlinear models the random parameters are first specified through the nonlinear predictor, and then attempt is made to specify the (marginal) covariance. However, such specification of random parameters is not necessary in view of the following three observations.
i) The target random parameters can be alternatively defined as the hierarchical differences of conditional means, and then they can be made additive to the fixed nonlinear predictor. Note that this may give rise to variance-mean relationships that should be accounted for in modelling the covariance structure. However, the additivity feature should help to overcome the estimation problem mentioned above.
ii) For BLUP-type optimal estimation of additive random parameters, it is sufficient to specify only the covariance structure, and later the random parameters can be specified (in a wide sense) to match the covariance structure.
iii) Although the usual BLUP is not designed to meet range restrictions, it being a Stein-type shrinkage estimator may often work well provided the fixed predictor does meet the restrictions. However, if necessary a suboptimal BLUP via ridge-modification of the shrinkage coefficient can be constructed to meet range restrications while preserving unbiasedness.
We, therefore, propose a wide sense specification of random parameters by first modelling the conditional covariances in a hierarchical manner while accounting for variance-mean relationships, instead of the customary reverse route in which the covariance structure is obtained after the nonlinear functional form of random parameters is specified. Estimating functions with suitable properties can be constructed for both fixed and random parameters. Illustrative examples for estimation with survey and nonsurvey data are presented.
Everyone is encouraged to attend. Statistics graduate students are expected to attend.
ABSTRACT: Observations on security prices, currency exchange
rates, interest rates, and many other financial indices are usually
recorded at the
beginning of a period (say the day), the end of the period, together with
the high and low for that period. When estimating parameters from
historical data, often only the closing prices, close and open prices are
used in the estimation of parameters and the subsequent analysis, although
information on the extremes of the process is highly informative,
particularly of the changes in volatility. Modelling the distribution of
extremes is also of importance in the valuation of certain derivative
products, hedging, risk management.
We discuss methods for simulation and estimation (model calibration).
Monte Carlo simulation and cross validation are used to estimating
parameters. Canadian Bank security prices are used as an example.
Everyone is encouraged to attend. Statistics graudate students are expected to attend.
ABSTRACT: This talk will discuss the construction of estimating function for parametric regression models in situations where information about responses or covariates may be missing, and where observation may be response-selective. Modelling of covariate distributions is avoided by the use of semiparametric likelihoods and pseudo likelihoods. Applications include multistage studies in epidemiology and medicine, field reliability studies, and broad classes of missing data and measurement error problems.
Everyone is encourged to attend. Statistics graduate students are expected to attend.
ABSTRACT: A complex distribution can be approximated by a mixture
of component
distributions with simpler forms. This is one way of producing a
"non-parametric" model that is flexible enough to accomodate many
distributions. In some contexts, the mixture components found can
also be interpreted as "latent classes"--for example, as different
species of organisms. Deciding how many components should be in the
mixture is a difficult problem. When Bayesian methods are used, this
problem can be bypassed by letting the number of components be
countably infinite, with an appropriate prior for mixing proportions.
This leads to what is called the "Dirichlet process" mixture model.
These models are easily implemented using Gibbs sampling when conjugate
priors are used for the parameters of the component distributions, but
when non-conjugate priors are used, Gibbs sampling involves a possibly
difficult numerical integration. In this talk, I will review past
work on this problem, and present two new Markov chain sampling
methods for Dirichlet process mixture models. One method uses a
well-chosen Metropolis-Hastings proposal distribution to bypass the
need to look at an infinite number of possibilities. The other method
is based on the temporary introduction of auxiliary variables. Both
methods are easily implemented and allow use of Dirichlet process
mixture models with non-conjugate priors.
Everyone is encouraged to attend. Statistics graduate students are expected to attend.
ABSTRACT: Based on the marginal likelihood approach, we develop a
model selection criterion, MIC, for
regression models with the general variance structure. These include
weighted regression models, regression
models with ARMA errors, growth curve models, and spatial correlation
models. We show that MIC is a
consistent criterion. For regression models with either constant or
non-constant variance, simulation studies
indicate that MIC not only provides better model order choices than
the asymptotic efficient criteria AIC, AICC,
FPE and C_{p}, but is also superior to the consistent criteria BIC
and FIC in both small and large sample sizes.
These results also hold for regression models with AR(1) errors,
except that BIC performs slightly better than
MIC in the large sample size case. Finally, Monte Carlo studies show
that the effectiveness of model selection
criteria decreases as the degree of heterogeneity (or the first order
autocorrelation) increases.
Some key words: AIC; AICC; BIC; FIC; MIC; Marginal likelihood.
Everyone is encouraged to attend. Statistics graduate students are expected to attend.
ABSTRACT: Fitting mixture distributions to samples from heterogeneous populations leads to some interesting statistical and computational issues. I will outline a likelihood-based approach that works well in practice, and illustrate the methodology with a variety of examples from biology, medicine and archeology. We will see that a model which fits well isn't necessarily correct, and a model that fits poorly isn't necessarily useless.
Reference: http://icarus.math.mcmaster.ca/peter/mix/mix.html
Everyone is encourged to attend. Statistics graduate students are expected to attend.
ABSTRACT: This paper provides nonparametric empirical Bayes (EB) solutions to two-tail tests in an exponential family under the standard product distance loss function. Based on empirical data and the present data EB test procedures are developed. These procedures are asymptotically optimal for all prior distributions which possess finite second moments. Rates of convergence are investigated for the EB procedures which are developed under some moment conditions on the prior distributions.
Everyone is encouraged to attend. Stat. grad. students are expected to attend.
ABSTRACT: Variance components in factorial designs with balanced data are commonly estimated by equating mean squares to expected mean squares. For unbalanced data, the usual extensions of this method are the Henderson methods, which require special formulas that are rather involved. We introduce a simpler method, that of creating a balanced data set by resampling from the original one. Revised formulas for expected mean squares are presented for the two-way case; they are easily generalized to larger factorial designs. Indications are that estimators of variance components for main effects and interactions are sometimes much more accurate than Henderson Method I estimators.
Everyone is encouraged to attend. Statistics graudate students are expected to attend.
ABSTRACT: The shape of an object or image, regarded as a subset of
IR^n, can be defined as the total of all properties which are
invariant under similarity transofrmation of the object in IR^n.
In practice, the shape of an object is encoded from the configuration of a
finite set of points called landmarks chosen at important
locations on the object. In this talk, we shall survey two of the main
approaches to shape analysis due to F.L. Bookstein and D.G. Kendall. Small
(1996) proveded an extension of the Bookstein representation by
representing the shapes of p-simplexes on manifolds. These
manifolds are quite distinct from those proposed by D.G. Kendall based
upon Procrutes distances. A formula for geodesic distance in simplex shape
space permits the implementation of multidimensional scaling methods for
the statistical shape analysis of 2- and 3-dimensional objects.
We apply this representation of simplex shapes to a shape analysis of
Iron Age brooches from modern day Switzerland. The brooches are then
grouped into five classes based upon chronological ordering. It is shown
that these five classes form relatively distinct groups when the first two
principal coordinates of shape variation are displayed. This supports the
assumptions used in archeological seriation where artefacts are given a
rough chronological ordering based upon stylistic features.
ABSTRACT: Quality of life (QOL) has been included as a major endpoint in addition to the traditional endpoints such as tumour response and survival in many oncological clinical trials. The development of quality of life measurements as an acceptable clinical endpoint has raised a lot of discussion among researchers. Most studies in the past assessed QOL retrospectively within randomized trials on a small subset of cancer patients or treated this as a secondary endpoint to the primary question. Recently, QOL endpoints become more important. The FDA also recommended that the beneficial effects on QOL and/or survival can be the basis of approval of new anti-cancer drugs. However, there are a many design and analysis problems arise from such studies. Further statistical research on various aspects of design and analysis of QOL data is needed. In this paper, a number of statistical issues associated with QOL endpoints will be discussed, they include: 1) design considerations; 2) study of psychometric properties; 3) aggregate score versus global QOL; 4) data management and compliance; 5) sample size calculation; 6) method of analysis with missing data with growth curve and other models; 7) the choice of measuring QOL domains longitudinally versus a utility measurement such as quality adjusted life years (QALY). A general review and examples of the above problems will be given.
Everybody is encouraged to attend. Stat grad students are expected to attend.
ABSTRACT: When measurements are taken from polished surfaces of minerals or resin-impregnated soils or from biopsy samples or microscope slides, the sampling unit is a 3-d specimen with hidden random structure. Stereology is a methodology for measuring these internal features. The talk reviews stereology, concentrating on non-standard statistical methods.
ABSTRACT: Construction and applications of biological interaction
automata (BIA) and of reactive lattice gas automata (LGA) will be
discussed. The statistical mechanical theory of BIA and LGA will be
presented. The macroscopic PDEs for BIA and LGA will be derived. An
important aspect of BIA and LGA method is that it allows to investigate
dynamics of interactive and reactive systems at a mesoscopic level that
goes beyond phenomenological PDEs description.
The BIA method will be illustrated by considering its applications to
study collective motion and aggregation in biological systems. The LGA
method will be illustrated by considering its applications to study Turing
pattern formation, chemical wave propagation and excitation processes as
well as the effects of the heterogeneity of the reacting medium on
reaction-transport dynamics.
ABSTRACT: There is a considerable literature on changepoint problems. Almost all of it has concentrated on the analysis of a single sequence of observations which, it is suspected, has undergone a change in distribution at some unknown instant. In contrast, very little work has been done on the "multipath setup" where the data consist of a number of sequences each of which may undergo a change. For example, in a clinical trial to assess the effect of a drug, patients may be monitored for a period after receiving a treatment which may only become effective after some unknown time interval. Typically, these intervals would vary randomly from subject-to-subject, and, possibly, systematically from subgroup-to-subgroup of subjects; for example, males may take longer to react than females. I shall discuss how one may introduce covariates into this type of multipath changepoint setting, an area hitherto not addressed in the changepoint literature. The approach allows one to assess whether different subgroups have different changepoint distributions. The approach is through maximum likelihood. A flavor of the proofs of consistency and asymptotic normality of the estimated regression parameters is given and some of the difficulties outlined; in particular, the role played by mixtures is emphasized.
Note: The speaker is a candidate for the department's Statistics position.
ABSTRACT: The sample size determination is an important issue in the conduction of a clinical trial. In this talk, we first review some standard methods in sample size determination for cancer clinical trials with survival endpoints. Some examples where these standard methods are not applicable and thus should be modified are then introduced. Finally, we present some results on the sample size determination in equivalence clinical trials demonstrating one treatment is not inferior to another and in single arm trials comparing a new treatment to a standard historical control.
ABSTRACT: In epidemiological studies, it is often of interest to evaluate the relationship of some covariate variables, such as age, sex or treatment, and a survival time that is defined as the elapsed time between two related events. A simple estimation method is proposed under the proportional hazards model for the regression analysis of such studies when the times of the occurrences of the two events defining the elapsed time are right- or interval-censored. The method does not involve any iteration among unknown parameters as some full likelihood approaches and gives the partial likelihood approach (Cox, 1972) when the time of the occurrence of the originating event is observed exactly. The asymptotic properties of the proposed regression coefficient estimates are derived and an AIDS cohort is analyzed to illustrate the proposed approach.
ABSTRACT: Robust designs against small departures from regression response and departures from the assumption that errors are uncorrelated are discussed. There are two approaches (minimax approach and infinitesimal approach) to derive these designs. For the minimax approach, the theory is well developed and commonly practiced in applications. Using the infinitesimal approach, we can define and obtain V-robust, B-robust and M-robust designs. Subject to satisfying a robustness constraint, those designs minimize the determinant of the mean squared error matrix of the least squares estimator at the ideal model. The robustness constraint is quantified in terms of boundedness of the Gateaux derivative of this determinant, in the direction of a contaminating response function or autocorrelation structure. Examples are given to compare robust designs with classical optimal designs.
Dr. J. Zhou is a job candidate for the statistics position.
ABSTRACT: Let $Y$ be a multivariate normal linear model with covariance $A\otimes\Sigma$ and mean $\mu\in S$, where $S=\{Xb:K'b=M'u \mbox{ for some }u \}$, and $A$ and $\Sigma$ may be singular. An explicit formula is obtained for the best quadratic estimator of $\Sigma$ with respect to a large class of loss function. Admissible invariant test for testing a linear hypothesis on the mean are obtained through a general decision-theoretic lemma. None of the matrices $A$, $\Sigma$, $X$, $K$ and $M$ are assumed to have full column rank.
Dr H. Cheng is a job candidate for the statistics position.
ABSTRACT: In magnetic resonance imaging a radio frequency pulse
is
applied to a material that has been
subjected to a magnetic field. The process yields a sequence of T_2
relaxation observations. In heterogeneous
material the observations are represented through an integral
equation with expoential kernel and a relaxation
time distribution dependent upon the underlying material. In this
talk we present methods that can be used to
estmate and make inferences about the relaxation time distribution.
In medical applications of magnetic resonance imaging, discrimination
between different tissue types based upon
the relaxation time distributions is desired. We will present
extensions of LDA that can be used in this setting for
discrimination based upon estimated relaxation time distrubtions. An
example application will be given for
breast tissue samples of differing tissue types.
Note: Edward Susko is a candidate for a postion in the Department of Mathematics and Statistics.
ABSTRACT: Consider the exact likelihood estimation of a multivariate Gaussian
autoregressive moving average (ARMA) model. Some old but still difficult
questions are:
- how to find the closed form of the exact likelihood function?
- how to derive the likelihood equation?
- how to deal with missing data?
In this talk solutions will be given in some cases, and open questions
will be mentioned. For the complete data case, a simple method is employed by Ma (1997,
Biometrika) to derive the explicit expressions for the determinant and
exact inverse of the covariance matrix, and thus to obtain the closed
form of the exact likelihood function. For a univariate ARMA model and a multivariate stationary AR model,
explicit formulas are given for the derivatives of the exact
log-likelihood function and the resulting likelihood equations.
Dr C. Ma is a job candidate for the statistics position.
ABSTRACT: A general method on statistical analysis of misspecfied regression models is developed, which can be applied to situations with random effects, omitted covariates and measurement error, for example. We illustrate the technique with a semi-parametric regression model for multiple occurrences of skin tumors in a chemoprevention clinical trial.
Dr Jiang obtained a PhD in statistics from Cornell University in 1996 and has been working at Northwestern University as an assistant professor since.
ABSTRACT: To measure the spread of HIV infection in a given
population, to estimate the waiting time
distribution from the time of HIV infection to the time when the
patient is first diagnosed with HIV through
laboratory testing, to determine the number of people who are HIV
infected but have not yet been tested, to
estimate the number of people living with HIV, to model the disease
progression from HIV infection to various
clinical stages and to assess the efficacy of the anti-viral therapy
and prophylaxis medication with respect to
disease progression and mortality are crucial questions in public
health management of the HIV/AIDS epidemic
in terms of prevention and planning.
The Division of HIV/AIDS Surveillance at Laboratory Centre for
Disease Control, Health Canada has been
traditionally collecting data based on AIDS diagnoses through AIDS
Case Surveillance Reporting System since
1982. The HIV/AIDS surveillance has also been expanded to collecting
laboratory testing data on HIV positive
tests. For population based disease reporting, there is often a
reporting delay which creates severe under
counting of disease incidence for the recent past due to diseases
which have been diagnosed but not yet
reported. Reporting delays can sometimes be recorded, but only after
the disease has been reported. This creates
a time-length selection bias in statistical analysis for reporting
delays, because the data is collected conditioning
on the reporting delay distribution itself.
Some clinical observational HIV/AIDS data are available as
supplementary information for the national
HIV/AIDS surveillance program. I would like to introduce a clinical
based long-term prospective cohort data
following 1348 patients who have been tested HIV positive. This
cohort is estimated to cover about 97%-99% of
all individuals who are known to be HIV infected living in eight
health regions from southern Alberta. Although
the true HIV infection time is not observable for these patients, it
has been retrospectively documented for each
patient that HIV infection time must lie within a time interval.
During the follow-up, onset of different clinical
symptoms, measurements on CD-4 markers and viral load for each
patient are documented in regular time
intervals. All patients are followed until their deaths or to
September 30 ,1997 if they are still alive. For most
prospective cohort data, the enrollment for each patient takes place
sometimes after the HIV positive tests and
is conditioning on the survival time distribution of patients after
HIV infection. This again creates time-length
selection bias.
I will present several statistical methodology issues, mainly through
analysis on the southern Alberta clinical
cohort data. I will also discuss some population based approaches to
estimate the historical trend of HIV
infections based on surveillance data. Some analyses are still in
progress and will lead to open discussions. I will
use a general illness-death process to illustrate the statistical
modeling. There are several challenges: (1) the
onset of HIV infection is uncertain, (2) time-length selection bias,
(3) the surveillance definition of AIDS as a
collection many clinical illnesses and the onset of the first AIDS
defining illness as a competing risk framework,
(4) efficacy the drug intervention. I hope this presentation will
stimulate interest for graduate students in
statistics regarding the potiential development of statistical
methods and their applications in public health
related to infectious diseases with long incubation.
KEY WORDS: illness-death process, hazard function, relative risk,
competing risk, time-dependent covariates,
left and right truncation, doubly censored survival data,
parametric/non-parametric/semi-parametric models,
EM-algorithm.
ABSTRACT: With the availabilty of several medications for the treatment of a disease, it is important to decide on the equivalence of the medications. In the case of two medications one is interested in the equality of two populations F and G. The classical hypothesis H_0:F=G allows the determination of difference in the case of rejecting the null hypothesis. However in bioequivalence one is interested in the case where the null hypothesis cannot be rejected. Therefore interval hypothesis tests based on a measure of distance between the two populations are more appropriate. I will introduce such a test based on the Mallow's distance between two populations. It is based on the distribution of the whole population and not only on the mean of the distribution. In this talk I will introduce this approach, discuss the small sample properties of the the test and illustrate the method on several examples.
This work is joint with A. Munk, Dept. of Mathematics, Bochum University, Germany.
References:
Munk, A. und Czado, C (1997). Nonparametric Validation of Similar Distributionsand Assessment of Goodness of Fit. To appear in JRSS B.
Czado, C. and Munk, A. (1997). Assessing the similarity of distributions- finite sample performance of the empirical Mallows distance. To appear in JSCS.
ABSTRACT: A number of generalizations
of the Kolmogorov Strong Law of Large Numbers (SLLN) are known including
convex
combinations of r.v's with
random coefficients.
In the case of pairs of i.i.d. r.v's
$(X_1,Y_1),\ldots ,(X_n,Y_n)$, with $\mu$ being the probability
distribution of
$X$'s, the averages of $Y$'s for which the accompanying $X$'s
are in a vicinity of a given point $x$ may converge with probability 1
(w.p.1)
and for $\mu$-a.e. $x$ to conditional expectation
$E\left( Y|X=x \right) $.
We consider Nadaraya-Watson estimator
of $E\left( Y|X=x \right) $ where the vicinities of $x$ are determined by
windows widths $h_n$. Its convergence w.p. 1 and for $\mu$-a.e. $x$ under
condition $E
|Y| \infty$ is called a Strong Law of Large Numbers for
Conditional
Expectations (SLLNCE). If its convergence holds true for all
probability
distributions of $X$ it is called Universal.
We investigate the minimal assumptions for the SLLNCE and for
the
Universal convergence and we improve the best known results in this
direction.
Dr Joanna Jasiak, Department of Economics, York Unviersity, will give a talk entitled: "Stochastic Volatility Duration Models" at 10:00a.m. in N638 Ross.
ABSTRACT: We propose a class of two factor dynamic models for duration data encountered in financial econometrics and insurance models. Empirical findings suggest that the first two conditional moments of times between stock trades, for example, feature distinct dynamics and different degrees of temporal dependence. Accordingly, durations appear to be driven jointly by movements in the conditional mean and in the conditional variance. The formal representation of this effect is obtained by introducing two factors, where the first one accommodates the conditional mean dynamics and the second the over-dispersion (or variance). This new framework is adequate for modelling processes involving time varying uncertainty and time related risk. In particular, in the stock market context, the conditional mean and variance of intra-trade durations can be interpreted as measures of the market liquidity and risk on the transaction time, respectively. Estimation and prediction of market activity requires hence an approach allowing for eventual temporal interactions between factors, or their absence depending on the market organization and the traded stock. We call these dynamic two factor duration models Stochastic Volatility Duration (SVD) models. The paper shows their distributional properties, inference procedures and examples based on waiting times to trade the IBM and Alcatel stocks.
See departmental calendar.