The wide availability of fast computers has had enormous impact on introductory statistics courses. Computers initially allowed students to apply numerical and graphical methods to realistic data sets, thereby reducing the emphasis on numerical algorithms and formulae. More recently, it has been recognised that computers also have great potential for teaching statistical concepts, adding to their role as sophisticated calculator.

Programs such as Minitab and SAS lack features required for teaching concepts. For example, we might want to ...

• graphically display standard distributions and show how changes to the parameters affect the shape of the distribution,
• repeatedly select samples from a model to show the variability of graphical and numerical summaries,
• build empirical distributions of summary statistics, confidence intervals and p-values with simulation in order to illustrate their properties, ...

While data-analysis programs can perform some of the above, they can rarely make the mechanism clear to students. Models, sampling and empirical distributions must be first-class citizens in software used for teaching statistical concepts.

This talk will demonstrate CAST, a computer-based resource that is designed to teach statistical concepts. CAST is accessed using a web browser and contains both expository text and over 300 small programs (applets) that do most of the teaching. The applets share an extensive framework of code that is designed for teaching statistical concepts.

CAST can be described as a textbook with dynamic, interactive diagrams. Since students must interact with each page, it is claimed that their attention is retained and learning is improved. Even in lectures, the diagrams are effective ways to teach most statistical concepts.

However, Cox's model may not fit the data well. A natural generalization of Cox's model is semi-parametric transformation model, under which an unknown transformation of the survival time is linearly related to the covariates. This class of regression models studied by Cheng et al. (1995, 1997), which includes the proportional hazards model, provides many useful alternatives to the Cox model in univariate survival analysis.

We consider semi-parametric transformation models with random effects for the analysis of the aforementioned correlated and possibly censored failure time observations. The inference procedures for the regression parameters and their large sample properties are presented. An alternative to this model is the marginal approach which does not impose any structure on the correlation. We model each individual failure time with subject- and cluster-specific time-dependent covariates using aformentioned linear transformation models, however no specific parametric correlation structure is imposed on the observations. Under this setting, regression methods are proposed to analyze such correlated observations. Furthermore, we showed how to construct pointwise and simultaneous confidence intervals for survival function of subjects with a specific set of covariates are developed. An ad hoc model selection procedure is also given.

In this presentation, the grouped parametric maximum likelihood approach to estimating the distribution of the reporting delay is discussed. The exact grouped likelihood method is developed for the Weibull and log-logistic distributions and applied to the Canadian AIDS surveillance data. To compare the performance of parametric and non-parametric adjustments, a few data sets were simulated and truncated at four different points in time. The two methods are found to agree for sufficiently large truncation times, and for early truncation times parametric estimation is found to provide valuable insight into the performance of non-parametric estimation.

Suggestions are made on how parametric estimation of delay can be used as a diagnostic measure for the appropriateness of the use of the non-parametric approach.

Suppose the treatment and placebo survival distributions follow a proportional hazards model only after an unknown shift in location. How shall we estimate the shift as well as the proportion in hazards? This motivates the envelope MLE method.

The nonparametric maximum likelihood estimators (NPMLE) and the empirical likelihood ratio statistic sometimes do not exist in semiparametric settings. However, the NPMLE often exist in an enlarged parameter space. We propose to gradually shrink the enlarged parameter space by putting more and more constraints. This results the envelope MLE. The approach is a counter part of the sieve MLE (Grenander 1981).

We shall present one case in detail: 1). location problem where several samples are from the same (unknown) distribution except different locations. We shall briefly mention other models that can be treated by envelope MLE.

A Wilks type theorem for the empirical envelope likelihood ratio statistic and the asymptotic distribution of the location estimator is provided.

Setting up models that capture the essential features of the dynamics of disease transmission is relatively easy. Simulations indicate that simple Markov process models work well. However, computing the implications of such models is surprisingly difficult. With strong assumptions, namely a closed population and homogeneous mixing (the probability of transmission from a given infected person to a susceptible in a small interval of time is the same for all susceptibles) efficient computation of the evolution of the epidemic is possible, and I will describe several related algorithms. An invariance theorem regarding the final size distribution makes it possible in principle to extend these methods to a much more general class of models that do not require the population to be closed and that incorporate realistic levels of heterogeneity in mixing. However, the feasibility of such computations is severely restricted by the necessity of storing the entire distribution of the state of the process, even though in the end one would be satisfied with knowing just the mean and the variance of the size. To illustrate the problem, I will present a model for transmission of hepatitis B in areas of high endemicity. Because of particular features of the epidemiology of this disease, computations for a realistic model are actually possible.

We propose robust confidence intervals and p-values for linear combinations of the parameters of a simple linear regression model. We search for robust procedures which are stable and yet informative. For instance, in the case of confidence intervals, we wish to construct intervals that are stable in the sense of achieving coverages near the nominal one even in the presence of outliers and other departures from the parametric model. Moreover, we wish to obtain intervals which are informative in the sense of having relatively short lengths. The problem of getting stable and informative confidence intervals has deserved attention in the literature (see Barnett and Lewis, Outliers in Statistical Data, 1994, p. 74 and references therein). However, the insofar seem to neglect the bias of the estimator which turns out to be crucial to accomplish stable confidence intervals.

To achieve the goals of stability and informativeness with our approach we need robust point estimates with the following properties: (1) asymptotic normality under general conditions; (2) known asymptotic bias bounds (for the intercept and slope parameters).

We consider some median-based estimates of the slope, for instance, Brown and Mood's estimate (1951), Siegel's repeated median of slopes (1982) and Theil and Sen's pairwise median of slopes (1951, 1968). We show that, to some extent, these estimates satisfy the requirements (1) and (2) above. We show that the proposed robust confidence intervals constitute an improvement over the intervals constructed around a robust point estimate using its asymptotic distribution under the ``target'' parametric model.

The location-scale model is also considered. In this case, we may take advantage of some additional information on the sign of the bias so as to yield shorter confidence intervals.

Note: The paper is available for copying in the UBC Statistics Mail Room. The journal is in the UBC library's e-journal collection, so getting it on-line should be possible for most of you.

Read it in advance, so you can contribute to the discussion.

In recent years statistical extreme value theory has matured to such an extent to contribute usefully to the study of substantial real problems, particularly in the area of environmental extremes. Examples include the design of off-shore structures (Coles and Tawn, 1994) and the study of reservoir flood safety (Anderson and Nadarajah, 1993).

A fairly commonly occurring characteristic is that the variables whose extremes are of interest are ordered. In hydro-meteorology one thing that is of interest is the dependence of extreme values of d-hour rainfall over a range of values of d. One approach is to fit a multivariate extreme value distribution over that range. If X(d) denotes rainfall aggregated over d hours, and if d' > d then X(d) <= X(d') <= (d'/d) X(d) for all (X(d), X(d')), so an order restriction in the multivariate extreme value model is needed. Similar order restrictions arise in the study of the joint distributions of large hourly mean wind speeds and large wind gusts.

The aim of this talk is to develop multivariate extremal models and associated statistical procedures for vector observations whose components are subject to an order relationship. We consider only the bivariate case. The results are applied to the joint analysis of rainfall extremes corresponding to different durations.

In this talk we introduce a notion of depth in the regression setting. It provides the `rank' of any line (plane), rather than ranks of observations or residuals. In simple regression we can compute the depth of any line by an O ( n log n ) algorithm. For any bivariate data set Z_n of size n there exists a line with depth at least n/3. The largest depth in Z_n can be used as a measure of linearity versus convexity. In both simple and multiple regression we consider the deepest fit, which generalizes the univariate median and is equivariant for monotone transformations of the response. Throughout, the errors may be skewed and non-identically distributed (e.g. heteroskedastic). We also construct depth-based regression quantiles. They estimate the quantiles of y given x, as do the L¹-based regression quantiles, but can withstand the effect of leverage points. Using the concept of regression depth, we obtain some new results of discrete geometry.

The B-WISE (Bayesian regression With Interactions and Smooth Effects) method of regression modelling was designed to improve upon other flexible regression schemes in the areas of model interpretability and ease of implementation. B-WISE models have been shown to have good predictive performance, and performance can be even further improved with a natural Bayesian model averaging scheme.

In this presentation I will outline some ways in which some of the constraints inherent in a B-WISE model can be relaxed, so that the technique can be used in more general situations and with even greater flexibility. We will see that much of the interpretability of B-WISE models is retained and implementation is still relatively straightforward.

Neurofibromatosis 2 (NF2) is a rare genetic disease that affects approximately 1 in 40000 people. Some of the characteristic features of this disease include the onset of multiple tumours on the cranial and spinal nerves, juvenile cataracts and hearing loss. Almost all affected individuals develop bilateral tumours of the schwann cells that line the vestibular nerves; these tumours are called as vestibular schwannomas (VS). Evidence from molecular genetic studies has suggested that a "2-hit" hypothesis is appropriate for the development of VS in patients with NF2; that is to say that a tumour cell develops from a normal schwann cell after the cell sustains two mutations to its genetic material. Several authors have proposed probabilistic models for this process and have shown that such models are consistent with incidence data for several different types of cancer.

We will discuss a selection of probabilistic models for a "2-hit" hypothesis and present the results from the fitting of such models to incidence data from NF2 patients. Molecular evidence does not exclude the possibility that additional hits are necessary for the development of VS. We will therefore discuss a "3-hit" model and compare this model's fit to both the data and to the fit of the "2-hit" models. Genotype-phenotype correlations have been reported in patients with NF2 and thus a model that incorporates a patient's genotype is presented. Finally, a bivariate model is proposed to estimate the distributions of the ages at onset of both the first and second VS. All of these models will be presented with minimal mathematical detail; emphasis will be on the application of such models to patient data and on the results.

Propp and Wilson's coupling from the past (CFTP) algorithm provides exact samples and, thus, an elegant alternative to convergence diagnostics for standard MCMC samplers. I shall explain how this method works and discuss some practicalities regarding its use in MCMC sampling.

Unfortunately the CFTP technique is only applicable when the distribution to be sampled possesses certain special properties. We propose a way to use the method's basic idea more generally and demonstrate that our algorithm works well in some quite challenging applications. Although our method is approximate, it comes with diagnostics to help assess and control the level of approximation.

The data sets in this analysis were two ~80MB files of individual packet headers representing 120 minutes of bidrectional Internet traffic. The number of time-stamped packet headers in the two files were 2,153,603 and 2,066,750. Each line corresponds to one packet, the information in each column is:
1. Timestamp in seconds [931406400 = 4pm]
2. Packet length
3. Protocol id [1=ICMP, 6=TCP, 17=UDP]
4. Flow id.

The flow id is a hex number such as 0d32d150 to associate packets with the same IP number, port, etc at origin and destination. I decided to cluster the TCP flows according to their packet length distribution. The packet length distribution for each flow was summarized into frequency counts for five packet length classes. Restricting attention to TCP flows containing 100 packets or more resulted in around 2000 frequency tables from each file.

In the talk I will describe my experience in fitting this data to finite mixtures of 5-category multinomial distributions using the EM algorithm. Certain strategies were important to avoid numerical difficulties: for example it was necessary to increase the dispersion of the component distributions in the early stages of the fitting. The final models fitted had 16 and 18 components.

Data from a reaction time experiment comes in the form of a bivariate point process realization. Flashes are presented to a subject according to a homogeneous Poisson process, and the subject responds by pressing a button each time the flash is seen. An objective of analysis of such data is to determine whether there are interaction effects due to pairs of consecutive flashes. We propose a simple parametric model for the eye-brain-hand system which underlies the data. In order to complete the specification, it is necessary to identify the probability density of the delay between the occurrence of a flash and the corresponding response. One way of making this identification is to examine the coherence and the cross-intensity function between the flashes and responses. Nonparametric estimators of point process intensity functions and coherence have been studied in a sequence of papers by Brillinger. These estimators exhibit the usual bias-variance trade-off. Choi and Hall (1999) have introduced data sharpening, a bias-reduction procedure for density estimation which reduces the order of magnitude of the bias while increasing the variance by a constant factor. We adapt this method to the point process problem, showing how it may be applied to the estimation of the intensity functions for one-dimensional stationary point processes as well as the coherence.

From these nonparametric estimates, it is possible to identify an adequate parametric model. Estimation of the mean delay and a parameter governing the above interaction effect can then proceed using maximum likelihood.

In this talk, we consider models that incorporate the second state into the analyses. The basic stochastic models are Markov chains, alternating renewal processes and marked point processes. For the Markov chains and alternating renewal process models, we consider simple fixed effects models as well as random effects models where the random effects are introduced to allow for heterogeneity between patients and correlation of data on one patient. For these models, the statistical inference is based on maximum likelihood. For the marked point process model, we use a generalized estimating equation approach.

We apply these models to a data set from a MS clinical trial. The aim of the analyses is to relate the available covariates to the disease process. We do not attempt a comprehensive analysis of the data set, rather the aim here is to see what can be achieved and which questions can be addressed with the different models.

Case A. The number of covariates appearing in the linear part is a priori given, say q.

Case B. We have available q possible regressors for the linear part, but only an unknown (possibly much smaller) subset of them are relevant for Y, hence we would like to select it.

We propose a penalized least squares approach and, in both cases, we obtain finite sample upper bounds for the risk of the estimator and, as a consequence, the consistency of the estimator of f at the optimal nonparametric rate. We also discuss the distributional properties of hat(ß) in cases A and B and show that sqrt(n) consistency of hat(ß) can be achieved in both cases.

The common denominator of these two very different applications is the use of nonparametric tools -- MARS (Multivariate Adaptive Regression Splines) and Neural Networks -- to deal with high dimensional data. The essence of each of these methods is data-driven model selection through cross-validation and a useful way to understand such methods is in a Bayesian context.

The Bayesian analogue, evaluates a posterior probability distribution over the space of models based on the data AND a prior probability distribution on that space. The fundamental question is "what prior is implicitly assumed by model selection criteria like GCV (Generalized Cross-Validation) and by adaptive tools like MARS and NNs?"

This question can be anwered in two ways. One is by following the mainstream of the Bayesian model selection industry. The other adopts a 'reverse engineering' perspective in reconstructing the prior assumptions via simulation with respect to the MARS and NN estimators.

We treat the observed data as conditional on the presence of an animal, and then use a Bayesian approach to estimate the probability of finding animals in any given location. This results in a method that allows for mapping the propensity of a certain area to be chosen by an animal in the future.