Showing papers in "The Statistician in 2000"

The Philosophy of Statistics

Abstract: This paper puts forward an overall view of statistics. It is argued that statistics is the study of uncertainty. The many demonstrations that uncertainties can only combine according to the rules of the probability calculus are summarized. The conclusion is that statistical inference is firmly based on probability alone. Progress is therefore dependent on the construction of a probability model; methods for doing this are considered. It is argued that the probabilities are personal. The roles of likelihood and exchangeability are explained. Inference is only of value if it can be used, so the extension to decision analysis, incorporating utility, is related to risk and to the use of statistics in science and law. The paper has been written in the hope that it will be intelligible to all who are interested in statistics.

Wavelet Analysis and its Statistical Applications

Abstract: Summary. In recent years there has been a considerable development in the use of wavelet methods in statistics. As a result, we are now at the stage where it is reasonable to consider such methods to be another standard tool of the applied statistician rather than a research novelty. With that in mind, this paper gives a relatively accessible introduction to standard wavelet analysis and provides a review of some common uses of wavelet methods in statistical applications. It is primarily orientated towards the general statistical audience who may be involved in analysing data where the use of wavelets might be effective, rather than to researchers who are already familiar with the field. Given that objective, we do not emphasize mathematical generality or rigour in our exposition of wavelets and we restrict our discussion to the more frequently employed wavelet methods in statistics. We provide extensive references where the ideas and concepts discussed can be followed up in greater detail and generality if required. The paper first establishes some necessary basic mathematical background and terminology relating to wavelets. It then reviews the more well-established applications of wavelets in statistics including their use in nonparametric regression, density estimation, inverse problems, changepoint problems and in some specialized aspects of time series analysis. Possible extensions to the uses of wavelets in statistics are then considered. The paper concludes with a brief reference to readily available software packages for wavelet analysis.

Prediction and retrospective analysis of soccer matches in a league

Abstract: A common discussion subject for the male part of the population in particular is the prediction of the next week-end's soccer matches, especially for the local team Knowledge of offensive and defensive skills is valuable in the decision process before making a bet at a bookmaker We take an applied statistician's approach to the problem, suggesting a Bayesian dynamic generalized linear model to estimate the time-dependent skills of all teams in a league, and to predict the next week-end's soccer matches The problem is more intricate than it may appear at first glance, as we need to estimate the skills of all teams simultaneously as they are dependent It is now possible to deal with such inference problems by using the Markov chain Monte Carlo iterative simulation technique We show various applications of the proposed model based on the English Premier League and division 1 in 1997–1998: prediction with application to betting, retrospective analysis of the final ranking, the detection of surprising matches and how each team's properties vary during the season

Complex stochastic systems

Abstract: A PRIMER ON MARKOV CHAIN MONTE CARLO, Peter J. Green Introduction Getting Started: Bayesian Inference and the Gibbs Sampler MCMC-The General Idea and the Main Limit Theorems Recipes for Constructing MCMC Methods The Role of Graphical Models Performance of MCMC Methods Reversible Jump Methods Some Tools for Improving Performance Coupling from the Past (CFTP) Miscellaneous Topics Some Notes on Programming MCMC Conclusions CAUSAL INFERENCE FROM GRAPHICAL MODELS, Steffen L. Lauritzen Introduction Graph Terminology Conditional Independence Markov Properties for Undirected Graphs The Directed Markov Property Causal Markov Models Assessment of Treatment Effects in Sequential Trials Identifiability of Causal Effects Structural Equation Models Potential Responses and Counterfactuals Other Issues STATE SPACE AND HIDDEN MARKOV MODELS, Hans R. Kunsch Introduction The General State Space Model Filtering and Smoothing Recursions Exact and Approximate Filtering and Smoothing Monte Carlo Filtering and Smoothing Parameter Estimation Extensions of the Model MONTE CARLO METHODS ON GENETIC STRUCTURES, Elizabeth A. Thompson Genetics, Pedigrees, and Structured Systems Computations on Pedigrees MCMC Methods for Multilocus Genetic Data Conclusion RENORMALIZATION OF INTERACTING DIFFUSIONS, Frank den Hollander Introduction The Model Interpretation of the Model Block Averages and Renormalization The Hierarchical Lattice The Renormalization Transformation Analysis of the Orbit Higher-Dimensional State Spaces Open Problems Conclusion STEIN'S METHOD FOR EPIDEMIC PROCESSES, Gesine Reinert Introduction A Brief Introduction to Stein's Method The Distance of the GSE to its Mean Field Limit Discussion

Batting strategy in limited-overs cricket

Abstract: The paper attempts to understand batting strategies that are employed in limited overs cricket games. The question of the optimum batting strategy is posed in a simplified dynamic programming representation. We demonstrate that optimum strategies may be expected to differ fundamentally in the first and second innings, typically involving an increasing run rate when setting a target but a run rate which may decline over the course of an innings when chasing one. Data on English county level limited overs games are used to estimate a model of actual batting behaviour. The statistical framework takes the form of an interesting variant on conventional survival analysis models.

Quantifying expert opinion in the UK water industry: an experimental study

Abstract: This paper reports an experiment in which 32 experts at three water companies quantified their opinions about two problems: the cost of refurbishing specified pumping stations and the length of unrecorded S24 sewers in four towns. Medians and both unconditional and conditional assessments of quantiles were elicited and assessments were compared with reality and given scores. The quantiles that experts were asked to assess varied and the experts differed in the extent of their relevant background knowledge. The influences of these factors on standard deviations and scores are examined. The main focus is on modelling methods and ways of using the elicited assessments to form subjective probability distributions. We consider fitting log-normal distributions to model asymmetric distributions and also examine models to relate a quantity's assessed standard deviation and interquantile range to its size, finding that interquantile ranges can be modelled more accurately. In addition, different approaches to separating opinions about components of variance are evaluated and compared.

A Simulation Comparison of Several Procedures for Testing the Poisson Assumption

Abstract: The importance of the Poisson distribution among the discrete distributions has led to the development of several hypothesis tests, for testing whether data come from a Poisson distribution against a variety of alternative distributions. An extended simulation comparison is presented concerning the power of such tests. To overcome biases caused by the use of asymptotic results for the null distribution of several tests, an extended simulation was performed for calculating the required critical points for all the tests. The results can be useful to researchers as a guide to selecting the appropriate test from several alternatives that are available.

Bayesian model selection approach to analysis of variance under heteroscedasticity

Abstract: The classical Bayesian approach to analysis of variance assumes the homoscedastic condition and uses conventional uniform priors on the location parameters and on the logarithm of the common scale. The problem has been developed as one of estimation of location parameters. We argue that this does not lead to an appropriate Bayesian solution. A solution based on a Bayesian model selection procedure is proposed. Our development is in the general heteroscedastic setting in which a frequentist exact test does not exist. The Bayes factor involved uses intrinsic and fractional priors which are used instead of the usual default prior distributions for which the Bayes factor is not well defined. The behaviour of these Bayes factors is compared with the Bayesian information criterion of Schwarz and the frequentist asymptotic approximations of Welch and Brown and Forsythe.

Student Selection of Random Digits

Abstract: The paper describes an investigation in which four groups of university students, of sizes 228, 111, 51 and 68, were asked to generate randomly a sequence of 25 digits from {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Previous studies of this kind have suggested that people have tendencies to avoid repetition, to respond serially and to cycle. The aim of our investigation was to study further the nature and extent of people's biases. Particular emphasis was put on the frequency and spread of digits in a selection, as well as on aspects of repetition and clustering. The distribution of the number of clusters of size k was obtained, and our analysis includes the use of this distribution. Our results support previous research showing the very special (and less favoured) status of 0 but also show a strong tendency of students to balance selections and to avoid clustering and sequentially repeating digits.

Inference for the poly-Weibull model

Abstract: Summary. We consider inference for the poly-Weibull model, which arises in competing risk scenarios when the risks have independent Weibull distributions and it is not known which is responsible for the failure. Real and generated data sets illustrate our approaches to inference, which in addition to standard likelihood methods include Bayesian inference by Laplace's method for analytical approximation of integrals and sampling-importance resampling. The poly-Weibull distribution is that of the minimum of several independent Weibull random variables. This arises in scenarios of competing risks (Cox and Oakes (1984), chapter 7), where failure may be due to several causes, each supposed to follow a Weibull law. As we shall see, its advantage over the Weibull model is that it allows not only increasing, constant or decreasing hazard functions with zero or non-zero asymptotes but also non-monotone hazard functions, for example with a 'bathtub' shape. This is an attractive property because such hazards are not uncommon in practice; they correspond to an initial high failure rate a 'bum-in' period followed by lower odds of failure, which eventually increase. See Blackstone et al. (1986) for an application of such hazards in the context of survival after heart surgery. This paper discusses inference for the model, which deserves to be better known. We show by example that standard likelihood or Bayesian inference is straightforward, despite the assertion by Berger and Sun (1993) that a computation of the likelihood is 'typically not feasible', a belief that led them to apply Gibbs sampling for Bayesian inference. In fact the likelihood is so simply obtained that Markov chain simulation is unnecessary, and either Laplace approximation (Tiemey and Kadane, 1986) or the Bayesian bootstrap (Smith and Gelfand, 1992) may be applied; neither has the difficulties of assessing convergence that are associated with Markov chain methods. This reduces considerably the computational burden in obtaining posterior probabilities, densities or moments and widens the class of prior densities that can easily be used because they need not be chosen mainly for computational convenience. The chief difficulty in using the likelihood is possible non-identifiability of the model parameters, but we give some practical suggestions for detecting this.

Bayesian modelling of prehistoric corbelled domes

Abstract: The field of archaeology provides a rich source of complex, non-standard problems ideally suited to Bayesian inference. We discuss the application of Bayesian methodology to prehistoric corbelled tomb data collected from a variety of sites around Europe. We show how the corresponding analyses may be carried out with the aid of reversible jump Markov chain Monte Carlo simulation techniques and, by calculating posterior model probabilities, we show how to distinguish between competing models. In particular, we discuss how earlier analyses of tomb data by Cavanagh and Laxton and by Buck and co-workers, where structural changes are anticipated in the shape of the tomb at different depths, can be extended and improved by considering a wider range of models. We also discuss the extent to which these analyses may be useful in addressing questions concerning the origin of tomb building technologies, particularly in distinguishing between corbelled domes built by different civilizations, as well as the processes involved in their construction.

On Estimating Standard Errors in Multilevel Analysis

Abstract: Standard errors of maximum likelihood estimators are commonly estimated from the inverse of the information matrix. When the information matrix is a function of parameters, some of which are estimated with little precision, the standard error may be estimated very poorly. This problem is discussed in the context of two-level (random-coefficient) models, and some remedies are proposed.

Bayes Prediction in a Pareto Lifetime Model With Random Sample Size

Abstract: The prediction of future ordered observations shows how long a sample of units might run until all fail in life testing. This paper deals with the problem of predicting the future ordered observations in a sample of size n from a Pareto distribution where the first r ordered observations have been observed. The analysis will depend mainly on assuming that the sample size n is a random variable having a Poisson or binomial distribution. Numerical examples are used to illustrate the procedure.

The Kelly criterion and bet comparisons in spread betting

Abstract: The Kelly strategy, risking a fixed fraction of one's gambling capital each time when faced with a series of comparable favourable bets, is known to be optimal under several criteria. We review this work, interpret it in the context of spread betting and describe its operation with a performance index. Interlocking spread bets on the same sporting event are frequently offered. We suggest ways of investigating which of these bets may be most favourable, and how a gambler might make an overall comparison of the bets from different firms. Examples illustrate these notions.

Sample size calculations for failure time random variables in non-randomized studies

Abstract: Summary. In non-randomized studies, sample size calculations for failure time random variables are often computed on the basis of the unadjusted log-rank test which assumes that the variable designating group membership is independent of other patient covariates. We show that by extending the methods proposed by Lipsitz and Parzen to time-to-event random variabes the power derived from the unadjusted log-rank test overestimates the true power of the comparison when the variable of interest is correlated with other covariates. We model the hazards by using exponential regression and derive the sample size formulae for both censored and uncensored data. We also present results of a simulation study to assess the validity of the derived formulae when the intended method of analysis is the proportional hazards regression model and the data continue to have an exponential distribution. We apply the methods proposed to the non-Hodgkin's lymphoma prognostic factors project data set.

Construction of Reference Centiles Using Mean and Dispersion Additive Models

Abstract: Summary. A procedure for constructing 'age'-related reference centiles by flexible modelling of the mean and variance of a dependent variable, assuming normal errors, is described. Both the mean and the variance of the dependent variable are modelled by using either parametric or smooth nonparametric functions of age. This model is a submodel of the more general mean and dispersion additive model (MADAM). Maximum (penalized) likelihood estimation is used for fitting the (non)parametric model. The problem of skewness in the data is dealt with by using a Box-Cox transformation. Flexible and interactive model fitting is provided by a menu-driven GLIM4 library macro called MADAM. A method of model selection is provided which uses the deviance as a measure of the goodness of fit to the data. The method is demonstrated by an example.

Prediction Intervals Based on Autoregression Forecasts

Abstract: The variability of parameter estimates is commonly neglected when constructing prediction intervals based on a parametric model for a time series. This practice is due to the complexity of conditioning the inference on information such as observed values of the underlying stochastic process. In this paper, conditional prediction intervals when using autoregression forecasts are proposed whose simple implementation will hopefully enable wide use. A simulation study illustrates the improvement over classical intervals in terms of empirical coverage.

Use of the bootstrap in robust estimation of location

Abstract: Robust measures of location may be used in estimation problems to help to mitigate the possible effects of outliers or asymmetry. A practical difficulty is how to determine reliable valid estimates of the precision of the corresponding sample estimates. Commonly, precision is expressed by means of standard errors or confidence intervals. Several data sets are used to make comparisons of the performance of standard errors and confidence intervals based on both asymptotic normal results and the bootstrap for a number of robust location measures, including trimmed means with a fixed trimming level and Huber's proposal 2. Bootstrap results and asymptotic results using a winsorized standard deviation are generally comparable, but there is a tendency for the precision to be overstated with the latter both for very small data sets and for Huber's proposal 2 with skewed data sets. In addition, precision with Huber's proposal 2 is found to be marginally superior to that with the corresponding trimmed mean in the presence of extreme outliers.

Modelling Rental Guide Data Using Mean and Dispersion Additive Models

Abstract: Summary. This paper analyses rent data obtained from a survey conducted for the 1994 rental guide for the city of Munich. We describe how mean and dispersion additive models can be applied to the situation where both the mean and the heterogeneity in the dependent variable, rent here, need to be modelled by using parametric and/or nonparametric functions of explanatory variables. The skewness in the rent variable is dealt with either by using a Box-Cox transformation to normality or by modelling the rent variable by using a family of skew distributions. A gamma distribution model was chosen to provide centiles estimates and intervals for the rent variable.

Cobweb Diagrams for Multiway Contingency Tables

Abstract: Cobwebs represent the categories of cross-classifying variables by nodes on the arcs of a circle. Linking two nodes by a line provides information about the corresponding two-variable marginal table. In this paper the widths of the lines are proportional to the square of the adjusted residuals corresponding to the fit of the independence model to the marginal table. Lines are shaded to indicate the signs of the residuals. The method can be used with variables of mixed types and for conditional independence models.

The Analysis of Contingency Tables with One Factor as The Response: Round Two

Abstract: For contingency tables with one factor as a response, log-linear models can be used provided that a minimal model, which constrains the predicted sample sizes of the response factor to equal the actual sizes, is fitted first. Only models that contain this minimal term make inferential sense. For response factors with two levels the results from fitting such log-linear models are identical with those from the corresponding logistic regression. The correspondence is illustrated with two examples from recent literature, both of which have been previously misanalysed.

Initial Stage Problem in Autoregressive Binary Regression

Abstract: When data are repeatedly measured over time, they are highly correlated. To account for serial correlation, we may use an autoregressive (AR) model, in particular the AR(1) model. However, when t= 1, the previous outcome y 0 is unobserved. To overcome such problems in binary regression, we may simply set Y 10 = 0 and the contribution of the possible errors from one or several such initial cases to the model is small when each time series is sufficiently long. Another approach is to model the initial cases separately by using a marginal model and the rest of the observations by using a conditional AR(1) model. We consider the Bayesian approach of modelling the outcomes Y i0 by a beta prior and the resulting model is a two-point mixture model such that the outcome Y i0 = k occurs with probability π k , k = 0, 1, and π 0 + π 1 = 1. These models are demonstrated and compared by using a methadone clinic data set. A simulation study reveals that the mixture model is better than the conditional AR(1) model as it gives smaller relative biases and mean-squared errors for most covariates. Also, the standard deviation to standard error ratios in the mixture model are closer to 1 than in the conditional AR(1) model for most covariates. The improvements are more profound when the time series for each patient is shorter. This demonstrates the importance of the mixture model when the sample size is small.