David R. Bellhouse

The Reverend Thomas Bayes, FRS: A Biography to Celebrate the Tercentenary of His Birth

Thomas Bayes, from whom Bayes Theorem takes its name, was probably born in 1701 so that the year 2001 would mark the 300 th anniversary of his birth. A sketch of his life will include his family background and education, as well as his scientific and theological work. In contras t to some, but not all, biographies of Bayes, the current biography is an attempt to cover areas beyond Bayes’s scientific work. When commenting on the writing of scientific biography, Pearson (1978) stated, “it is impossible to understand a man’s work unless you understand something of his character and unless you understand something of his environment. And his environment means the state of affairs social and political of his own age.” The intention here is to follow this general approach to biography. There is very little primary source material on Bayes and his work. For example, only three of his letters and a notebook containing some sketches of his own work, almost all unpublished, as well as notes on the work of others were known to have survived. Neither the letters, nor the notebook, are dated, and only one of the letters can be dated accurately from internal evidence. This biography will contain new information about Bayes. In particular, among the papers of the 2 nd Earl Stanhope, letters and papers of Bayes have been uncovered that were previously not known to exist. The letters indirectly confirm the centrality of Stanhope in Bayes’s election to the Royal Society. They also provide evidence that Bayes was part of a network of mathematicians initially centered on Stanhope. In addition, the letters shed light on Bayes’s work in infinite series.

A review of optimal designs in survey sampling

Results in five areas of survey sampling dealing with the choice of the sampling design are reviewed. In Section 2, the results and discussions surrounding the purposive selection methods suggested by linear regression superpopulation models are reviewed. In Section 3, similar models to those in the previous section are considered; however, random sampling designs are considered and attention is focused on the optimal choice of j. Then in Section 4, systematic sampling methods obtained under autocorrelated superpopulation models are reviewed. The next section examines minimax sampling designs. The work in the final section is based solely on the randomization. In Section 6 methods of sample selection which yield inclusion probabilities 7T = n/N and rij = n(n - 1)/N(N - 1), but for which there are fewer than N C, possible samples, are mentioned briefly.

Banishing Fortuna : Montmort and De Moivre

In 1708 Pierre Rémond de Montmort published Essay d’analyse sur les jeux de hazard, an analysis of contemporary games of chance using probability theory. Three years later Abraham de Moivre published De Mensura Sortis in which various probability problems were solved. Montmort felt that De Moivre had plagiarized his work. In 1718 De Moivre expanded his work under the title The Doctrine of Chances. Montmort remained unhappy with this new work. Both the Essay d’analyse and The Doctrine of Chances contain allegorical engravings that describe in pictures the nature and importance of their respective author’s work.

Maty’s Biography of Abraham De Moivre, Translated, Annotated and Augmented

November 27, 2004, marked the 250th anniversary of the death of Abraham De Moivre, best known in statistical circles for his famous large-sample approximation to the binomial distribution, whose generalization is now referred to as the Central Limit Theorem. De Moivre was one of the great pioneers of classical probability the- ory. He also made seminal contributions in analytic geometry, complex analysis and the theory of annuities. The first biography of De Moivre, on which almost all subsequent ones have since relied, was written in French by Matthew Maty. It was published in 1755 in the Journal britannique. The authors provide here, for the first time, a complete translation into English of Matys biography of De Moivre. New mate- rial, much of it taken from modern sources, is given in footnotes, along with numerous annotations designed to provide additional clarity to Matys biography for contemporary readers.

Symbolic operators for multiple sums

Abstract Operators for expectation and the derivation of unbiased estimates for multiple sums are given. Calculations automated are functions of outer products of full partitions. An operator FP for the derivation of a full partition is iterated across the dimensions of a sum to automate a calculation. Applications include multi-stage sample surveys and r -way contingency tables. The procedures have been defined in Mathematica .

Linear Models for Randomized Response Designs

Abstract Linear models, based on random permutation models, are developed to include a wide class of one-sample randomized response designs. The general form of the “optimal” estimator of the finite population mean or proportion is obtained. Most of the conventional randomized response estimators are seen to be optimal, in terms of minimum average mean squared error, within their associated designs. The optimality results are obtained for sampling without replacement and include extensions of randomized response designs to unequal probability sampling.

Systematic sampling of periodic functions

Using Fourier series expansions of functions of one and two variables, we find the variances of the sample mean for random and systematic sampling methods in one and two dimensions.

A Public Health Controversy in 19th Century Canada

From 1858 to 1860, the English naturalist and social activist Philip P. Carpenter toured North America. In April of 1859 he visited Montreal, Canada. Shocked by the sanitary conditions of the city, he wrote a paper that used statistical arguments to call for health reforms. Six years later he settled in Montreal and quickly became an active promoter of this cause. He began accumulating additional numerical evidence in support of his views. In the aftermath of a cholera scare in 1866, Carpenter became the driving force behind the creation of the Montreal Sanitary Association. That same year he published a second, more detailed article that took advantage of the 1861 census data to analyze mortality rates in Montreal. He made further statistical investigations in 1869. Unfortunately, Carpenter did not understand some of the subtleties as- sociated with the analysis of vital statistics. An obscure bookkeeper, Andrew A. Watt, made a scathing public attack on both Carpenters data and his interpretation thereof. In a series of newspaper articles, Watt scrutinized systematically all of Carpenters writings, showing his faults and correcting them wherever he could. Although Watts arguments were correct, the public was slow to under- stand them. The controversy continued through 1870. When the nature of Watts criticisms finally became better understood and Carpenter persisted with statistical arguments, the latter lost credibility and was abandoned by his own association.

The deification of Newton in 1711

The mathematician William Jones obtained a number of Isaac Newton’s manuscripts and letters through the acquisition of papers owned by John Collins. Jones published them in 1711 in a book entitled Analysis per quantitatum series, fluxiones, ac differentias. It was one of the small events in the priority controversy between Newton and Leibniz over the calculus. Inserted in the book, as well as on the title page, are a number of allegorical engravings, almost certainly commissioned by Jones. This article discusses some interpretations of the engravings. As with Halley’s dedicatory poem to Newton in Principia mathematica, the engravings endow Newton with a god-like status. At the same time, the engravings also show some of Newton’s activities as a mortal and place him in a superior position to Leibniz with respect to the discovery of the calculus and as a mathematician.

Joint Survival Analysis of Time to Drug Change and a Terminal Event with Application to Drug Failure Analysis using Transplant Registry Data

Statistical approaches for drug effectiveness studies after liver transplant have used a survival model with changes in treatment as a time-dependent covariate. However, the approach requires that changes in the time-dependent covariate be unrelated to survival outcome. Usually this is not the case, as one drug may be discontinued and an alternative chosen due to the declining health status of the patient. Other approaches examine only subjects who remain on the same drug over a time window, which discards valuable data and may lead to biased effects since this excludes data related to early deaths and to individuals who perform poorly on the drug and had to switch treatments. Because of these issues there are conflicting results seen in the evaluation of immunosuppressive drug effectiveness after liver transplant. We propose a joint survival outcome model with a time-to-drug-change event and a terminal event in graft failure that is useful in drug effectiveness studies where subjects are discontinued from an immunosuppressant (in favour of alternative treatment) due to health reasons. We also include a longitudinal biomarker component. The model takes account of the dependencies across out- comes through shared random effects. Using a Markov chain Monte Carlo approach, we fit the joint model to data from liver transplant recipients from the Scientific Registry for Transplant Recipients.