David J. Bartholomew

A goodness of fit test for sparse 2p contingency tables

When a model is fitted to data in a 2p contingency table many cells are likely to have very small expected frequencies. This sparseness invalidates the usual approximation to the distribution of the chi-squared or log-likelihood tests of goodness of fit. We present a solution to this problem by proposing a test based on a comparison of the observed and expected frequencies of the second-order margins of the table. A chi2 approximation to the sampling distribution is provided using asymptotic moments. This can be straightforwardly calculated from the expected cell frequencies. The new test is applied to several previously published examples relating to the fitting of latent variable models, but its application is quite general.

What is statistics

Definitions of statistics abound but many fail to capture adequately the essential interplay of data and theory. It is argued that the ubiquity of variability and uncertainty, which characterize a statistical problem, establish the subject as a major player in science and all rational enquiry. Emphasis is placed on the breadth and unity of the subject and questions are raised for the Royal Statistical Society about its balance, publications and role in education

The origin of factor scores: Spearman, Thomson and Bartlett.

The indeterminacy of factor scores has been a perennial source of debate since the time of Spearman. The main purpose of this paper is to show that, in spite of his inadequate tools and concepts, Sir Godfrey Thomsons approach of 70 years ago was on the right lines. His thinking was constrained by the primitive state of his statistical understanding but it is illuminated by his substantial exchange of the correspondence with M. S. Bartlett in the 1930s, most of which has survived in the Godfrey Thomson archive at the University of Edinburgh. In order to justify our claim and clarify the issues, we have found it necessary to fill in some of the gaps in the original derivations of Spearman, Thomson and Bartlett and to express their work in terms which are intelligible today. The opportunity is taken to relate this earlier work to contemporary debates.

Embedding the new science of research: the organised culture of Scottish educational research in the mid‐twentieth century

Educational research was established in the early decades of the twentieth century in many parts of Europe. The early years were the crucial years as they established dominant forms of inquiry, pioneer sites, and related artefacts, the tools and texts. This paper focuses on the early growth of research culture in education in Scotland, its subjects of study, and its key workers, texts and innovations, to illuminate one site of research development and, in doing so, to engage with a limited but developing field, the histories of educational research. Scotland had an inventive and novel approach to research as well as an urgency to its tasks. It was shaped by close connections with the USA, but its style of work was its own, reflecting local cultures of cooperation and meritocracy. Its culture was organic and systematic, network based, non‐hierarchical, public and national. It was a leading site of empirical and psychologically based large‐ and small‐scale research, outside North America. It was an exporter and importer of techniques, data and people, and was both national and international at the same time. The Scottish case, and its North American scientific links, illuminates the ways in which the national and the international begin to be closely interwoven in the early twentieth century.

Principal Components Analysis

The main purpose of principal-components analysis is to reduce the dimensionality of multivariate data to make its structure clearer. It does this by looking for the linear combination of the variables which accounts for as much as possible of the total variation in the data. It then goes on to look for a second combination, uncorrelated with the first, which accounts for as much of the remaining variation as possible – and so on. If the greater part of the variation is accounted for by a small number of components, they may be used in place of the original variables.

Godfrey Thomson and the Rise of University Pedagogical Study: A Recorded Lecture Delivered at the University of Edinburgh in November 1950 by Godfrey H. Thomson--A Transcript with Commentary.

Professor Sir Godfrey Thomson is one of the key foundational actors in the history of the educational sciences in the UK. At a time when educational studies and the study of educational psychology were very closely linked, in the decades of the mid‐twentieth century, Thomson was a crucial figure in education research. He is known for his work on intelligence, factorial analysis and the validation and production of intelligence tests (the Moray House tests). However, he viewed himself as a teacher in his work as a professor at the University of Edinburgh and as director of Moray House teachers’ college. He managed closely an ambitious plan to develop an advanced school of education, combining the university department of education, teacher training and a demonstration school, and supervised and taught on many of its courses. This paper is based on a unique resource, an audio recording of Thomson teaching in the early 1950s. It considers the distinctiveness and the research value of this audio source in relation to complementary oral and documentary sources.

Measurement, Estimation and Prediction

Measurement is commonly taken for granted in statistical work but, in the fields where missing observations occur, it is often the main objective. This is because the quantities to be ‘measured’ turn out to be represented by the parameters or random variables of a statistical model. Measurement then becomes a matter of predicting the values of random variables or of estimating the parameters of a distribution. When the unobserved variables are latent and, possibly indeterminate in number, the key idea is to determine their conditional distribution given what has been observed. This is essentially a routine matter involving the manipulation of probability functions. However, it is necessary to make clear what has to be defined and what are the constraints imposed by the logic of probability theory. This is important because much controversy, for example in relation to factor scores, has resulted from a failure to appreciate this point. We also introduce the one-parameter exponential family of distributions. This achieves a substantial simplification without incurring a serious loss of generality. In fact, it permits a considerable degree of unification of existing models and the development of new ones.

Models for Time Series

In a discrete time series the unobserved variables are latent only in the sense that they lie in the future and are therefore unknown. Any model may thus reasonably begin with the joint distribution of all variables—past and future—from which the relevant conditional distribution may be determined. Two types of model will be described. The first specifies the mean values of the joint distribution and the second, its covariance structure. The former will be described as ‘regression-type’ models and the latter as ‘autoregressive’ models. A regression-type model assumes that what we observe is the sum of a systematic part and an ‘error’. The systematic part specifies the mean value at successive points in time and the errors are assumed independent. Over sufficiently short periods of time one may be willing to assume that the systematic part is a simple function of time, possibly linear or cyclical, but whatever form is chosen, it is part of the input. An autoregressive model involves an assumption about the covariance structure of the data and, in particular, about the serial correlations of members of the time series. We illustrate this by supposing that any member of the series is correlated with one or two immediate predecessors. The results correspond, as they should, with standard results.

Help will be welcomed from every quarter: the work of William Boyd and the Educational Institute of Scotland’s Research Committee in the 1920s

This paper discusses evidence, collected during an ESRC‐funded project (‘Reconstructing a Scottish School of Educational Research, 1925–1950’), of a remarkable vision to involve teachers in educational research in Scotland by the Educational Institute of Scotland in the 1920s through the work of its Research Committee. Led by William Boyd, the Committee thought that involvement in research was a crucial stepping stone towards achieving professional status for teachers. It conducted a number of detailed investigations involving teachers, thereby introducing research into the consciousness and practice of teachers. This paved the way for Scotland to make significant contributions to educational research on the international stage.

Maxwell, Albert Ernest

A.E. Maxwell (1916–1996) spent most of his career at the Institute of Psychiatry, University of London, latterly as head of the Biometrics Unit. He is best known for his co-authorship (with D.N. Lawley) of the book Factor Analysis as a Statistical Method. This established factor analysis as a branch of statistical methodology. His other publications include several books on psychological statistics and he also engaged in extensive consulting and advisory work. Keywords: factor analysis; psychological statistics; applied multivariate methods