Murray Aitkin Introduction to Statistical Modelling and Inference CRC Press 2022
This book develops a new introductory course in statistics, aimed at students in a wide range of programmes including Data Science.
It is suitable for students in a general Statistics programme, for whom Data Science can be one of many application areas.
Students are expected to have a basic mathematical background, of algebra, coordinate geometry and calculus.
The book presents both the Bayesian and the frequentist approaches to statistical analysis.
This may sound confusing, but these approaches, while apparently different, have very close connections and in many cases use the same data quantities to draw conclusions which are very similar.
The frequentist approach has two conflicting possible interpretations.
One is based on the idea of hypothetical replications of the sample data, and the relation between the observed data and the hypothetical replications.
The other does not depend on hypothetical replications, but is generally based on a quadratic assumption about the fundamental evidence function of statistical
theory.
Bayesian theory does not rely on replications or the quadratic assumption, and is more generally relevant to the analysis of the complex data structures which are increasingly
common, especially in Data Science.
The frequentist analysis is still relevant however for one major class of models, those for Gaussian-based regression and ANOVA (analysis of variance), important applications in
many fields.
The frequentist analysis is also the basis, in this book, for the Bayesian analysis of generalised linear models, the extension of regression models to response distributions in the exponential family.
Murray and Irit Aitkin Statistical Modeling of the National Assessment of Educational Progress
Springer 2011
This book describes our joint contract work for the US National Center for Educational Statistics, on an integrated psychometric and statistical model for the large-scale NAEP surveys of the US Department of Education. The model incorporates the clustered survey design through multiple model levels, and allows for stratification and oversampling of minority ethnic groups. We allow for guessing through a two-class latent class model with different item parameters in the classes.
We are extending this work with a current Institute of Education Sciences grant to examine Bayesian model comparisons for the competing possible models for the survey data, and to obtain the full information from records with incomplete convariate information.
Murray Aitkin Statistical Inference: an Integrated Bayesian/Likelihood
Approach Chapman and Hall/CRC Press 2010
This book sets out an integrated Bayesian and likelihood approach to statistical inference, using the likelihood as the primary measure of evidence for statistical model parameters, and for the statistical models themselves.
To assess the strength of evidence from the data for competing parameter values or competing models, it uses likelihood ratios between the parameter values or the models. To interpret the likelihood ratios it uses a Bayesian approach which requires in general only the non-informative priors that are widely used in posterior
inference about model parameters, though it can accommodate informative priors.
Comparison of different statistical models requires a treatment of the unknown model parameters. This problem is usually treated by Bayes factors; the book gives a different approach which uses the full posterior distribution of the likelihood for each model. This quite small change to standard Bayesian analysis allows a very general and unified approach to a wide range of apparently different inference problems.
A further contribution of the book is to develop a general Bayesian approach to finite population inference. This approach, using the multinomial distribution and a non-informative Dirichlet prior, can also be adapted to provide a general Bayesian "non-parametric" analysis.
The book is intended to provide both an exposition of an alternative to standard Bayesian inference, and the foundation for a course sequence in modern Bayesian theory at the graduate or advanced undergraduate level.
I have been working since 2003 with Irit Aitkin, in a series of research contracts with the US National Center for Education Statistics and several research grants from the Institute of Education Sciences, on the development and implementation of fully model-based approaches to the analysis of the large-scale US National Assessment of Educational Progress (NAEP) surveys of primary and high-school students' achievement. We have developed multi-level model-based approaches for the clustered and stratified survey designs, examined alternative distributions for the latent ability, and formulated alternative latent class models for guessing, or "engagement" on the test items. For each of these developments, we have compared the results of the model-based approach with the current methods of analysis using both simulated and actual NAEP data.
Bayesian Course
Professional Associations, Memberships & Awards:
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Current and Recent Funded Research
Australian Research Council (ARC) 2012-2015
The ARC supported a
Discovery project on latent class modelling approaches to sub-group
structure in social networks, which supported Duy Vu as Research
Associate and visits from Brian Francis. Analyses in this project
(Social Networks 2014 - hotkey) recovered the known group structure in a
well-known social network of women in Natchez Mississippi in the 1930s
(a classic sociological data set) and the known leadership structure in
the Noordin Top terrorist network (Journal of the Royal Statistical
Society - hotkey), which other extensive social network analyses had
failed to identify. A simulation study (Metron - hotkey) supported the
integrated Bayesian/likelihood approach to model comparison, as applied
to finite mixture models.
US Department of Education 2003-2011
Australian Research Council (ARC) 2005-2008
The ARC supported a Discovery project 2005-8 on Bayesian and likelihood approaches to finite mixture, random effect and multinomial models, on which Irit Aitkin worked part-time with me. The project also supported Charles Liu briefly. This project led to the development of the integrated Bayesian/likelihood approach to statistical inference set out in the book
and in research papers Journal of Mathematical Psychology 2008 and Annals of Applied Statistics 2009 with Charles Liu.
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