The monthly seminar of the SSA in WA was presented by Brenton Clarke, the current President of the Branch. The seminar was held via ZOOM, the web-based video conferencing tool, because of the COVID-19 pandemic restrictions.

Since I live very close to Brenton, I made the short journey to his home for the live seminar in his study.

About 20 people were on the video conference. I sat in the front seat with Brenton, together with my M&M chocolate coated peanuts to help me enjoy the video! The title of the talk was ‘Generalisations of Orthogonal Components in Analysis of Variance (ANOVA)’. I had a feeling from the abstract that we were going to be bombarded with ‘elegant’ matrix algebra.

The seminar reminded me of when I was a student when it was imperative to see and suffer through the construction of much theory if you were serious in understanding statistical methodology.

We were treated to an entertaining historical talk from the 1870s to the 2020s on the orthogonal Helmert matrices used in an example of a 2-way ANOVA (randomised complete block design) to relate the decomposition of the sums of squares in the ANOVA table to uncorrelated random variables. Brenton showed us the long-hand way versus the ‘elegant’ mathematical matrix algebra approach involving Kronecker products. Brenton showed remarkable knowledge and recollections of the use of many references in his talk spanning 1860s to 2020.

He showcased work from his 2002 paper and his 2008 book on ‘Linear Models’.

I think we witnessed a marvelous exposition of theory illustrating Brenton’s breadth of work he has done in the last 20 years of his research on the theory of linear models. One could say we saw the beauty of statistics with echoes of Beethoven’s 5th symphony ‘Eroica’ in the background.

A good series of comments in the discussion followed the talk which left us pondering on the use of the elegant theory in practice. Brenton even alluded to work on the Hasse or lattice diagram of the ANOVA table showing the effects of the terms in the model as shown by Rosemary Bailey work (2020).

Well done Brenton on a very informative discussion and summary history of work with various authors on the linear models and their matrix algebra investigating the various theoretical relationships in the analysis of variance table.

During the night I was pondering how the sweep algorithm used in the ‘beautiful’ ANOVA module of the GenStat (https://www.vsni.co.uk/software/genstat) program related to this talk. We will leave that for another talk.

Brenton and I enjoyed a meal at the local Thai restaurant to celebrate the night observing the small numbers of people at at the restaurant due to the COVID-19 restrictions.

Three cheers for my great friend for a great talk!

by Mario D’Antuono

References

Bailey, R.A. (2020) Hasse diagrams as a visual aid for linear models and analysis of variance. Communications in Statistics- Theory and Method. https://doi.org/10.1080/03610926.2019.1676443

Clarke, B.R. (2002) A representation of orthogonal components in analysis of variance, International Mathematical Journal, 1, 133-147. https://www.semanticscholar.org/paper/A-representation-of-orthogonal-components-in-of-Clarke/a7b2d8f31ac011c13f222741b8f97f6ced801a68

Clarke, B.R. (2008) Linear Models The Theory and Application of Analysis of Variance, Wiley, Hoboken, N.J. https://www.amazon.com/Linear-Models-Application-Probability-Statistics/dp/0470025662

Farhadian, R. and Clarke, B.R. (2020) A note on the Helmert transformation, Communications in Statistics- Theory and Method, submitted

Irwin, J.O. (1934) On the independence of constituent items in the analysis of variance, J. Roy. Statist. Soc., Suppl., 1, 236-251