New Zealand Statistical Association 2024 Conference

Marti Anderson – ANZJS Read Paper

Massey University and PRIMER-e

The incremental progression from fixed to random factors in the analysis of variance: a new synthesis (with discussion)

This is joint work with Ray N. Gorley, Antonio Terlizzi

The preprint of this ANZJS Read Paper is available at the following web page (opens in a new tab): Anderson, Gorley and Terlizzi: ANZJS Read Paper (pdf)

Ah, the well-known and perplexing phenomenon faced by every practicing researcher who embarks on an analysis of variance (ANOVA) … are my factors fixed or random? And then (yikes!) the rather uncomfortable realisation that, yes, it does matter. The choice affects: (i) the expectations of mean squares (EMS); (ii) the estimates of variance components; (iii) the construction of suitable statistics for testing hypotheses; and (iv) the nature and extent of the inferences arising from such tests. Well, good research often starts where there is a fight, and how to calculate the “correct” EMS arising from mixed-model ANOVA designs has been the source of a long-standing tussle. Isn’t it amazing that Cornfield & Tukey long ago (1956) suggested that the difference between a fixed and a random factor was not a dichotomy, but rather, a gradation, which depends only on your sampling effort relative to your inference space. When combined with the (little-bit-later) work of Hartley and Rao, we not only get a lovely extension to the most general cases (balanced or unbalanced designs, any types of factors along the gradation, multivariate, etc.), but also achieve a sweet resolution to the dispute. In this talk, I will briefly outline all of this and showcase its virtues in an ecological example: the responses of 151 species of mollusc to a sewage outfall on the Italian coast. I’ll unveil the desirable increase in power that taking this synthetic approach can afford.