# What does “model-based” mean for multivariate statistics?

What does "model-based" mean for multivariate statistics? E.g. mvabund– an R package for model-based analysis of multivariate abundance data.

How is "non-model-based multivariate statistics" then?

Thanks!

Multivariate abundance data (for ecology: observations as rows and species as columns) is usually analyzed using an algorithmic approach. For example, a summarizing distance measure (eg. Bray Curtis) is used for each observation. This matrix of distances is then analyzed using an algorithm. PERMANOVA, for example, uses permutations of the data in a Bray Curtis matrix to create a null distribution, and then test the F-ratio against a distribution of F-ratios to determine significance (Anderson 2001). This does not fit a statistical model.

Model based analyses of multivariate data have been more recently advocated for (see Warton et al. 2014) due to the challenges of distance based multivariate approaches confounding dispersion and location effects (Warton et al 2012). A model based approach "involves specifying a statistical model for the observed multivariate abundance data". I highly recommend reading all of the cited publications, they are excellent.

Anderson, M. J. 2001. A new method for non-parametric multivariate analysis of variance. Austral Ecology:32–46.

Warton, D. I., S. D. Foster, G. De’ath, J. Stoklosa, and P. K. Dunstan. 2014. Model-based thinking for community ecology. Plant Ecology.

Warton, D. I., S. T. Wright, and Y. Wang. 2012. Distance-based multivariate analyses confound location and dispersion effects. Methods in Ecology and Evolution 3:89–101.

According to the world-renowned statistician Peter McCullagh (of generalized linear models fame), a statistical model can be defined as "a set of probability distributions on the sample space" (2002, p. 1225). Therefore, regardless of being multivariate or univariate, I would define a model-based statistics as a set of statistical statements, which includes corresponding assumptions or facts in regard to data probability distributions, as well as specification of relationship between variables, representing (modeling) the data. Thus, a non-model-based multivariate (or univariate) statistics is the one, lacking some elements, comprising statistical model, for example, distribution assumptions. This is, of course, my own simplistic approach to the topic. Much more detailed and formal treatise on the subject can be found the above-referenced paper by McCullagh.

References

McCullagh P. (2002). What is a statistical model? Annals of Statistics, 30(5), 1225-1310. Retrieved from https://galton.uchicago.edu/~pmcc/pubs/AOS023.pdf