Title: Reconciling Least Squares and Robustness in the Two-Way Layout Fixed Effects Experimental Design with an Example from R. A. Fisher’s book "The Design of Experiments."

Abstract: In the last 18 years, the speaker has written two international books in the Wiley Series in Probability and Statistics in essentially diametrically opposing areas of statistical practice. The first book was published in 2008 and is titled Linear Models the Theory and Application of Analysis of Variance. That book begins with a historical development of least squares analysis and normal likelihood theory analysis of the linear model through the auspices of experimental design and analysis of variance, basically as was developed by the giant R.A. Fisher and his contemporaries. The second book, published 10 years later, describes the functional approach to robustness, particularly through the story of M-estimation. The title of the book is "Robustness Theory and Application." This book details a modern theory of statistical inference developed since the 1960s. So, what is the “right” way to approach statistics when analyzing data?

Some insight is offered by appealing to a classical analysis of a linear model representing the data used to form a two-way layout that is proffered in the book “The Design of Experiment” by R.A. Fisher for the purposes of illustrating the analysis of variance for a Randomized Complete Block Design. It is a summarized data set of Immer et al. of yields of five barley varieties for 12 different years, by location combinations. The data reveal outliers and heteroscedasticity.

For these data, we review the appropriate least squares and robust approaches to the analysis.

The history of the robust approach is explained by recounting the development of Huber's M-estimators, the influence function of Hampel that is drivable for any estimator, and why the Fréchet differentiability of M-estimators as described by the speaker in the 1980s established under certain conditions for M-estimators guaranties the robustness of an M-estimator. The early example of a Fréchet differentiable M-estimators afforded by the Tukey Bi-square Functional M-estimator are supported in robust computer packages and are applicable to the linear model representing the two-way layout. Different choices of constraints under parameter redundancies used in different models require one to first represent the analyzes using the same model, before making comparisons. The approach is similar to that of H. Scheffé.

Digressions are made to discuss non-smooth versus smooth M-estimation and to emphasize that the least squares diagnostic approach of transformation is also fraught with danger. Otherwise, we show that both least squares, properly done, and robust inferences essentially agree for these data. This observation reinforces the results resolving what potentially could have been a schizophrenic like situation.