Towards robust analysis of variance

Seheult, A.H.; Tukey, J.W.

Abstract

A distinctive feature of analysis of variance is the common occurrence of more than one error term. This feature calls attention to the two distinct potential roles of a single mean square. As a “numerator” it measures the variability visible at a given level in a design hierarchy, and as a “denominator” it measures how much variability has been “passed up” to higher levels, and may, if appropriate, serve as part of an error term. We propose a straightforward multiphase procedure that explicitly recognizes these two roles, and argue that, in general, such considerations preclude naive use of robust regression techniques for analysis of factorially designed experiments. Instead, an upsweeping-by-medians decomposition of the data is followed by a comparison-within-subtable analysis to flag exotic (“relatively large”) entries in each of the subtables associated with the different sorts of variation. A classical analysis by means, after replacing each identified exotic entry by an algorithmically specified value, yields a decomposition of the data that is used to construct an analysis of variance table in which for each sort of variation there is both a list of any exotic entries and an inner (“denominator”) mean square that ‘excludes’ those exotic entries. The analysis can then be completed by downsweeping the inner subtables that are insufficiently prominent, and providing (formally) appropriate error terms for analyzing table entries that remain. The results are displayed as a decomposition of the data into exotic values and those inner subtables, both simple and composite, that survive downsweeping. The exploratory nature of the approach is emphasized, and the method is applied to an example of a factorial experiment in which all factors have three or more versions.

Citation

Seheult, A., & Tukey, J. (2001). Towards robust analysis of variance. In A. Saleh (Ed.), Data analysis from statistical foundations : a festschrift in honour of the 75th birthday of D.A.S. Fraser (217-244). Nova Science Publishers

• Online Publication Date: Dec 25, 2001
• Publication Date: Dec 25, 2001
• Deposit Date: Oct 21, 2008
• Publicly Available Date: Oct 21, 2008
• Publisher: Nova Science Publishers
• Pages: 217-244
• Book Title: Data analysis from statistical foundations : a festschrift in honour of the 75th birthday of D.A.S. Fraser.
• Chapter Number: 15
• Keywords: Downsweeping, Exotic values, Factorial experiments, Fibian,; Half-Winsorizing, Inner mean squares, Mean polish, Median polish, ; Middle-Median, Midmedian,; Robust ANOVA, Upsweeping, Winsorizing.
• Public URL: https://durham-repository.worktribe.com/output/1674393
• Publisher URL: https://www.novapublishers.com/catalog/product_info.php?products_id=656