The Decomposition of Effects in Full Factorial Experimental Design into Individual Treatment Combinations

ABSTRACT

Traditional experimental design gives valuable information on main effects and interactions, but may be difficult to interpret because average effects, or contrasts, are being calculated. In order to make sense of the data, experimental scientists employ a number of techniques, including examination of cell means, interaction plots, and transforming the data. Higher-order interactions, where one treatment combination (TC) is active (critical mix), may be difficult to identify. This article examines the decomposition of all effects into their individual treatment combinations (TC effects), and discusses the resulting linear equations. This enables significant effects to be identified that could otherwise be missed, and higher-order effects can be identified without the need to transform the data. Surprisingly simple models can be derived through the use of decomposed effects, even when complex systems are being analyzed. The true values of effects are estimated rather than contrasts, enabling main effects to be evaluated without interference from even strong interactions and providing a solution to a “critical mix.”

KEYWORDS:

• critical mix
• effects
• experimental design
• interactions
• treatment combinations

ACKNOWLEDGEMENT

The author is grateful to Dr Bobbie Leon Foote, US Military Academy, West Point, for invaluable help with significance testing.