**Sankhya:
The Indian Journal of Statistics**

2006, Volume 68, Pt. 3, 392--408

**Testing Treatment Effects in Two-Way Linear Models: Additive or Full Model?**

Bin Cheng, Columbia University, New York, USA

Jun Shao, University of Wisconsin, Madison, USA

SUMMARY. Under a two-way analysis of variance/covariance model, we consider the problem of testing the main treatment effect (a fixed effect of primary interest) when the interaction between the treatment and the other factor (which is either fixed or random) is practically negligible but not exactly zero. Although the theory for analysis of variance/covariance is well-developed (at least for the fixed effects models), practitioners are not clear on whether the test based on additive model (assuming no interaction) or the test based on full model (including interaction terms) should be adopted. The use of additive model is motivated by a possible gain in the power of the test. On the other hand, the use of full model addresses the concern of having an inflated size of the test when the interaction is not exactly zero. Under balanced fixed effects models, we show that the test based on additive model has correct size even if the additive model is wrong but its power may be very low in the presence of a small interaction effect; contrary to common beliefs, in many practical situations the gain in power by using the additive approach is not substantial even if the additive model is correct. Under unbalanced fixed effects models or balanced/unbalanced mixed effects models, the test based on additive model generally has an inflated size unless the additive model is correct.

*AMS (2000) subject classification. *Primary 62J10.

*Key words and phrases. *Analysis of variance/covariance, mixed
effects, unbalanced models, interaction effect, exact tests, size
and power.