Analyzing Designed Experiments with Multiple Responses

Abstract

This paper is an overview of a unified framework for analyzing designed experiments with univariate or multivariate responses. Both categorical and continuous design variables are considered. To handle unbalanced data, we introduce the so-called Type II∗ sums of squares. This means that the results are independent of the scale chosen for continuous design variables. Furthermore, it does not matter whether two-level variables are coded as categorical or continuous. Overall testing of all responses is done by 50-50 MANOVA, which handles several highly correlated responses. Univariate p-values for each response are adjusted by using rotation testing. To illustrate multivariate effects, mean values and mean predictions are illustrated in a principal component score plot or directly as curves. For the unbalanced cases, we introduce a new variant of adjusted means, which are independent to the coding of two-level variables. The methodology is exemplified by case studies from cheese and fish pudding production.

Keywords:

• 50-50 MANOVA
• general linear model
• least-squares means
• multiple testing
• principal component
• rotation test
• unbalanced factorial design

Acknowledgements

The work in this paper is a part of the IBION project (Industrial Biostatistics Network) financed in part by the Research Council of Norway through grant 145456/130. The work was also partially supported by The Norwegian Research Council's grant 132975/112 (Development of Methods for Recipe optimization in the Food Industry). Dr Fr⊘ydis Bjerke at Matforsk is thanked for her helpful comments.