Academic article

Analyzing designed experiments with multiple responses

Langsrud, Øyvind; Jørgensen, Kjetil; Ofstad, Ragni; Næs, Tormod

Publication details

Journal: Journal of Applied Statistics, vol. 34, p. 1275–1296, 2007

Issue: 10

International Standard Numbers:
Printed: 0266-4763
Electronic: 1360-0532

Open Access: none


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.