50-50 MANOVA

Multivariate Analysis of Variance for Collinear Responses

Introduction

Multiple responses are common in industrial and scientific experimentation and a multivariate alternative to ordinary analysis of variance (ANOVA) is often required. Significance tests based on classical multivariate ANOVA (MANOVA) are, however, useless in many practical cases. The tests perform poorly in cases with several highly correlated responses and the method collapses when the number of responses exceeds the number of observations.

50-50 MANOVA is a new method which handles this problem. Principal component analysis is an important part of the new methodology.

References with abstracts

Langsrud, Ø. (2002), 50-50 Multivariate Analysis of Variance for Collinear Responses, The Statistician, 51, 305-317.

ABSTRACT: Classical multivariate analysis-of-variance tests perform poorly in cases with several highly correlated responses and the tests collapse when the number of responses exceeds the number of observations. This paper presents a new method which handles this problem. The dimensionality of the data is reduced by using principal component decompositions and the final tests are still based on the classical test statistics and their distributions. The methodology is illustrated with an example from the production of sausages with responses from near infrared reflectance spectroscopy. A closely related method for testing relationships in uniresponse regression with collinear explanatory variables is also presented. The new test, which is called the 50-50 F-test, uses the first k components to calculate SS_MODEL. The next d components are not involved in SS_ERROR and they are called buffer components.

KEY WORDS: Multiresponse, Significance testing, Principal component, Hotelling's T ², Experimental design, Stabilized multivariate tests.

Langsrud Ø. (2000), Fifty-Fifty MANOVA: Multivariate Analysis of Variance for Collinear Responses, Proceedings of The Industrial Statistics in Action 2000, vol. 2, p. 250-264, University of Newcastle upon Tyne.

ABSTRACT:

The performance of classical MANOVA tests is often poor in cases with several highly correlated responses. These tests collapse when the number of responses exceeds the number of observations. This problem is closely related to the problem of collinear explanatory variables in uniresponse regression. In such cases principal component regression (PCR) is a powerful tool which handles the estimation problem. Based on PCR one can also test whether there is any relationship between the response variable and the explanatory variables. The fifty-fifty F-test uses the first k components to calculate SS_MODEL. The next d components are not involved in SS_ERROR and they are called buffer components.

Similar ideas are used to construct new and powerful MANOVA tests. The new tests are still based on the classical test statistics and their distributions. The dimensionality of the data is, however, reduced in two different ways. In each test, a principal component decomposition is performed on data that are adjusted for other model terms. The first k components are used as response variables and therefore the number of responses is reduced. In addition the next d components are used as buffer components. This can be viewed as a reduction of the number of error degrees of freedom. The principal component decomposition depends on the observed data, but the tests are still correct. The parameters, k and d, can also be calculated from the data based on explained variance criteria. This paper suggests a rule of thumb for this - the fifty-fifty rule.

The methodology is illustrated with an example from sausage production. Fat and salt were varied according to a 6x3 design and the sausage-mixes were analysed by near infrared reflectance spectroscopy. Based on equally spaced wavelengths from 1100 to 2500 nm, there are in this case 351 response variables. The results clearly demonstrate the benefits of the new MANOVA method.