Canonical Variate Analysis and Related Methods with Longitudinal Data

Abstract

Canonical variate analysis (CVA) is a widely used method for

analyzing group structure in multivariate data. It is

mathematically equivalent to a one-way multivariate analysis of

variance and often goes by the name of canonical discriminant

analysis. Change over time is a central feature of many

phenomena of interest to researchers. This dissertation extends

CVA to longitudinal data. It develops models whose purpose is to

determine what is changing and what is not changing in the group

structure. Three approaches are taken: a maximum likelihood

approach, a least squares approach, and a covariance structure

analysis approach. All methods have in common that they

hypothesize canonical variates which are stable over time.

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The maximum likelihood approach models the positions of the group

means in the subspace of the canonical variates. It also

requires modeling the structure of the within-groups covariance

matrix, which is assumed to be constant or proportional over

time. In addition to hypothesizing stable variates over time,

one can also hypothesize canonical variates that change over

time. Hypothesis tests and confidence intervals are developed.

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The least squares methods are exploratory. They are based on

three-mode PCA methods such as the Tucker2 and parallel factor

analysis. Graphical methods are developed to display the

relationships between the variables over time.

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Stable variates over time imply a particular structure for the

between-groups covariance matrix. This structure is modeled

using covariance structure analysis, which is available in the

SAS package Proc Calis.

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Methods related to CVA are also discussed. First, the least

squares methods are extended to canonical correlation analysis,

redundancy analysis, Procrustes rotation and correspondence

analysis with longitudinal data. These least squares methods

lend themselves equally well to data from multiple datasets.

Lastly, a least squares method for the common principal

components model is developed.