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Multivariate Statistics

The term multivariate statistics may be defined as the collection of methods for analyzing multivariate data. Data are said to be multivariate when each observation has scores for two or more random variables. Although this definition could be construed as including any statistical analysis including two or more variables (e.g., correlation, ANOVA, multiple regression), the term multivariate is usually reserved for analyses that include two or more dependent variables. For example, multiple regression with several predictors and one dependent variable would not qualify as a multivariate technique, but multivariate multiple regression with two or more dependent variables would.

Multivariate statistical methods are essential in communication research and research in many other areas because research questions and hypotheses often include more than one dependent variable, and it is common practice for investigators to measure multiple dependent variables in each study. Multivariate methods allow investigators to formulate research questions and test hypotheses that are more complex than those that can be addressed using only univariate methods.

This entry describes issues relevant to the application of multivariate statistics without focusing on any particular type of multivariate analysis. The emphasis is on the unique parts of multivariate analyses that differ from univariate analyses. For example, the use of multivariate statistical methods entails the more prominent use of vectors, matrices, and linear combinations. The simultaneous analysis of multiple dependent variables also means that it is important to distinguish between univariate versus multivariate hypotheses and univariate versus multivariate hypothesis tests. To illustrate the breadth of multivariate statistical methods, this entry surveys issues in three areas of multivariate statistical applications: multivariate extensions of commonly used univariate methods, methods for data reduction, and methods for examining how a set of variables relate to one another.

Important Aspects of Multivariate Analysis

A number of multivariate methods can be seen as direct extensions of commonly applied univariate models. Canonical correlation is an extension of the Pearson correlation analysis. Hotelling’s T2 statistic can be used as a t-test comparing sets of means. Multivariate multiple regression extends multiple regression and multivariate analysis of variance extends the analysis of variance. Other multivariate methods are not direct extensions of univariate methods, but instead allow researchers to address new and different ends not possible using univariate statistical methods. For example, principal components analysis allows one to represent the information contained in a large number of variables in a reduced subset of variables, cluster analysis allows one to classify individuals into distinct groups based on their scores on a set of variables, and one can use exploratory factor analysis to examine how a set of variables relate to one another. Regardless of whether one uses a method that extends a commonly used univariate approach or a method that is unique to multivariate applications, the simultaneous analysis of multiple dependent variables requires the researcher to think differently.

Matrices and Matrix Operations

Informally, one can think of a matrix as a container used to organize information. Such a matrix has dimensions. For example, a 2 × 2 matrix has two rows and two columns and will contain four elements. One may organize the data collected in a data matrix with dimensions n × p, where n indicates the number of individuals in our sample and p the number of variables measured for each. Subscripts allow one to note the exact location of an element in a matrix. For example, element a35 is the value from the third observation for the fifth measured variable in matrix A. A matrix with one of the dimensions equal to 1 is called a vector and a matrix with dimension of 1 × 1 is called a scalar. For example, the following shows a 3 × 3 matrix X, a 4 × 1 vector y, and a scalar value z.

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