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These techniques are important because, despite all their differences, they all involve the same fundamental mathematical operations, and provide insights into the underlying structure of data (Manly, 2005). The goal of MVAs is to decompose the observed variables into those that are statistically independent and those that are statistically related. Statistical independence is a concept that is easy to understand; mathematical independence is a more abstract concept. Statistical independence implies that if two variables are linked, then the probability of a value of one (or all) of the variables is related to a value of the other (or all) variables. This has different meanings for categorical variables (which are discrete) and quantitative variables (which are continuous). For example, in comparing two values of a categorical variable, the probability of one of those values is related to a value of the other, whereas for quantitative variables, the probability of one of those values is related to a value of the other. The mathematical (and statistical) relationship between two variables can be represented as a correlation coefficient, which can be positive, negative, or zero; positive correlation coefficients imply that the variables in the relationship tend to move in the same direction and move together, negative correlation coefficients imply that the variables in the relationship tend to move in opposite directions and move apart, and zero correlation coefficients imply that the variables in the relationship tend to move together and move in the same direction, although at different speeds. In addition, there are more elaborate techniques for relating variables that involve the use of two or more correlation coefficients.
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