multivariate analysis

Univariate analysis consists in describing and explaining the variation in a single variable. Bivariate analysis does the same for two variables taken together (covariation). Multivariate analysis (MVA) considers the simultaneous effects of many variables taken together. A crucial role is played by the multivariate normal distribution, which allows simplifying assumptions to be made (such as the fact that the interrelations of many variables can be reduced to information on the correlations between each pair), which make it feasible to develop appropriate models. MVA models are often expressed in algebraic form (as a set of linear equations specifying the way in which the variables combine with each other to affect the dependent variable) and can also be thought of geometrically. Thus, the familiar bivariate scatter-plot of individuals in the two dimensions representing two variables can be extended to higher-dimensional (variable) spaces, and MVA can be thought of as discovering how the points cluster together.

The most familiar and often-used variants of MVA include extensions of regression analysis and analysis of variance, to multiple regression and multivariate analysis of variance respectively, both of which examine the linear effect of a number of independent variables on a single dependent variable. This forms the basis for estimating the relative (standardized) effects of networks of variables specified in so-called path (or dependence or structural equational) analysis-commonly used to model, for example, complex patterns of intergenerational occupational inheritance. Variants now exist for dichotomous, nominal, and ordinal variables.

A common use of MVA is to reduce a large number of inter-correlated variables into a much smaller number of variables, preserving as much as possible of the original variation, whilst also having useful statistical properties such as independence. These dimensionality-reducing models include principal components analysis, factor analysis , and multi-dimensional scaling . The first (PCA) is a descriptive tool, designed simply to find a small number of independent axes or components which contain decreasing amounts of the original variation. Factor analysis, by contrast, is based on a model which postulates different sources of variation (for example common and unique factors) and generally only attempts to explain common variation. Factor analysis has been much used in psychology, especially in modelling theories of intelligence.

Variants of MVA less commonly used in the social sciences include canonical analysis (where the effects are estimated of a number of different variables on a number of-that is not just one-dependent variables); and discriminant analysis (which maximally differentiates between two or more subgroups in terms of the independent variables).

Recently, much effort has gone into the development of MVA for discrete (nominal and ordinal) data, of particular relevance to social scientists interested in analysing complex cross-tabulations and category counts (the most common form of numerical analysis in sociology). Of especial interest is loglinear analysis (akin both to analysis of variance and chi-squared analysis) which allows the interrelationships in a multi-way contingency table to be presented more simply and parsimoniously.