Multivariate Analysis

Multivariate analysis is a set of statistical techniques that analyze the characteristics of a population observing the statistical dependence of multiple variables simultaneously. Multivariate data analysis groups a large variety of different statistical techniques that are commonly used in applied social sciences, having practical applications in economics, psychology, marketing, and sociology as well in a large variety of other disciplines. The common feature of these techniques is the ability to analyze the population with respect to a variety of different attributes simultaneously. Multivariate techniques differ essentially on the statistical assumptions on which they are built, and they generally target different research questions. The common feature is that they include several variables simultaneously in the analysis of the problem. Specific differences are presented in the remaining of this entry.

There are several techniques for multivariate analysis (Figure 1). They are grouped into two main categories: dependence and interdependence techniques. Dependence techniques explore the dependence between one or more dependent variables and one or more independent variables, that is, how the mean of these variables relate to each others. This category is further differentiated in terms of the number of dependent and independent variables, as well as in terms of the metric used in the dependent variable—essentially if the variable is continuous (metric) or discrete (binary or ordered). Examples are multiple regression analysis, discriminant analysis, multivariate analysis of variance (MANOVA), and conjoint analysis.

Interdependence techniques analyze the interrelation among variables but assuming no hierarchical dependence among them. This category is further differentiated in terms of the subject of the statistical analysis, which can refer to variables, cases (e.g., consumers), or objects in analysis, as well in the metric of the variables. Examples of interdependence techniques are factor (and principal components) analysis, cluster analysis, and multidimensional scaling and correspondence analysis.

Regression Analysis

Regression analysis is a multivariate technique that aims at estimating the conditional mean of a [Page 1016]population while measuring the impact of influencing covariates on the average. In regression analysis, several independent variables are simultaneously related to a dependent variable. The methodology, started in the early 1800s (Stigler 1981), was later named regression by Francis Galton in an anthropological study. On that occasion, the name regression was used to describe the biological phenomenon of extremes regressing toward the mean of the population: in Galton's example, descendants of tall ancestors tended to be shorter, while shorter parents tended to have taller descendants, indicating an overall convergence toward the average of the population.

Figure 1 Classification of Multivariate Techniques

Source: Hair et al. 2006, 14–15.

The basic principle of regression analysis is the estimate of a line that best fits a series of data. Essentially, in the basic case where a variable Y has a mean conditionally dependent on X, regression analysis enables the researcher to estimate the line that describes Y as a function of X, identifying the relation Y = α + β · X + ε (Figure 2). In the equation, the unknown parameters are estimated, where α indicates the mean of the population (the intercept), and β indicates the increase in Y for each unit increase in X (the coefficient of the slope). The residual of the equation is ε, which is a stochastic component and indicates the difference between the real value and the estimated value. In the case of multivariate regression, X corresponds to a matrix of at least two columns.

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