The correlation coefficient completely defines the dependence structure only in very particular cases, for example when the distribution is a multivariate normal distribution.
One researcher validated the ability of multivariate regression models to forecast the impact on sales of a product of many variables including price, discount, visual merchandizing, etc.
Small multiple designs, multivariate and data bountiful, answer directly by visually enforcing comparisons of changes, of the differences among objects, of the scope of alternatives.