A consistent and general modified Venn diagram approach that provides insights into regression analysis

Venn diagrams are used to provide an intuitive understanding of multiple regression analysis and these diagrams work well with two variables. The area of overlap of the two variables has a one-to-one relationship to the squared correlation between them. This approach breaks down, however, with three-variables. Making the overlap between the pairs of variables consistent with their squared bivariate correlations often results in the overlap of two of these variables with the third variable that is not the same as the variance of the third variable accounted for by the other two variables. I introduce a modified Venn diagram approach that examines the relationships in multiple regression by using only two circles at a time, provides a new and consistent reason why the circles need to be of the same size, and designates a "target variable" whose overlap with the other circle corresponds to the variance accounted for by the other variable or variables. This approach allows the visualization of the components involved in multiple regression coefficients, their standard errors, and the F-test and t-test associated with these coefficients as well as other statistics commonly reported in the output of multiple regression programs.