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Multiple Regression v. Canonical Correlation


Consider the standard linear model for multiple regression, in which a single variable (Y) is a function of multiple independent variables (xi):
Y = b1*x1 + b2*x2 + ... + bqxq

Suppose that the dependent variable, Y, is an attitudinal variable such as "Satisfaction with Life."
Why are some people are more satisfied with their lives than others?
You theorize that one's happiness depends on one's socioeconomic status, usually measured by
occupational prestige

Normally, these three variables do not incorrelate sufficiently to form a reliable scale.
So you treat them as independent variables predicting to Life Satisfaction in a regression equation.

Research, however, shows that "satisfied" people often lead "dull" not "exciting" lives.

You could treat "Excitement in Life" as a dependent variable and run another regression equation.
Then you would compare the equations and the coefficients.
Alternatively, you could seek to explain "Happiness in Life" measured by both variables:

This reconceptualization of the research fits canonical analysis, which computes the maximum correlation between two sets of variables:

a1*y1 + a2*y2 + ... + apyp = b1*x1 + b2*x2 + ... + bqxq