How does Jack decide whom to go for ??
Research has shown that Jack makes a descion based on many factors and looks is just one of them:
"Stereotypes are the problem. Different people like different things. Plenty of guys like smart or smarter women. Now,... controlling women...that's a different story."
When choosing which media to go in for and how much to spend there are many brand / competitor parameters as well
Which leads us to Econometric modelling
However, we are often interested in testing whether a dependent variable (y) is related to more than one independent variable (e.g. x1, x2, x3).
We could perform regressions based on the following models:
y = ß0 + ß1x1 + e
y = ß0 + ß2x2 + e
y = ß0 + ß3x3 + e
And indeed, this is commonly done. However it is possible that the independent variables could obscure each other's effects. For example, an animal's mass could be a function of both age and diet. The age effect might override the diet effect, leading to a regression for diet which would not appear very interesting.
One possible solution is to perform a regression with one independent variable, and then test whether a second independent variable is related to the residuals from this regression. You continue with a third variable, etc. A problem with this is that you are putting some variables in privileged positions.
A multiple regression allows the simultaneous testing and modelling of multiple independent variables. (Note: multiple regression is still not considered a "multivariate" test because there is only one dependent variable).
The model for a multiple regression takes the form:
y = ß0 + ß1x1 + ß2x2 + ß3x3 + ..... + e
And we wish to estimate the ß0, ß1, ß2, etc. by obtaining
y1 = b0 + b1x1 + b2x2 + b3x3 + .....
The b's are termed the "regression coefficients". Instead of fitting a line to data, we are now fitting a plane (for 2 independent variables), a space (for 3 independent variables), etc.