 With simple linear regression we saw how we could use a straight line as an approximate model of our data (R^2 tells us just how approximate it is). We then saw that we could use the equation that describes this straight line to explore the relationship between the two variables in a few different ways. In particular, we could use it to predict the scores associated with new cases in the future.

Multiple regression is just the same, except instead of using a one-dimensional line to describe the data, we use an n-dimensional hyperplane (lawks!). Simple linear regression placed a one-dimensional line in a two-dimensional space (these two dimensions are of course the ones represented on a graph as x and y). Multiple regression takes this further by placing an n-dimensional hyperplane in an (n+1)-dimensional space. So linear regression is just the simplest possible form of multiple regression, where n = 1.

Recall that the straight-line formula we used was:

y = ax + b

This is saying that y, the dependent variable, is reached by multiplying x, the observed variable, by a certain number, a, and then adding another number, b.

If we want to take this further and predict y from more than one independent variable, then we need a different regression coefficient for each. Let's say we want to predict y from two other variables, x1 and x2, we would have two different coefficients, b1 and b2. Our regression formula would therefore be:

y = a + b1x1 + b2x2

and would therefore describe a two-dimensional slice of three-dimensional space. (These three dimensions can still just about be plotted on a graph, but beyond this point you can no longer easily visualize what is going on.) More x's can be added to the equation - as many as we want, in fact. So you could quite easily have a situation where y is predicted from five other variables, giving this equation:

y = a + b1x1 + b2x2 + b3x3 + b4x4 + b5x5

As with simple regression, the computer will calculate the a and b values for you. However, there are various ways in which they can be added to the equation. All of these are accessed from the analyze > regression > linear menu.