On this page, I read about multiple ways to deal with multicollinearity. I determined that the last method suggested may be best for my purposes. I copied it below:
Treat the common variance as a separate variable and decontaminate each covariate by regressing them on the others and using the residuals. That is, analyze the common variance as a separate variable.
I do not know how to do it in SPSS or R. If someone could explain how to do it in SPSS or R, I'd appreciate it!
One approach that may be what the quote is talking about is to orthogonalize the covariates. You keep $x_1$ as it is then fit a regression model with $x_2$ as the response and $x1$ as the predictor variable and store the results, call this $x_2^*$, then fit another regression model with $x_3$ as the response and $x_1$ and $x_2^*$ as the predictors and store the residuals from this model, call this variable $x_3^*$, continue until you have done this with all the variables. These new variables will be orthogonal and not have any covariance problems, use them as the predictors in your final regression with $y$ as the response.
Note that the order of the $x$ variables become very important in this proceedure. The slope on $x_1$ is the effect of $x_1$ as if it had been the only thing in the model, the slope of $x_2^*$ is the effect of $x_2$ adjusting for $x_1$ but not any of the other variables, etc.
If you change the order of the $x$'s in the process then the interpretations will change as well.
If that is not what is being referred to (it sounds right based on the concept of decontaminate using regression and residuals, but I don't know what they mean by common variance) then we will need more detail, background to be able to help.
