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I think it is especially important to note that linear regression assumes that the relationship between the variables is, well, linear, and that in the real world, it very rarely actually is.

At best, a big asterisk should come from any of these results if you didn't have someone with actual experience validate your design/proposed analyses first.

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You can use the same techniques used in Linear Regression (with multiple features) to do Polynomial Regression. For example suppose you have two features x1 and x2, you can add higher order features like x1x2 or x1^2 or x2^2 or a combination of these. While doing linear regression you treat these terms as individual features so x1x2 is a feature say x3. This way you can fit non-linear data with a non-linear curve. However there is a problem of overfitting, i.e your curve may try to be too greedy and fit the data perfectly, but that's not what you want. So Regularization is used to lower the contributions of the higher order terms.

Wikipedia has an article on Polynomial Regression: http://en.wikipedia.org/wiki/Polynomial_regression

P.S I'm doing this course https://www.coursera.org/course/ml so my knowledge may not be entirely correct so take everything I've said with a pinch of salt. :)

Someone once said that all models are wrong, but some are more useful than others (paraphrasing).

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