Hi Janet,

That's a very good question, and a complete answer could be a math course all by itself. But let's try to get some ideas. I suggest that you start by searching the web for "linear regression", and check some of the references -- such as,

In particular, search the articles for "polynomial regression". I think you will find that polynomial regression, such as

y = b_{0} + b_{1}*x + b_{2}*x^{2}

is considered to be "linear regression" because "y" is a linear function of all 3 predictor variables -- 1, x, and x^{2}. (The "1" comes from b_{0}; that is we can think of b_{0} as b_{0}*x^{0}. x^{0}, and anything to the zero power, is always 1.) Such a polynomial could have a u-shape over a range of x, and could fit the data quite well.

But if we assume we can only use "simple linear regression" (that is, an intercept and one term), then we can only have a model of the form

y = b_{0} + b_{1}*x

[although technically x could be to any power, or even something like e^{x} or log(x); the restriction is we can only have one term using x]. Such a model could exhibit a u-shape, but generally, because we are restricted to a single x-term, it will not fit the data very well. For example, the equation y = b_{0} + b_{1}*x, will not fit u-shaped data very well. Even though the calculations will give an answer that crosses through the u-shaped data, it will fit poorly.

Therefore, to improve the fit, you must use "multiple linear regression" (such as polynomial regression) rather than "simple linear regression". And just as important is the fact that when using regression, never blindly fit a model to your data -- the math will always give an answer, even a bad answer! You must inspect the results, such as the scatter-plot you mention, to see if the model is appropriate.

I think you can find some other ideas in the articles you find with the web search.

I hope this helps you. Best wishes.

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