# Design of Experiments with existing data

#### lorenzovonmt

##### New Member
Ok, I see what you mean. What is the reasoning behind including a squared X1 term in the regression?

#### katxt

##### Active Member
Regression assumes that all the relationships are straight lines. Equal increments to a variable produce equal increments to the response. However, as we increase X1, at first Y goes down until X1 is about 100 and after that Y starts to increase. The X1sq term make the straight line into a parabola and allows for the curve.

#### lorenzovonmt

##### New Member
Thanks for your insight. Would that be considered a non-linear regression then?

#### katxt

##### Active Member
Oddly, no. It is still technically a linear regression. See Spunky's recent post at http://www.talkstats.com/threads/linear-vs-non-linear-functions.77498/#post-229519
You have four columns of predictor variables (including X1sq). The response Y = constant + A.variable 1 +B.variable 2 +C.variable 3 +D.variable 4 It is this pattern that makes it a linear regression. The regression finds A, B, C and D.
Y =A.(variable 1 +B.variable 2)/variable 3 would be non linear.

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#### lorenzovonmt

##### New Member
I see. How would you interpret the coefficient of X1sq though?
Also, do you happen to have examples of this technique used in practice either in a paper or textbook off the top of your head? If not, don't worry about it.

#### katxt

##### Active Member
The coefficients of X1 and X1sq together define the parabola. Between them you can determine the X1 value that gives minimum (or maximum) Y. I have no idea what Y is in your experiments but if it something that is good when it is low (like Y is cost or curing time or resistance) then your experiments have found the optimum X1 value to minimize Y. Y is a minimum when X1 = -coeff of X/coeff of X1sq/2.
Google curve straightening or linearizing curves. It is a very common thing to do in physics and engineering.

#### lorenzovonmt

##### New Member
Yes, Y is a response to be minimized. Thank you for all your help
I'll go ahead and do some reading on curve straightening and linearizing curves.