What does the term X1*X1 mean in an RSM design?

Question

Hello,

When I'm using DOE and I click 2nd order interactions, the model will add combinations of every interaction of my factors, e.g. X1*X2, X2*X3 and X1*X3. This makes sense to me because these terms tell me about the combined effect of two factors on my response. However, if I click the RSM button it will add quadratic terms such as X1*X1, X2*X2 and X3*X3. My question is, what physical meaning do these terms have? What do they imply for my response? Let's say one of my factors is time, does this then mean that time^2 might have an effect on my response while time alone won't?

I'm trying to grasp the concept of these quadratic terms with respect to the interpretation of possible results.

Many thanks in advance.

statman · Answer

Adding to MRB, think of the quadratic term as evidence of a departure from the linear relationship.

Linear

The quadratic term introduces curvature to the relationship between the predictor variable and the response variable.

Quadratic

It allows the model to capture non-linear relationships that may not be adequately represented by linear terms alone. The coefficient associated with the quadratic term indicates the direction and magnitude of the curvature. A positive coefficient implies an upward-opening curve, while a negative coefficient implies a downward-opening curve. The presence of a quadratic term may affect the interpretation of the associated linear term. The linear coefficient represents the slope of the relationship while the quadratic coefficient provides insight into the departure from linear.

Always plot your data.

MRB3855 · Answer

Hi @MannSamplesCrow : Your question "Let's say one of my factors is time, does this then mean that time^2 might have an effect on my response while time alone won't?". Yes, with some caveats.  There is an hierarchy; if X^2 is significant, then we leave X in the model as well, even if it has a high p-value. The squared term(s) basically allows for curved (quadratic)  response wrt to that factor...over and above the linear term alone.