Hi,

Before making general conclusions on the adequacy of the resulting regression models, I recommend you to review the following two links about R-squared statistical measure.

http://blog.minitab.com/blog/adventures-in-statistics-2/regression-analysis-how-do-i-interpret-r-squared-and-assess-the-goodness-of-fit

http://blog.minitab.com/blog/adventures-in-statistics-2/how-to-interpret-a-regression-model-with-low-r-squared-and-low-p-values

Generally, RSM is based on an assumption of a quadratic surface. Using a higher order curvature is overfitting, but I haven't seen your specific DOE or results. Look carefully at your residuals.

Like Prof. Jonas wrote, you must be careful. RSM quadratic polynomials have a fundamental reason. RSM is supported in multivariate Taylor series expansion, that is, any smooth model may be represented by a linear model at the neighborhood of a point, and by a quadratic polynomial at the neighborhood of the neighborhood. A cubic polynomial may be overfitting. As example if you fit a quadratic polynomial to three points (each one with different independent variable) you will obtain a perfect fit (r2=1), but without prediction capacity, because you fit three parameter to three points. First, I recommend to you to evaluate the significance of parameter of your quadratic polynomial and the significance of parameter of your cubic polynomial. It is possible that cubic polynomial parameters would not be significant. It is more important the parameter significance that r2. R2 is a numerical value, significance is a statistical estimation. Additionally, you must be performed replication in each treatment of your experimental design. May be with replication you can increase the power of test and your model will have significance in their parameters.    

thanks for your Answers , Dr. Miguel the factors were significant in the cubic polynomial even there was  a factor (which is must be significant) wasn't significant in quadratic but it was in cubic 

Be careful Hadeel. I suggest that you must to confirm the results. A cubic model may be the result of an outsider point as the example attached for a single-variable model.

hadeeeeel????

This is an interesting trend of Q&A. I agree with Prof Alvarado in evaluating the significance of prediction capacity. It is important to validate the polynomial equation far out of data range rather than inside the data range. extreme caution must be exercised before generalizing a particular fitting trend. R square value alone cannot really validate the external predictions.

Hi,

I know that when using a central composite design it can be possible to estimate cubic effects, but I never had the occasion to test it.

If you uses cubic effects, check for VIF. If they are high, it means you have too high correlations. I think that you may calculate prediction R². Adding a variable in your model will increase necessarily the R², but if this variable brings only noise, prediction R² will decrease.

R²=(Explained sum of squares)/(total sum of squares) = 1 - (Residual sum of squares)/(Total sum of squares). To obtain prediction R² you have to replace (Residual sum of squares) by PRESS obtained by cross validation (leave one out).

If the cubic effect as an important effect, the lack of fit (LOF) is probably significant.

In conclusion, if the cubic effect increases significantly (typically more than 5-10%) prediction R² keep it in the model. If not, keep the simplest model.

Hope this will help you.

What is your response variable? What transform does the software suggest?

A lot of times, (OK always) a linear term will be highly correlated with cubic term, A vs A^3. If by cubic term you mean a 3-way interaction, ABC, DEF, etc, then you can use these in a RSM. If the R^2(adj) and R^2(pred) are pretty close to each other, then you're probably on to something.    

En diseños de experimentos compuestos, y en general en diseños factoriales, los términos de orden tres, ya sean variables cúbicas o interacciones de orden tres, se consideran parte del error experimental si no se han realizado réplicas. Y también un término cúbico no es un aporte, dado que es difícil de explicar

Of course, cubic term (A^3) is highly correlated with main effect, but not with quadratic (A^2) effect, in coded units, but not too much (VIF = 4.8 for a 3 factors CCD). For 3 factors interactions (ABC), it depends on the resolution of the base factorial design you used. To estimate these effects, the minimum resolution is 7 which means a very expensive DOE in terms of number of experiments if you have more than 4 or 5 factors.

Thank you all for your answers, I think I will stay with the Quadratic model with removing some interaction terms that is insignificant and thanks Dr.

The R^2 is not the defining term. Check the significance (p-values) of each term in the correlation and most probably you are over fitting the data. Yes the RSM can fit both quadratic as well as cubic data but also keep in mind the residuals as you need a minimum number of residuals for making the entire model significant.

The backward method doesn't help as in the quadratic method it eliminates the terms which doesn't have the required alpha. so more or less you will reach the same R^2 in quadratic with and without backward elimination.

thanks for the responses so far. Am going through the same challenge because i have constructed a third order rotatable design and would like to simulate data to fit a cubic model, someone to help