If you choose the model in the manner your described, you'll get a linear regression model (linear in parameters, not in the X's). So if you want to have non-linear (e. g. quadratic) X's you have to calculate a new variable with X²-values and make another regression with X² instead of X.
Developing a linear regression model includes a few steps:
1. Make a scatterplot matrix of all X vs. Y to detect the kind of dependencies. If you see a non-linear connection, transform X in a manner that the connection is afterwards (nearly) linear. (Nice candidates to start are log(x), 1/x, x², x³,..)
2. Let the software build your model and take a deep look at the residual plots! Minitab gives you four: qq-plot, residuals vs. fitted, histogram and a run chart. The values should all be near the line in the qq-plot. There shouldn't be a pattern at the residuals vs. fitted plot, the histogram should be bell-shaped and the run chart shouldn't have patterns either.
3. If patterns exist: Go back to the process (or the source of data) and look for systematic influences, perhaps there are factors which were forgotten to track. If there are time-dependencies, a time-series model could be more appropriate than a simple linear regression model. If you have a heteroscedastic residuals vs. fitted plot (looks like a trumpet), the variance of the response Y is not constant. You have to find a transformation for Y there.
4. If patterns are absent: R² should be "good enough" (for production higher than .80). If the linear model is appropriate and despite this R² is poor, the variation among the X's is too high.
To answer your first question: The default is that all X's are modeled linear. A search for better modelling as a standard procedure will perhaps give you the best fitted model from a mathematical viewpoint, but it will be more difficult to interpret it and it could be an over-fitted model. E. g. for 8 X's and the number of possible transformations restricted to 6 the software has to build 6^8=1'679'616 different models and select the best one.
Regards,
Barbara
This also leads to next question, if Std. Dev in regression output (written in MiniTab output as s) can not be fully explained by linear terms, how do we find out, which X is to be made quadratic ?
for the caution of overfit...
