Preferred method for identifying curvilinear effect in multi-variable regression framework Say some previous findings identified a curvilinear effect of X on Y, (specifically that X had a positive effect on Y, and that X^2 had a negative effect). You want to see if the same holds for your entirely different sample (although everything else between studies, constructs/measures, are exactly the same). Neither the previous study nor my study are experimental (so I am not manipulating X, merely observing it).  There is no explicit theoretical reason why a curvilinear effect would occur.
I will give some examples of how I would currently go about seeing if this is true, but I would like some input and peoples opinions on what they believe would be a preferrable method, if any of my suggestions are innapropriate, and of course if their are other alternatives.
Examples:
1) Simply examine the bivariate scatterplot and fit some type of smoothing line to the mean of Y over bins of X (e.g. LOESS). Although confounding could be an issue, if a curvilinear effect exists their will likely be some evidence in its distribution.
2) Examine the partial regression plot or other visualization techniques to identify the effect of X on Y independent of other confounding variables.
3) Use some type of model selection criteria (e.g. BIC), and determine whether a model including X^2 is preferred over a model without X^2
4) Include a model with X^2 and see if X^2 has a statistically significant regression coefficient.
Like I said any other suggestions are welcome as well. 
Edit: In this context, my main concern is to identify if the effect of X on Y is best represented in a similar manner to the previous study. While their could be other substantively interesting points to compare between studies (like the magnitude of the effect of X on Y), this is not my main concern.
 A: It sounds as though you are interested in formal inference and for that method 4 is best.  Add X^2 to a model containing terms you wish to control for and conduct a test to assess the streght of evidence for the quadratic term given the terms in the model.  Note however that "absence of evidence is not evidence of absence" and statistical power will come into play (this is of interest if you fail to reject or the CI contains zero).  You willof course also want to perform diagnostics of model assumptions prior to drawing conclusions.   
Methods 1 and 2 are excellent exploratory tools and I would encourage exploring the relationship in as many meaningful ways as you wish (since you know a priori what formal test you will conduct -- a test of the quadratic term -- this will not lead to data-driven hypothesis testing).  Other methods of exploration include plotting a fitted LOESS smoother or spline to the (possibly partial) relationships, fitting smoothers or parametric fits within subsets of the data (eg using conditioning plots), a 3d scatterplot with a fitted surface (particularly if you include continuous interactions), etc.  These plots will not only help you understand the data better but can also be used as part of a less formal case for/against the quadratic (keeping in mind that humans are excellent at spotting trends in noise).
I'm not sure what model selection methods you refer to in 3, but generally automated model selection and testing do not mix.  If you are referring to using information criteria (AICc, BIC, ...) note that the theory behind these is based on prediction rather than testing.  So, number 4 is the most rigorous way to test the quadratic.
Finally, 2 comments on terminology:


*

*'multivariate' models are those whose response is a matrix, and 'multivariable' models are those with a vector response and multiple terms on the RHS.

*Partial residual plots differ from partial regression plots..
A: Thanks again for the response, and any other responses in the future will be much appreciated. I think I personally prefer using exploratory tools to identify the relationships, especially since the original researcher did not give any real reason why a curvilinear relationship would exist theoretically. Although exploration would identify if the relationship was not properly identified (e.g. if it should have been a cubed polynomial term instead of squared), this seems unlikely given theres no reason why it would have a curve in it to begin with.
The attached image is a plot of several of their finals models, and I have plotted the expected value of Y given their X and X^2 coefficients reported, holding everything else in the models constant. If their results are representative, I should observe similar findings in my sample controlling for other confounders. It also is enlightening just to plot their findings, as I see the squared term dominates in several of the models, and so for all practical purposes it has a negative relationship that reaches the bottom of realistic Y values fairly quickly.

