I would like to research the influence of two independent variables, which together determine the curvilinear relationship with one dependent variable (meaning certain constellations of the two independent variables achieve a certain outcome). Which options do I have to approach this problem? The dependent variable is adjustment to the host-country. The two independent variables are cultural difference as measured with the indices of the GLOBE study and perceived cultural difference (as indicated by the participants in the study on a five-point likert scale). Cultural difference is assumed to be expected by the participant prior to the stay abroad, while perceived cultural difference is what he experiences during the stay. The delta between his expectations about the difference to the host-country and his actual experiences is what determines the relationship with adjustment.

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    $\begingroup$ The best option is definitely to give more details on your problem. Show us some data. Explain what the three variables are. What is your goal? Interpretability? Predictive accuracy? $\endgroup$ – DeltaIV Jun 3 '17 at 8:06
  • $\begingroup$ These details are very important, especially the fact that all your variables seem to be ordinal You should include all this stuff in the question, instead than in a comment. Also, consider adding the ordinal-data tag and/or the likert tag. If "adjustment to host country" is also measured on a Likert scale, then this is a case for ordinal regression. Search the site for "ordinal regression" and you'll find hundreds of useful answers. $\endgroup$ – DeltaIV Jun 3 '17 at 8:51

The nonlinear regression you want to perform can be seen in a supervised learning way

  • starting with the nonlinear model definition $ f(\vec x; \vec \theta) $ (it’s a prior choice) which in your case is $ f : \mathbb{R}^{2} \times \Theta \rightarrow \mathbb{R} $ whose parameters $ \theta $ you want to fit using a Training Set $ D = \{(x_{0}, x_{1}, y)_{i}\}_{i=1,...,N} $ of $ N $ samples
  • defining a cost function $ C(D; \theta) $ e.g. the squared sum of residuals
  • cast the parameters fitting as minimization problem $ \min_{\theta \in \Theta}C(D; \theta) $
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