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I have been trying to run a multilevel model with both a linear and a quadratic term for income as my main variables of interest. It looks something like: \begin{eqnarray} &&y_{ij}=\beta_{0j}+\beta_{1j}\text{Income}+\beta_{2j}\text{Income}^2+\epsilon_{ij}\nonumber\\ &&\beta_{0j}=\gamma_{01}\text{Level-2 variables}+u_{0j}\nonumber\\ &&\beta_{1j}=\gamma_{11}\text{Level-2 variables}+u_{1j}\nonumber\\ &&\beta_{2j}=\gamma_{21}\text{Level-2 variables}+u_{2j}\nonumber \end{eqnarray} It has suggested to me by someone that it is useful to center the income variables for interpretation, yet I am not entirely sure how to do that when a quadratic term if present. I tried to first demean the linear term and then square it separately for the positive and negative part, but this would fix the tipping point at 0, which is not something I want. Is there a better way of doing this? Or should I give up on centering at all?

Thanks JY

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    $\begingroup$ Related questions might provide a little insight (but do not directly address multilevel models): stats.stackexchange.com/questions/47178, stats.stackexchange.com/questions/34488. $\endgroup$
    – whuber ♦
    Commented Aug 15, 2013 at 19:05
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    $\begingroup$ I think the centering makes the interpretation more difficult. You can compute differences in mean for any two income levels, and get confidence limits for same. $\endgroup$
    – Frank Harrell
    Commented Aug 15, 2013 at 19:47

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