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How do you interpret an odds ratio of an interaction between two continuous variables $x_1, x_2$? Would you fix $x_2$ and let $x_1 = x$ and $x_1 = x+1$?

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Whether it's an odds ratio, or a survival analysis, or a typical continuous response variable, you don't directly interpret interaction terms. When all covariates are categorical (i.e., in an ANOVA), you interpret 'simple effects'. That is, the effect of a factor on the dependent variable at each level of the other factor with which it interacts. When you have two continuous covariates, as in your case, this is a little more subtle (in that there aren't specific levels of the other covariate that are obviously the 'right' levels to condition on). You still need to assess the association between the odds ratio and one of your covariates at pre-specified levels of the other covariate, so if there are some specific values that are salient, or particularly meaningful for some reason, you could use those. More commonly however, you assess the relationship at the mean and +/- 1 SD of the other covariate, and interpret that.

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  • $\begingroup$ Does it matter what covariate you fix? $\endgroup$ – ross Feb 1 '12 at 0:14
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    $\begingroup$ Fix whichever one makes more sense to you to fix. $\endgroup$ – gung Feb 1 '12 at 1:52
  • $\begingroup$ (+1) You've made an important point that you can't interpret the value of the interaction term in isolation - only in the context of the 'main effects'. It's also relevant to point out that one may interpret interactive effects as capturing non-linear effects of the predictors, similar to how, for example, polynomial terms in a regression model act. $\endgroup$ – Macro Feb 1 '12 at 2:33
  • $\begingroup$ @Macro: For a multiple regression model with higher order terms would it make sense to take the partial derivative to interpret the coefficients? $\endgroup$ – ross Feb 1 '12 at 2:42
  • $\begingroup$ The partial derivative of...? It is true that, you think of the regression function as a taylor approximation of the "true" objective function (see my answer to stats.stackexchange.com/questions/21271/…), the coefficients of second order terms are estimates of the partial derivative of the true objective function, but I'm not sure what you mean. $\endgroup$ – Macro Feb 1 '12 at 2:54

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