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We are wondering whether it would be possible to use the same variable twice in a model: Once dichotomized, once continuous. The variable are questionnaire scores and participants are pre-screened to fall into extreme groups (so the middle is missing). The idea is that the two groups are qualitatively different from each other and do not represent a linear trend. To be precise, we expect a negative relationship with the DV in one group, but not in the other.

So we could model it using some non-linear model, but it might be difficult to interpret. If we use only the categorical variables we could compare these different groups well, but we lose all the variance in the predictor/questionnaire. So we were wondering if it is possible to get separate regression lines for each group by simply including the interaction of the two? It does feel very wrong to use the same variable twice. Also, it might be the same as using a quadratic predictor, maybe?

So the model would be DV = IVgrouped * IVcont.

Any thoughts? Thanks!

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    $\begingroup$ A nonlinear model (linear or cubic spline for example) will be easier to interpret than what you suggested, I think. $\endgroup$ – Frank Harrell Jul 18 '17 at 16:21
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Of course, your software will let you do this. And as long as there are (approximately) linear trends within each of these groups, you would also get reasonable predictions from the model (at least within the range where you have a decent number of observed values). However, you would have some serious problems to interpret the coefficients without looking at the predicted values for different values, because the coefficients for the linear variable in the interaction would be relative to the zero value of the variable and the coefficients of the main and interaction effects are correlated (which particularly matters for confidence/prediction intervals). You may also get funny discontinuities in the piece-wise linear "curve" you use at the ends of each group.

If you need to look at the predicted values anyway, perhaps it becomes preferrable to use e.g. splines. Almost all standard statistical software will quite easily do this for you (and often also give you useful plots of the non-linear relationship between expected response and continuous covariate). I only specifically mention splines, because they are a commonly used approach and because, If you want, you could then force the knots to be on the boundaries of the groups you have a-priori identified (similar to the idea you had).

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