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How would I interpret a transformed dependent variable (4th root) with some of its predictor variables transformed as well? In our study, we transformed our dependent variable to 4th root, $Y^\frac{1}{4}$, and one of our predictor variables to $\ln(X)$. How would I interpret that?

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marked as duplicate by whuber May 22 '14 at 15:44

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For the sake of comparison, in the case of a simple linear regression model with no transformations, i.e. $Y = \beta_{0}+\beta_{1}X$, our interpretation is $$\textit{a one unit increase in X results in an increase of Y by $\beta_{1}$ units}$$ Now, if we are modeling $Y = \beta_{0} + \beta_{1}ln(X)$, our interpretation is $$ \textit{a 1% increase in X results in an increase of Y by $\frac{\beta_{1}}{100}$ units} $$ Unfortunately, in the case of most power transformations, there is not really an intuitive interpretation (as far as I know) of the effect, such as % change with logarithms. So, for $Y^{.25} = \beta_{0}+\beta_{1}ln(X)$, I would interpret this as $$ \textit{a 1% increase in X results in an increase of $Y^{.25}$ by $\frac{\beta_{1}}{100}$ units} $$ If you have more than one dependent variable in your model, these interpretations still apply, but in the context of holding the other dependent variables fixed (i.e. examining the unique impact of $X_{1}$ on $Y$).

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