I am a running a regression with variables that are transformed by the inverse hyperbolic sine and take and interaction of the effects. Here I present it with some random effect sizes.

$\operatorname{asinh}(y) = b_0 + 0.2 \operatorname{asinh}(x_1) + 2 \operatorname{asinh}(x_2) + 1.5 \operatorname{asinh}(x_1)*\operatorname{asinh}(x_2)$

Say I am interested mainly in $x_1$, and want to see the impact of the continuous variable $x_2$ on the effect of $x_1$ on $y$. How can I interpret this saying more than "for a given level of $x_1$, a higher level of $x_2$ increases the effect of $x_1$ on $y$"?

Can I interpret the not back-transformed coefficients directly saying something like "for a given level of $x_1$, a 1% increase in $x_2$ increases the effect of $x_1$ on $y$ increases by 1.5%" or am I making a mistake here?

Thank you!


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