# Solved – log transformation vs square root transformation, Can I do both?

I am analyzing ARIMA or multiple regression model. \$Y= X_1 + X_2 + X_3 + X_4\$

Can I have the log on Y and square root on X1 and square root on X2, at the same time? That is,

\$log Y = sqrt{X_1} + sqrt{X_2} + X_3 + X_4\$

If yes, how can I interpret the results?

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Yes; you can do this and still achieve a valid model.

Interpretation, however, is complicated by this transformation. Your model then looks like

\$\$log(E(Y_i)) = beta_0+ beta_1sqrt{X_{i1}} + beta_2sqrt{X_{i2}} + beta_3X_{i3} + beta_4X_{i4}\$\$

This means that e.g. a doubling of \$X_1\$ is associated with an expected increase of \$Y\$ by a factor \$exp(beta_1sqrt{2})\$, all other factors kept equal. For a unit increase in \$X_1\$ there is no easy interpretation; that depends on the current value of \$X_1\$.

A unit increase of \$X_3\$ is associated with an expected increase of \$Y\$ by a factor \$exp(beta_3)\$. So, interpretation is easier without transforming the variables. However, if square root transformation improves the fit or is necessary in some other way, then it is mathematically perfectly correct.

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