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I am using a replication dataset for a research, and one variable (GDP per capita) is included in the dataset only as a natural logarithm. Is it possible to transform the ln(GDP per capita) back into the original GDP per capita through SPSS? And if I use ln(GDP per capita) for a linear regression, does that change my results?

Thank you in advance!

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    $\begingroup$ Apparently SPSS has an exp function which will undo ln $\endgroup$
    – Henry
    Commented Mar 6, 2021 at 20:34
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    $\begingroup$ When you transform variables in a regression, and that transformation is not a simple change in the units of measure, you can expect the results to change. But how they change and how to interpret it depends on how the variable appears in your procedure (whether it is an explanatory "independent" variable or the response "dependent" variable). You ought to provide more specific information about your situation. $\endgroup$
    – whuber ♦
    Commented Mar 6, 2021 at 20:56
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    $\begingroup$ @whuber nice catch, the answer I wrote assumes the $log(GDP)$ is the response but it might not be. I'll add a note to answer that I'm assuming the $log(GDP)$ is the response. $\endgroup$
    – Lucas Roberts
    Commented Mar 6, 2021 at 23:31

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First I'll answer this part:

And if I use ln(GDP per capita) for a linear regression, does that change my results?

I am assuming in my answer that the $ln(GDP per capita)$ is the response variable or target, denoted $y_i$.

Transforming the numeric values of the $y_i$s will change the results of your analysis and interpretation.

Caveat: Unless all $y_i$ values are the same value and are the fixed point of the logarithm, your regression will be changed. Note that it's highly unlikely that real data would all be the same value in the $y_i$.

The more subtle aspect of the change is that the logarithm transformation is non-linear. Given that linear regression has a linearity (not the same as affine) baked into the Gauss-Markov theorem you cannot simply transform the results via anti-logs (e.g. $exp(\cdot)$ ) and still claim that you are modeling the expected value. The actual quantity you will be modeling after exponentiation of predictions would depend on the distribution you are assuming for the underlying data.

And now this one:

Is it possible to transform the ln(GDP per capita) back into the original GDP per capita through SPSS?

As mentioned in a comment to the original post, SPSS does seem to support the antilog or exponential function as it is more commonly referred. Be careful that both log and exponential are to the same base when doing your calculations.

Also, while the question ostensibly asks about exponential, my guess is the question is more about how the log tranform impacts modeling assumptions. For how this impacts modeling assumptions my comments to first part above apply.

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    $\begingroup$ Jensen’s Inequality implies that mere exponentiation is insufficient. You can convince yourself of that by logging y, calculating the mean of log y, and exponentiating it. You will not recover the mean of y, but something smaller. $\endgroup$
    – dimitriy
    Commented Mar 7, 2021 at 17:21

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