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I am using linear and generalised linear models, and have transformed my explanatory variables using $log10(\bullet)$ and $sqrt(\bullet)$ transformations, and my response variable using an arcsine square root transformation ($\arcsin(\mathrm{sgn}(x)\sqrt{|x|})$ as I had negative values in $x$). For the latter, the justification is to get the data normally distributed.

What is the justification (or point!) of transforming explanatory variables, as they do not need to be normally distributed?

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The idea is that by transforming old variables into new variables, you might get the data plotted in a more easily recognizable pattern, so that you can analyze the new variables more effectively than you can analyze the old variables. For instance, let's say you have plotted data for two variables X and Y, but the data doesn't suggest a linear relationship between X and Y. If you do a log transformation on one of the variables, say Y, and get a strong linear relationship, then you might conclude that log Y is probably a linear function of X. This implies that Y is exponentially related to X. Also, you are probably familiar with statistical analysis of linear models, which you can exploit if you transform Y using a log transformation. If you had not transformed the variable, and all you had was techniques of linear model analysis, you wouldn't have been able to detect that Y might roughly be an exponential function of X. The same reasoning goes for transforming variables to get a normally distributed data set: you are probably familiar with properties of normal distributions, and by analyzing the transformed variables that are normally distributed, you can learn more about the original variables than you could just by plotting the original variables themselves. So basically you are uncovering more information about the original variables X and Y indirectly.

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