# Transforming data with positive, negative, and zero values

I have a multiple linear regression model with several dependent variables that have positive, negative, and zero values, and are not normally distributed. I can't do a natural log transformation because of the 0 and negative values, can't square or cube it due to 0 values, and the Box-Cox transformation works only for positive and 0 values. Is there a transformation I can do that works for all of these? I've seen log(x+minimum value) as one option, but not so much here on this forum—is this a valid transformation?

• You can certainly do a log transformation by adding a constant, but why do you want to transform these data ? – Robert Long Feb 1 '19 at 15:32
• Would something like sign(x) * log(1 + abs(x)) work? It's a one-to-one transformation that has a log effect on both positive and negative values. It doesn't have issues handling x=0 either. – jjet Feb 1 '19 at 15:41
• @RobertLong some of the variables are positively skewed. – user10831611 Feb 1 '19 at 15:42
• Squares and cubes of zero are perfectly well defined; the problem is different with squares, namely that $-x$ and $x$ produce the same square so the transformation is not one-to-one. Cubes are often a bad idea because they will typically increase (the magnitude of) skewness and exaggerate outliers. – Nick Cox Feb 1 '19 at 15:53
• Concerning your actual problem of multiple regression, please investigate the threads on this site related to transformation of variables in multiple regression: there appear to be hundreds of them. You might begin by reviewing the highest voted and answered questions. – whuber Feb 1 '19 at 15:58