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?
Solved – Transforming data with positive, negative, and zero values
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I see no need to answer the general question, as I think your post just arises from a misunderstanding of logarithms.
The logarithm of any positive number to any base is negative when that number is below 1. log10(0.01) is -2, for example. By eye the smallest logarithm on the graph is about -1.7, which would be about 0.02 metres.
I am not impressed by the graph. It is easy enough in decent software to use a log scale and also show values in the original units. As your question shows, that will be clearer to many statistics users.
[LATER] Matt Krause seems very likely to be right about miles, not metres. What meaning is there in saying that agriculture is 20 mm away? I suppose you could be standing at the farm gate....
Determine the smallest number in your time series , say -10. for example . The constant you would need is then 10.0000000001 or larger in order to make all the adjusted values positive. It doesn't make make any difference as the reverse transformation needed to obtain forecasts will use the same adjustment factor.
Please see https://www.ime.usp.br/~abe/lista/pdfm9cJKUmFZp.pdf where Box & Cox suggested an alternative to deal with negative numbers without adding a constant .
EDITED after OP's question re logs ...
From the above reference ...perhaps this helps ... ..logs are used
Best Answer
Yes, you can add a constant and then take a logs.
There are many ways to transform data.
There is nothing inherently invalid about doing this, but very often such transformations are misguided. It is not necessary for the dependent variable to be normally distributed. The assumption about normality concerns the residuals, not the response variable itself. If the residuals are not plausibly normally distributed then of course some transformation may be warranted.
One major downside of such transformations is that it makes sensible model interpretation much more difficult.