Backtransforming log(x+1) transformed SE I am back-transforming y=log(x+1) data that represent individuals/m^2 of area. Data are in the form of X ± SE. I back-transformed the means, but I am wondering if it's possible to directly back-transform the standard errors and, if so, how? I need to back-transform these data because I am comparing to non-transformed data. Obtaining the raw data is not possible.
 A: In general, if you use a smooth nonlinear transformation $f(x)$ to transform a random variable $x$, its standard deviation will be approximately scaled (multiplied) by a factor $\phi=\left. \frac{\partial f}{\partial x} \right|_{x=\bar x}$, i.e. $\sigma(f(x)) = \phi \cdot \sigma(x)$, or $\sigma(x) = \sigma(f(x))/\phi$ (you can derive this using the first term in a Taylor expansion). In this case $\partial[\log(x+1)]/\partial x = 1/x$. So you can back-transform the value you have to get $x$, then multiply the standard error you have by that $x$ value.
This is a rather crude approach, but I think it's what I would recommend in the absence of any more information.
I might actually recommend that you transform your other (non-log-transformed) values in the other direction, to work on the log scale; if you are implicitly assuming Normally distributed values in your analysis, that's more likely to be true for abundances on the log scale than on the original (abundance/m$^2$) scale.
PS.  If you are using $\log_{10}$ rather than  natural log in your transformation, you should be aware that $\partial \log_{10}(x)/\partial x$ is $1/(\ln(10) \cdot x)$ (where $\ln$ is the natural logarithm; same as $\log$ above).
