lognormal distributionmeanmedian
My data is non-normal, I want to show my raw data, in a scientific journal, by median +/- mad, to show the true nature of the data.
However, if log-transformed, the data is normal. Can I then indicate significance based on calculation on log-transformed data? Which results in median+/-mad presented along with results from parametric tests.
You can report a confidence interval for the median. In R, you can use wilcox.test with the argument conf.int=TRUE. There's a tiny discussion of this in John Verzani's simpleR notes: see here.
Regarding plots: I don't really like using bar plots even for representing a set of means. I'd prefer to just plot little line segments for the CI:
The plot on the right was made with errbar() from the Hmisc package [CRAN page].
You could make the same sort of plot for the medians and the related confidence intervals, or you could use box plots (which, in the same amount of space, describe the entire distribution).
Apparently the concept of unbiasedness has already been discussed a long time ago. I feel it is a topic worth of dicussion as mean-unbiasedness is a standard requirement for a good estimator but for small sample it does not mean as much as in large sample estimations.
I post these two references as an answer to my second question in the post.
Brown, George W. "On Small-Sample Estimation." The Annals of Mathematical Statistics, vol. 18, no. 4 (Dec., 1947), pp. 582–585. JSTOR 2236236.
Lehmann, E. L. "A General Concept of Unbiasedness" The Annals of Mathematical Statistics, vol. 22, no. 4 (Dec., 1951), pp. 587–592. JSTOR 2236928
Best Answer
My rule of thumb is that if you do a statistical test on a transformation of the data that you will plot, then you should plot that transformation of the data.
Ideally the transformation should be motivated by the data type; for example, suppose you are looking cell counts in a Petri dish. Since these grow exponentially (at least until they hit the limit allowed by the dish), a log transformation is well justified, both scientifically and in terms of making the data look more normal. In this case, the log transformed data better answers the scientific question of interest and eases the statistical methodology, so it's clearly the form of the data the reader should be interested in.
In the real world, sometimes log transformations are used almost entirely because doing so simplifies the analysis (e.g. taming outliers, etc). In this case, I would still plot the transformed data: this is the form of the data for which you are answering questions about (i.e. "are the means of the log-transformed data different?") and as such, the reader should be most interested in this form of the data.