John Hayes writes:

I am a fan of the quarter root transform ever since reading about it on your blog. However, today my student and I hit a wall that I’m hoping you might have some insight on.

By training, I am a psychophysicist (think SS Stevens), and people in my field often log transform data prior to analysis. However, this data frequently contains zeros, so I’ve tried using quarter root transforms to get around this. But until today, I had never tried to back transform the plot axis for readability. I assumed this would be straightforward – alas it is not.

Specifically, we quarter root transformed our data, performed an ANOVA, got what we thought was a reasonable effect, and then plotted the data. So far so good. However, the LS means in question are below 1, meaning that raising them to the 4th power just makes them smaller, and uninterpretable in the original metric.

Do you have any thoughts or insights you might share?

My reply:

I don’t see the problem with predicted values less than 1, but you can get negative values, which is kind of weird. I remember we had similar issues when using the square root power for missing-data imputation. For some examples (such as imputation of income) the square-root power worked well, but because of the negative values we couldn’t just throw it into our imputation program as a default.

Another approach would be to think more seriously about what those zeros really imply, and perhaps use some latent-variable model.