# What is the statistical logic behind using logarithms in analyzing threshold elevations?

In my field, a lot of experiments are done comparing performance between a baseline condition, and a more difficult condition. Typically, thresholds are measured at each of these two conditions, and a threshold elevation is calculated.

So, suppose for a particular observer, the contrast detection threshold (e.g. the luminance of a circular patch on perfectly black background at which an observer is able to detect whether it is present or absent 75% of the time) is found to be 0.001 cd/m^2. And suppose that in a more challenging condition, where before each trial, a bright flash occurs, that same observer's threshold rises to 0.01 cd/m^2, meaning that more light is required for successful detection.

The threshold elevation can be expressed in at least two ways:

1) as a ratio between the baseline and more challenging condition: in this case it would be 0.01/0.001 = 10.

2) as a difference of the logs, so log(0.01)-log(0.001) = (-2) - (-3) = 1.

From what I have been told, the advantage of using the log transform is that it stabilizes the variances. So, if you had several observers performing this experiment, there would be a certain spread of values for each of the two conditions. If you were to use the ratio approach (as in (1) above), then the variances also get multiplied. But with a log transform, the variances do not grow, since the two conditions are added, rather than multiplied.

Am I on the right track with my interpretation? I'm still not comfortable with this yet, for the following reasons:

1) so what if the variances are multiplied - why does this threaten the integrity of any statistical calculations done later (such as whether the threshold elevation is significantly different from a value of "1")?

2) Doesn't doing statistics on a log transform somehow mean that you're not actually doing statistics on the underlying concept of the threshold elevation (which, to me, seems to be more of a ratio concept)? Or have I got it backwards, and is a log space actually more representative of the logarithmic nature of perception?

I'd really appreciate any guidance here - I've tried looking for resources that explain, in a textbook fashion, the logic behind this approach, but I'm coming up short.

thanks.