In psychological experiments, it is common to measure reaction time in the response of a subject (an experimental participant). In inferential statistics, reaction times are typically log-transformed because raw reaction times are skewed to the high. This is because there is a lower physical limit to how fast participants can respond, but not an upper one. Many statistical tests assume a normal distribution of the dependent variable, and thus, reaction times are log-transformed to reduce skew.
What happens to the units when we take the logarithm of a reaction time? A reaction time is measured in some unit of time, [T], e.g. seconds. But we cannot take the logarithm of a dimensioned quantity! The logarithm is defined as the inverse of exponentiation:
Dimensional homogeneity must be preserved under equality; that is, the units of must be the same as the units of and the units of must be the same as the units of . Thus, must all be unitless: in particular, the logarithmic function does not admit a dimensioned quantity as an argument, and a log-transformed quantity is unitless. Note that dimensional analysis is essentially type checking, where the types are physical units.
So when we log transform a reaction time, e.g. , what we actually mean is which we can also write as . Since the logarithmic function admits only unitless arguments, we must take the logarithm of a ratio of reaction times. In physical systems, there is often a natural standard reference value to take the ratio to, as for pressure (standard atmospheric pressure), but there isn’t such a natural standard that I know of for reaction times. So one can take the ratio with respect to a unit quantity in the units the reaction time was measured in, as shown above.
Thus, in labeling a plot or table of log-transformed reaction times, it is incorrect to write log RT (s). Instead, one should write log (RT/s) or log (RT/[s]) or maybe log RT (RT in s). We still want to know what units the raw reaction times were measured in, since they scale the log-transformed values!
For an expanded discussion of these topics, see Can One Take the Logarithm or the Sine of a Dimensioned Quantity or a Unit? Dimensional Analysis Involving Transcendental Functions by Chérif F. Matta and Lou Massa and Anna V. Gubskaya and Eva Knoll, to appear in the Journal of Chemical Education.
I noticed when I was trying to display the summary of an lmer object using display() from the arm() package, I was getting this error:
Error in UseMethod("fixef") :
no applicable method for 'fixef' applied to an object of class "mer"