# Units for log reaction times

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:

$y = \log_{b}x \,\,\,\mbox{if}\,\,\, x = b^{y}$

where $x, y, b \in \mathbb{R}$.

Dimensional homogeneity must be preserved under equality; that is, the units of $x$ must be the same as the units of $b^{y}$ and the units of $y$ must be the same as the units of $\log_{b}x$. Thus, $x, y, b$ 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. $\log(0.024 s)$, what we actually mean is $\log(\frac{0.024 s}{1 s})$ which we can also write as $\log(0.024 s/s)$. 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.

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