Weber-Fechner law
In the early 19th century, psychologist Ernst Weber
noted that in order for people to notice a given stimulus, the amount of
the stimulus must increase (or decrease) by a fraction of its
physical intensity to yield “just noticeable differences”
(jnd) .19,21 For example, Weber found that people
could not discriminate between 20.5 and 20.0 g weights but could usually
discriminate between 21 and 20 g.20 When baseline
weight was 40, 60, 80, and 100 g, the required increased in stimulus to
represent jnd was 42, 63, 84, and 105 g,
respectively.20 That is, to appreciate the differences
between weights (jnd), the weight (i.e., stimulus) should
increase by at least 5% of the original weight.20Gustav Fechner, another 19th century psychologist,
proposed that “jnd” can be conceptualized as units of
psychological intensity instead of physical
intensity.19-21 Subsequently, the relationship between
the intensity of a signal and how much more intense the signal needs to
increase before a person can reliably tell that the signal has
truthfully changed has become known as the Weber-Fechner Law of
psychophysics. As expected for physiological systems, the law is valid
within certain domains of stimulus-response ratios. In other words, it
is approximately correct for a wide variety of sensory dimensions,
although it may deviate at the extremes of the spectrum of stimuli.20,21 Other authors have tried to improve upon the
Weber-Fechner Law. Notably, Stevens27 proposed a power
law according to which a relationship between stimulus intensity and the
magnitude of sensation can be plotted on a log-log axis as a straight
line with a slope of the exponent.
Regardless of the exact mathematical description of the relationship
between “jnd” and stimulus, the reproducible relationships
between signal and perception was subsequently documented in other
fields as well 28: from influencing human behaviors by
specific marketing stimuli29 to the way people
experience the value of money19 to making risky
choices 30 to the mental line for
numbers.22 In addition, psychological research31 has demonstrated that people often use a simple
heuristic in decision-making, based on the prominent numbers as
powers of 10- defined as the powers of ten, their doubles, and their
halves [e.g., 1,2, 5, 10, 20, 50, 100, 200…] that approximate
the Weber-Fechner function. 31 32For example, when presented with monetary choices, people often make
judgments according to the “1/10 aspiration level” (rounded to the
closest number) in such a way that if the gains, losses, or
probabilities change by one order of magnitude (or more), they will stop
further examination of the observed results and accept the
findings.31 Thus, the most common heuristics defined
in the literature to categorize “dramatic” effects as
RR>2 8, RR ≥517, or
RR≥109 appear to be consistent with the Weber-Fechner
law. However, a fundamental property of the Weber-Fechner law is that
“jnd” occurs only when the increase or change in stimuli are aconstant percentage of the stimulus itself19,21; this is also directly applicable to evaluation
of treatment effects. Indeed, treatments effects are commonly assumed to
remain constant over a range of predicted risks,33providing further justification for the application of Weber-Fechner law
to assess the likelihood of approval of new therapeutics without testing
them in RCTs. Appendix 1 demonstrates how a stimulus (effect size) and
response [probability of not requiring further RCTs when\(jnd=\log{(OR)}]\) is derived from the Weber-Fechner law as:
\begin{equation}
\text{logit}\left(p\left(\text{no}n_{\text{RCT}}\right)\right)=A*\log\left(\text{OR}\right)+B\nonumber \\
\end{equation}where OR is odds ratio, A and B are fitted constants, respectively.
It is important to note that the response – i.e., not requiring further
testing in RCTs – is not linearly related to the size of treatment
effect (i.e., OR), but rather to the logarithm of the effect size
[log(OR)].