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)].