Effects may both increase or decrease in the long run: A mathematical illustration

It has been argued that small effects observed at a single instance should not be dismissed, because they can be expected to accumulate in the long term. This follows a classic argument by Abelson (1985). However, it has also been argued that such long-term accumulation is entirely speculative, and need not happen. Here this is shown with a simple mathematical formulation. This formulation shows that the Abelson example is in fact maximally biased towards effects accumulating, in a way that real psychological effects are unlikely to ever fulfill. Even small deviations can have large consequences for the long-term effects. Further, it is shown how under simple conditions, long-term effects may be smaller than those observed at a single timepoint. If the theoretical value of an effect lies in the long term, that effect should therefore be examined in the long term, rather than merely speculated about in the short term.