invertedU

InvertedU (often written inverted-U) denotes an upside-down U-shaped relationship in which the dependent variable increases with the independent variable up to an optimal point and then decreases as the independent variable continues to rise.

One of the most cited instances is the arousal–performance relationship described by the Yerkes–Dodson law: performance improves with arousal to an optimal level, after which further arousal reduces performance. Similar nonmonotonic patterns are observed in neuroscience, for example in the effect of dopamine on prefrontal cortex function, where too little or too much dopamine impairs working memory relative to an optimal intermediate level.

Outside psychology, inverted-U shapes appear in workload management, where moderate cognitive load yields best performance; in economics or marketing, where certain incentives may increase effort up to a point but excessive incentives can backfire; in ecology, where an optimal level of a resource supports maximum growth before crowding reduces it.

Modeling commonly uses quadratic or non-linear regression; a simple representation is f(x) = a - b(x - c)^2, with a the peak value at the optimum x=c. Real data may deviate from symmetry and require more flexible fits, and wide confounding factors may influence the apparent relationship.

Origin and usage: The inverted-U concept has a long history in empirical research; the term emphasizes non-monotonic relationships and the existence of an optimal moderate level of the influencing variable. See also U-shaped curve and nonmonotonic relationship.