Threshold model

In mathematical or statistical modelling a threshold model is any model where a threshold value, or set of threshold values, is used to distinguish ranges of values where the behaviour predicted by the model varies in some important way. A particularly important instance arises in toxicology, where the model for the effect of a drug may be that there is zero effect for a dose below a critical or threshold value, while an effect of some significance exists above that value.^[1] Certain types of regression model may include threshold effects.^[1]

Classic threshold models were developed by Schelling, Axelrod, and Granovetter to model collective behavior. Schelling attempted to model the dynamics of segregation motivated by individual interactions in America (Schelling, 1971) by constructing two simulation models. Schelling demonstrated that “there is no simple correspondence of individual incentive to collective results,” and that the dynamics of movement influenced patterns of segregation. In doing so Schelling highlighted the significance of “a general theory of ‘tipping’”.

Mark Granovetter, following Schelling, proposed the threshold model (Granovetter & Soong, 1983, 1986, 1988), which assumes that individuals’ behavior depends on the number of other individuals already engaging in that behavior (both Schelling and Granovetter classify their term of “threshold” as behavioral threshold.). He used the threshold model to explain the riot, residential segregation, and the spiral of silence. In the spirit of Granovetter’s threshold model, the “threshold” is “the number or proportion of others who must make one decision before a given actor does so”. It is necessary to emphasize the determinants of threshold. A threshold is different for individuals, and it may be influenced by many factors: social economic status, education, age, personality, etc. Further, Granovetter relates “threshold” with utility one gets from participating collective behavior or not, using the utility function, each individual will calculate his cost and benefit from undertaking an action. And situation may change the cost and benefit of the behavior, so threshold is situation-specific. The distribution of the thresholds determines the outcome of the aggregate behavior (for example, public opinion).

Regression analysis

The models used in segmented regression analysis are threshold models.

Fractals

Certain deterministic recursive multivariate models which include threshold effects have been shown to produce fractal effects.^[2]

Time series analysis

Several classes of nonlinear autoregressive models formulated for time series applications have been threshold models.^[2]

Toxicology

A threshold model used in toxicology posits that anything above a certain dose of a toxin is dangerous, and anything below it safe. This model is usually applied to non-carcinogenic health hazards.

Edward J. Calabrese and Linda A. Baldwin wrote:

The threshold dose-response model is widely viewed as the most dominant model in toxicology.^[3]

An alternative type of model in toxicology is the linear no-threshold model (LNT), while hormesis is a general term covering the type of response there may be to a drug.

References