Threshold model
Encyclopedia
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 differs 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. Certain types of regression model may include threshold effects.
Classic threshold models were developed by Schelling, Axelrod, and Granovetter to model the collective behaviors. Schelling,T.C. tried to model the dynamics of segregation by ‘color’ in the America motivated by individual interactions (Schelling, 1971), by constructing two simulation models, Schelling succeeded to demonstrate that “there is no simple correspondence of individual incentive to collective results”, the dynamics of movement results in the pattern of segregation. And Schelling highlights the significance of “a general theory of ‘tipping’”.
Mark Granovetter follows Schelling’s steps and proposes the threshold model (Granovetter & Soong, 1983, 1986, 1988), which assumes that individuals’ behavior depends on the number of people already engaging in that behavior (Both Schelling and Granovetter classify their term of “threshold” as behavioral threshold.). He uses the threshold model to explain the riot, the residential segregation and the spiral of silence, etc. In the spirit of Granovetter’s threshold model, the “threshold” means that “the number or proportion of others who must make one decision before a given actor does so”.
It’s necessary to emphasize the determinants of threshold. Threshold is different from people, and it’s 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, the public opinion).
effects.
applications have been threshold models.
posits that anything above a certain dose of a toxin
is dangerous, and anything below it safe. This model is usually applied to non-carcinogen
ic health hazards.
Edward J. Calabrese and Linda A. Baldwin wrote:
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.
Classic threshold models were developed by Schelling, Axelrod, and Granovetter to model the collective behaviors. Schelling,T.C. tried to model the dynamics of segregation by ‘color’ in the America motivated by individual interactions (Schelling, 1971), by constructing two simulation models, Schelling succeeded to demonstrate that “there is no simple correspondence of individual incentive to collective results”, the dynamics of movement results in the pattern of segregation. And Schelling highlights the significance of “a general theory of ‘tipping’”.
Mark Granovetter follows Schelling’s steps and proposes the threshold model (Granovetter & Soong, 1983, 1986, 1988), which assumes that individuals’ behavior depends on the number of people already engaging in that behavior (Both Schelling and Granovetter classify their term of “threshold” as behavioral threshold.). He uses the threshold model to explain the riot, the residential segregation and the spiral of silence, etc. In the spirit of Granovetter’s threshold model, the “threshold” means that “the number or proportion of others who must make one decision before a given actor does so”.
It’s necessary to emphasize the determinants of threshold. Threshold is different from people, and it’s 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, the public opinion).
Fractals
Certain deterministic recursive multivariate models which include threshold effects have been shown to produce fractalFractal
A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
effects.
Time series analysis
Several classes of nonlinear autoregressive models formulated for time seriesTime series
In statistics, signal processing, econometrics and mathematical finance, a time series is a sequence of data points, measured typically at successive times spaced at uniform time intervals. Examples of time series are the daily closing value of the Dow Jones index or the annual flow volume of the...
applications have been threshold models.
Toxicology
A threshold model used in toxicologyToxicology
Toxicology is a branch of biology, chemistry, and medicine concerned with the study of the adverse effects of chemicals on living organisms...
posits that anything above a certain dose of a toxin
Toxin
A toxin is a poisonous substance produced within living cells or organisms; man-made substances created by artificial processes are thus excluded...
is dangerous, and anything below it safe. This model is usually applied to non-carcinogen
Carcinogen
A carcinogen is any substance, radionuclide, or radiation that is an agent directly involved in causing cancer. This may be due to the ability to damage the genome or to the disruption of cellular metabolic processes...
ic 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.
An alternative type of model in toxicology is the linear no-threshold model
Linear no-threshold model
The linear no-threshold model is a method for predicting the long term, biological damage caused by ionizing radiation and is based on the assumption that the risk is directly proportional to the dose at all dose levels....
(LNT), while hormesis
Hormesis
Hormesis is the term for generally favorable biological responses to low exposures to toxins and other stressors. A pollutant or toxin showing hormesis thus has the opposite effect in small doses as in large doses...
is a general term covering the type of response there may be to a drug.