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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. Certain types of regression model may include threshold effects.
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).
Segmented Regression analysis
The models used in segmented regression analysis are threshold models.
Time series analysis
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 (LNT), while hormesis correspond to the existence of opposite effects at low vs. high dose, which usually gives a U- or inverted U-shaped dose response curve.
Liability threshold model
The liability-threshold model is a threshold model of categorical (usually binary) outcomes in which a large number of variables are summed to yield an overall 'liability' score; the observed outcome is determined by whether the latent score is smaller or larger than the threshold. The liability-threshold model is frequently employed in medicine and genetics to model risk factors contributing to disease.
In a genetic context, the variables are all the genes and different environmental conditions, which protect against or increase the risk of a disease, and the threshold z is the biological limit past which disease develops. The threshold can be estimated from population prevalence of the disease (which is usually low). Because the threshold is defined relative to the population & environment, the liability score is generally considered as a N(0, 1) normally distributed random variable.
Early genetics models were developed to deal with very rare genetic diseases by treating them as Mendelian diseases caused by 1 or 2 genes: the presence or absence of the gene corresponds to the presence or absence of the disease, and the occurrence of the disease will follow predictable patterns within families. Continuous traits like height or intelligence could be modeled as normal distributions, influenced by a large number of genes, and the heritability and effects of selection easily analyzed. Some diseases, like alcoholism, epilepsy, or schizophrenia, cannot be Mendelian diseases because they are common; do not appear in Mendelian ratios; respond slowly to selection against them; often occur in families with no prior history of that disease; however, relatives and adoptees of someone with that disease are far more likely (but not certain) to develop it, indicating a strong genetic component. The liability threshold model was developed to deal with these non-Mendelian binary cases; the model proposes that there is a continuous normally-distributed trait expressing risk polygenically influenced by many genes, which all individuals above a certain value develop the disease and all below it do not.
The first threshold models in genetics were introduced by Sewall Wright, examining the propensity of guinea pig strains to have an extra hind toe, a phenomenon which could not be explained as a dominant or recessive gene, or continuous "blinding inheritance". The modern liability-threshold model was introduced into human research by geneticist Douglas Scott Falconer in his textbook and two papers. Falconer had been asked about the topic of modeling 'threshold characters' by Cyril Clarke who had diabetes.
An early application of liability-threshold models was to schizophrenia by Irving Gottesman & James Shields, finding substantial heritability & little shared-environment influence and undermining the "cold mother" theory of schizophrenia.
- Robertson & Lerner 1949, "The heritability of all-or-none traits: viability of poultry"
- Dempster & Lerner 1950, "Heritability of threshold characters"
- Carter 1961, "The inheritance of congenital pyloric stenosis"
- Carter 1963, "The genetics of common malformations"
- Carter 1969, "Genetics of common disorders"
- Crittenden 1961, "An interpretation of familial aggregation based on multiple genetic and environmental factors"
- Smith 1970, "Heritability of liability and concordance in monozygous twins"
- James 1971, "Frequency in relatives for an all-or-none trait"
- Curnow 1972, "The Multifactorial Model for the Inheritance of Liability to Disease and Its Implications for Relatives at Risk"
- Smith 1974, "Concordance in twins - methods and interpretation"
- Fraser 1976, "Multifactorial-threshold concept - uses and misuses"
- Gianola 1982, "Theory and analysis of threshold characters"
- McGue et al. 1983, "The transmission of schizophrenia under a multifactorial threshold-model"
- Roff 1996, "The Evolution of Threshold Traits in Animals"
- Chevin & Lande 2013, "Evolution of discrete phenotypes from continuous norms of reaction". Am Nat 182:13–27
- Visscher & Wray 2015, "Concepts and Misconceptions about the Polygenic Additive Model Applied to Disease",
- Dodge, Y. (2003) The Oxford Dictionary of Statistical Terms, OUP. ISBN 0-19-850994-4
- Tong, H. (1990) Non-linear Time Series: A Dynamical System Approach, OUP. ISBN 0-19-852224-X
- Calabrese, E.J., Baldwin, L.A. (2003) "The Hormetic Dose-Response Model Is More Common than the Threshold Model in Toxicology". Toxicological Sciences, 71, 246–250 
- "An analysis of variability in the number of digits in an inbred strain of guinea pigs", Wright 1934a
- "The results of crosses between inbred strains of guinea pigs, differing in number of digits", Wright 1934b
- ch18, "Threshold characters", Introduction to Quantitative Genetics, Falconer 1960
- "The inheritance of liability to certain diseases, estimated from the incidence among relatives", Falconer 1965
- "The inheritance of liability to diseases with variable age of onset, with particular reference to diabetes mellitus", Falconer 1967
- "D. S. Falconer and Introduction to Quantitative Genetics", Hill & Mackay 2004
- "A polygenic theory of schizophrenia", Gottesman & Shields 1967