Hawthorne effect

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The Hawthorne effect (also referred to as the observer effect) is a type of reactivity in which individuals modify an aspect of their behavior in response to their awareness of being observed. This can undermine the integrity of research, particularly the relationships between variables.

The original research at the Hawthorne Works in Cicero, Illinois, on lighting changes and work structure changes such as working hours and break times was originally interpreted by Elton Mayo and others to mean that paying attention to overall worker needs would improve productivity.

Later interpretations such as that done by Landsberger suggested that the novelty of being research subjects and the increased attention from such could lead to temporary increases in workers' productivity. This interpretation was dubbed "the Hawthorne effect". It is also similar to a phenomenon that is referred to as novelty/disruption effect.

History

Aerial view of the Hawthorne Works, ca. 1925

 

The term was coined in 1958 by Henry A. Landsberger when he was analyzing earlier experiments from 1924–32 at the Hawthorne Works (a Western Electric factory outside Chicago). The Hawthorne Works had commissioned a study to see if its workers would become more productive in higher or lower levels of light. The workers' productivity seemed to improve when changes were made, and slumped when the study ended. It was suggested that the productivity gain occurred as a result of the motivational effect on the workers of the interest being shown in them.

This effect was observed for minute increases in illumination. In these lighting studies, light intensity was altered to examine its effect on worker productivity. Most industrial/occupational psychology and organizational behavior textbooks refer to the illumination studies. Only occasionally are the rest of the studies mentioned.

Although illumination research of workplace lighting formed the basis of the Hawthorne effect, other changes such as maintaining clean work stations, clearing floors of obstacles, and even relocating workstations resulted in increased productivity for short periods. Thus the term is used to identify any type of short-lived increase in productivity.

Relay assembly experiments

In one of the studies, researchers chose two women as test subjects and asked them to choose four other workers to join the test group. Together the women worked in a separate room over the course of five years (1927–1932) assembling telephone relays.

Output was measured mechanically by counting how many finished relays each worker dropped down a chute. This measuring began in secret two weeks before moving the women to an experiment room and continued throughout the study. In the experiment room they had a supervisor who discussed changes with their productivity. Some of the variables were:

• Giving two 5-minute breaks (after a discussion with them on the best length of time), and then changing to two 10-minute breaks (not their preference). Productivity increased, but when they received six 5-minute rests, they disliked it and reduced output.
• Providing food during the breaks.
• Shortening the day by 30 minutes (output went up); shortening it more (output per hour went up, but overall output decreased); returning to the first condition (where output peaked).

Changing a variable usually increased productivity, even if the variable was just a change back to the original condition. However it is said that this is the natural process of the human being adapting to the environment, without knowing the objective of the experiment occurring. Researchers concluded that the workers worked harder because they thought that they were being monitored individually.

Researchers hypothesized that choosing one's own coworkers, working as a group, being treated as special (as evidenced by working in a separate room), and having a sympathetic supervisor were the real reasons for the productivity increase. One interpretation, mainly due to Elton Mayo, was that "the six individuals became a team and the team gave itself wholeheartedly and spontaneously to cooperation in the experiment." (There was a second relay assembly test room study whose results were not as significant as the first experiment.)

Bank wiring room experiments

The purpose of the next study was to find out how payment incentives would affect productivity. The surprising result was that productivity actually decreased. Workers apparently had become suspicious that their productivity may have been boosted to justify firing some of the workers later on. The study was conducted by Elton Mayo and W. Lloyd Warner between 1931 and 1932 on a group of fourteen men who put together telephone switching equipment. The researchers found that although the workers were paid according to individual productivity, productivity decreased because the men were afraid that the company would lower the base rate. Detailed observation of the men revealed the existence of informal groups or "cliques" within the formal groups. These cliques developed informal rules of behavior as well as mechanisms to enforce them. The cliques served to control group members and to manage bosses; when bosses asked questions, clique members gave the same responses, even if they were untrue. These results show that workers were more responsive to the social force of their peer groups than to the control and incentives of management.

Interpretation and criticism

Richard Nisbett has described the Hawthorne effect as "a glorified anecdote", saying that "once you have got the anecdote, you can throw away the data." Other researchers have attempted to explain the effects with various interpretations.

Adair warns of gross factual inaccuracy in most secondary publications on Hawthorne effect and that many studies failed to find it. He argues that it should be viewed as a variant of Orne's (1973) experimental demand effect. So for Adair, the issue is that an experimental effect depends on the participants' interpretation of the situation; this is why manipulation checks are important in social sciences experiments. So he thinks it is not awareness per se, nor special attention per se, but participants' interpretation that must be investigated in order to discover if/how the experimental conditions interact with the participants' goals. This can affect whether participants believe something, if they act on it or do not see it as in their interest, etc.

Possible explanations for the Hawthorne effect include the impact of feedback and motivation towards the experimenter. Receiving feedback on their performance may improve their skills when an experiment provides this feedback for the first time. Research on the demand effect also suggests that people may be motivated to please the experimenter, at least if it does not conflict with any other motive. They may also be suspicious of the purpose of the experimenter. Therefore, Hawthorne effect may only occur when there is usable feedback or a change in motivation.

Parsons defines the Hawthorne effect as "the confounding that occurs if experimenters fail to realize how the consequences of subjects' performance affect what subjects do" [i.e. learning effects, both permanent skill improvement and feedback-enabled adjustments to suit current goals]. His key argument is that in the studies where workers dropped their finished goods down chutes, the participants had access to the counters of their work rate.

Mayo contended that the effect was due to the workers reacting to the sympathy and interest of the observers. He does say that this experiment is about testing overall effect, not testing factors separately. He also discusses it not really as an experimenter effect but as a management effect: how management can make workers perform differently because they feel differently. A lot to do with feeling free, not feeling supervised but more in control as a group. The experimental manipulations were important in convincing the workers to feel this way: that conditions were really different. The experiment was repeated with similar effects on mica-splitting workers.

Clark and Sugrue in a review of educational research say that uncontrolled novelty effects cause on average 30% of a standard deviation (SD) rise (i.e. 50%–63% score rise), which decays to small level after 8 weeks. In more detail: 50% of a SD for up to 4 weeks; 30% of SD for 5–8 weeks; and 20% of SD for > 8 weeks, (which is < 1% of the variance).

Harry Braverman points out that the Hawthorne tests were based on industrial psychology and were investigating whether workers' performance could be predicted by pre-hire testing. The Hawthorne study showed "that the performance of workers had little relation to ability and in fact often bore an inverse relation to test scores...". Braverman argues that the studies really showed that the workplace was not "a system of bureaucratic formal organisation on the Weberian model, nor a system of informal group relations, as in the interpretation of Mayo and his followers but rather a system of power, of class antagonisms". This discovery was a blow to those hoping to apply the behavioral sciences to manipulate workers in the interest of management.

The economists Steven Levitt and John A. List long pursued without success a search for the base data of the original illumination experiments, before finding it in a microfilm at the University of Wisconsin in Milwaukee in 2011. Re-analysing it, they found slight evidence for the Hawthorn effect over the long-run, but in no way as drastic as suggested initially. This finding supported the analysis of an article by S R G Jones in 1992 examining the relay experiments. Despite the absence of evidence for the Hawthorne Effect in the original study, List has said that he remains confident that the effect is genuine.

It is also possible that the illumination experiments can be explained by a longitudinal learning effect. Parsons has declined to analyse the illumination experiments, on the grounds that they have not been properly published and so he cannot get at details, whereas he had extensive personal communication with Roethlisberger and Dickson.

Evaluation of the Hawthorne effect continues in the present day. Despite the criticisms, however, the phenomenon is often taken into account when designing studies and their conclusions. Some have also developed ways to avoid it. For instance, there is the case of holding the observation when conducting a field study from a distance, from behind a barrier such as a two-way mirror or using an unobtrusive measure.

Trial effect

Various medical scientists have studied possible trial effect (clinical trial effect) in clinical trials. Some postulate that, beyond just attention and observation, there may be other factors involved, such as slightly better care; slightly better compliance/adherence; and selection bias. The latter may have several mechanisms: (1) Physicians may tend to recruit patients who seem to have better adherence potential and lesser likelihood of future loss to follow-up. (2) The inclusion/exclusion criteria of trials often exclude at least some comorbidities; although this is often necessary to prevent confounding, it also means that trials may tend to work with healthier patient subpopulations.

Secondary observer effect

Despite the observer effect as popularized in the Hawthorne experiments being perhaps falsely identified (see above discussion), the popularity and plausibility of the observer effect in theory has led researchers to postulate that this effect could take place at a second level. Thus it has been proposed that there is a secondary observer effect when researchers working with secondary data such as survey data or various indicators may impact the results of their scientific research. Rather than having an effect on the subjects (as with the primary observer effect), the researchers likely have their own idiosyncrasies that influence how they handle the data and even what data they obtain from secondary sources. For one, the researchers may choose seemingly innocuous steps in their statistical analyses that end up causing significantly different results using the same data; e.g., weighting strategies, factor analytic techniques, or choice of estimation. In addition, researchers may use software packages that have different default settings that lead to small but significant fluctuations. Finally, the data that researchers use may not be identical, even though it seems so. For example, the OECD collects and distributes various socio-economic data; however, these data change over time such that a researcher who downloads the Australian GDP data for the year 2000 may have slightly different values than a researcher who downloads the same Australian GDP 2000 data a few years later. The idea of the secondary observer effect was floated by Nate Breznau in a thus far relatively obscure paper.

Although little attention has been paid to this phenomenon, the scientific implications are very large. Evidence of this effect may be seen in recent studies that assign a particular problem to a number of researchers or research teams who then work independently using the same data to try and find a solution. This is a process called crowdsourcing data analysis and was used in a groundbreaking study by Silberzahn, Rafael, Eric Uhlmann, Dan Martin and Brian Nosek et al. (2015) about red cards and player race in football (i.e., soccer).

Illusory correlation

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In psychology, illusory correlation is the phenomenon of perceiving a relationship between variables (typically people, events, or behaviors) even when no such relationship exists. A false association may be formed because rare or novel occurrences are more salient and therefore tend to capture one's attention. This phenomenon is one way stereotypes form and endure. Hamilton & Rose (1980) found that stereotypes can lead people to expect certain groups and traits to fit together, and then to overestimate the frequency with which these correlations actually occur.

History

"Illusory correlation" was originally coined by Chapman and Chapman (1967) to describe people's tendencies to overestimate relationships between two groups when distinctive and unusual information is presented. The concept was used to question claims about objective knowledge in clinical psychology through Chapmans' refutation of many clinicians' widely used Wheeler signs for homosexuality in Rorschach tests.

Example

David Hamilton and Robert Gifford (1976) conducted a series of experiments that demonstrated how stereotypic beliefs regarding minorities could derive from illusory correlation processes. To test their hypothesis, Hamilton and Gifford had research participants read a series of sentences describing either desirable or undesirable behaviors, which were attributed to either Group A or Group B. Abstract groups were used so that no previously established stereotypes would influence results. Most of the sentences were associated with Group A, and the remaining few were associated with Group B. The following table summarizes the information given.

Behaviors	Group A (majority)	Group B (minority)	Total
Desirable	18 (69%)	9 (69%)	27
Undesirable	8 (30%)	4 (30%)	12
Total	26	13	39

Each group had the same proportions of positive and negative behaviors, so there was no real association between behaviors and group membership. Results of the study show that positive, desirable behaviors were not seen as distinctive so people were accurate in their associations. On the other hand, when distinctive, undesirable behaviors were represented in the sentences, the participants overestimated how much the minority group exhibited the behaviors.

A parallel effect occurs when people judge whether two events, such as pain and bad weather, are correlated. They rely heavily on the relatively small number of cases where the two events occur together. People pay relatively little attention to the other kinds of observation (of no pain or good weather).

Theories

General theory

Most explanations for illusory correlation involve psychological heuristics: information processing short-cuts that underlie many human judgments. One of these is availability: the ease with which an idea comes to mind. Availability is often used to estimate how likely an event is or how often it occurs. This can result in illusory correlation, because some pairings can come easily and vividly to mind even though they are not especially frequent.

Information processing

Martin Hilbert (2012) proposes an information processing mechanism that assumes a noisy conversion of objective observations into subjective judgments. The theory defines noise as the mixing of these observations during retrieval from memory. According to the model, underlying cognitions or subjective judgments are identical with noise or objective observations that can lead to overconfidence or what is known as conservatism bias—when asked about behavior participants underestimate the majority or larger group and overestimate the minority or smaller group. These results are illusory correlations.

Working-memory capacity

In an experimental study done by Eder, Fiedler and Hamm-Eder (2011), the effects of working-memory capacity on illusory correlations were investigated. They first looked at the individual differences in working memory, and then looked to see if that had any effect on the formation of illusory correlations. They found that individuals with higher working memory capacity viewed minority group members more positively than individuals with lower working memory capacity. In a second experiment, the authors looked into the effects of memory load in working memory on illusory correlations. They found that increased memory load in working memory led to an increase in the prevalence of illusory correlations. The experiment was designed to specifically test working memory and not substantial stimulus memory. This means that the development of illusory correlations was caused by deficiencies in central cognitive resources caused by the load in working memory, not selective recall.

Attention theory of learning

Attention theory of learning proposes that features of majority groups are learned first, and then features of minority groups. This results in an attempt to distinguish the minority group from the majority, leading to these differences being learned more quickly. The Attention theory also argues that, instead of forming one stereotype regarding the minority group, two stereotypes, one for the majority and one for the minority, are formed.

Effect of learning

A study was conducted to investigate whether increased learning would have any effect on illusory correlations. It was found that educating people about how illusory correlation occurs resulted in a decreased incidence of illusory correlations.

Age

Johnson and Jacobs (2003) performed an experiment to see how early in life individuals begin forming illusory correlations. Children in grades 2 and 5 were exposed to a typical illusory correlation paradigm to see if negative attributes were associated with the minority group. The authors found that both groups formed illusory correlations.

A study also found that children create illusory correlations. In their experiment, children in grades 1, 3, 5, and 7, and adults all looked at the same illusory correlation paradigm. The study found that children did create significant illusory correlations, but those correlations were weaker than the ones created by adults. In a second study, groups of shapes with different colors were used. The formation of illusory correlation persisted showing that social stimuli are not necessary for creating these correlations.

Explicit versus implicit attitudes

Two studies performed by Ratliff and Nosek examined whether or not explicit and implicit attitudes affected illusory correlations. In one study, Ratliff and Nosek had two groups: one a majority and the other a minority. They then had three groups of participants, all with readings about the two groups. One group of participants received overwhelming pro-majority readings, one was given pro-minority readings, and one received neutral readings. The groups that had pro-majority and pro-minority readings favored their respective pro groups both explicitly and implicitly. The group that had neutral readings favored the majority explicitly, but not implicitly. The second study was similar, but instead of readings, pictures of behaviors were shown, and the participants wrote a sentence describing the behavior they saw in the pictures presented. The findings of both studies supported the authors' argument that the differences found between the explicit and implicit attitudes is a result of the interpretation of the covariation and making judgments based on these interpretations (explicit) instead of just accounting for the covariation (implicit).

Paradigm structure

Berndsen et al. (1999) wanted to determine if the structure of testing for illusory correlations could lead to the formation of illusory correlations. The hypothesis was that identifying test variables as Group A and Group B might be causing the participants to look for differences between the groups, resulting in the creation of illusory correlations. An experiment was set up where one set of participants were told the groups were Group A and Group B, while another set of participants were given groups labeled as students who graduated in 1993 or 1994. This study found that illusory correlations were more likely to be created when the groups were Group A and B, as compared to students of the class of 1993 or the class of 1994.

Spurious relationship

From Wikipedia, the free encyclopedia

 

In statistics, a spurious relationship or spurious correlation is a mathematical relationship in which two or more events or variables are associated but not causally related, due to either coincidence or the presence of a certain third, unseen factor (referred to as a "common response variable", "confounding factor", or "lurking variable").

Examples

A well-known case of a spurious relationship can be found in the time-series literature, where a spurious regression is a regression that provides misleading statistical evidence of a linear relationship between independent non-stationary variables. In fact, the non-stationarity may be due to the presence of a unit root in both variables. In particular, any two nominal economic variables are likely to be correlated with each other, even when neither has a causal effect on the other, because each equals a real variable times the price level, and the common presence of the price level in the two data series imparts correlation to them.

An example of a spurious relationship can be seen by examining a city's ice cream sales. These sales are highest when the rate of drownings in city swimming pools is highest. To allege that ice cream sales cause drowning, or vice versa, would be to imply a spurious relationship between the two. In reality, a heat wave may have caused both. The heat wave is an example of a hidden or unseen variable, also known as a confounding variable.

Another commonly noted example is a series of Dutch statistics showing a positive correlation between the number of storks nesting in a series of springs and the number of human babies born at that time. Of course there was no causal connection; they were correlated with each other only because they were correlated with the weather nine months before the observations. However Höfer et al. (2004) showed the correlation to be stronger than just weather variations as he could show in post reunification Germany that, while the number of clinical deliveries was not linked with the rise in stork population, out of hospital deliveries correlated with the stork population.

In rare cases, a spurious relationship can occur between two completely unrelated variables without any confounding variable, as was the case between the success of the Washington Redskins professional football team in a specific game before each presidential election and the success of the incumbent President's political party in said election. For 16 consecutive elections between 1940 and 2000, the Redskins Rule correctly matched whether the incumbent President's political party would retain or lose the Presidency. The rule eventually failed shortly after Elias Sports Bureau discovered the correlation in 2000; in 2004, 2012 and 2016, the results of the Redskins game and the election did not match.

Hypothesis testing

Often one tests a null hypothesis of no correlation between two variables, and chooses in advance to reject the hypothesis if the correlation computed from a data sample would have occurred in less than (say) 5% of data samples if the null hypothesis were true. While a true null hypothesis will be accepted 95% of the time, the other 5% of the times having a true null of no correlation a zero correlation will be wrongly rejected, causing acceptance of a correlation which is spurious (an event known as Type I error). Here the spurious correlation in the sample resulted from random selection of a sample that did not reflect the true properties of the underlying population.

Detecting spurious relationships

The term "spurious relationship" is commonly used in statistics and in particular in experimental research techniques, both of which attempt to understand and predict direct causal relationships (X → Y). A non-causal correlation can be spuriously created by an antecedent which causes both (W → X and W → Y). Mediating variables, (X → W → Y), if undetected, estimate a total effect rather than direct effect without adjustment for the mediating variable M. Because of this, experimentally identified correlations do not represent causal relationships unless spurious relationships can be ruled out.

Experiments

In experiments, spurious relationships can often be identified by controlling for other factors, including those that have been theoretically identified as possible confounding factors. For example, consider a researcher trying to determine whether a new drug kills bacteria; when the researcher applies the drug to a bacterial culture, the bacteria die. But to help in ruling out the presence of a confounding variable, another culture is subjected to conditions that are as nearly identical as possible to those facing the first-mentioned culture, but the second culture is not subjected to the drug. If there is an unseen confounding factor in those conditions, this control culture will die as well, so that no conclusion of efficacy of the drug can be drawn from the results of the first culture. On the other hand, if the control culture does not die, then the researcher cannot reject the hypothesis that the drug is efficacious.

Non-experimental statistical analyses

Disciplines whose data are mostly non-experimental, such as economics, usually employ observational data to establish causal relationships. The body of statistical techniques used in economics is called econometrics. The main statistical method in econometrics is multivariable regression analysis. Typically a linear relationship such as

${\displaystyle y=a_{0}+a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{k}x_{k}+e}$

is hypothesized, in which ${\displaystyle y}$ is the dependent variable (hypothesized to be the caused variable), ${\displaystyle x_{j}}$ for j = 1, ..., k is the j^th independent variable (hypothesized to be a causative variable), and ${\displaystyle e}$ is the error term (containing the combined effects of all other causative variables, which must be uncorrelated with the included independent variables). If there is reason to believe that none of the ${\displaystyle x_{j}}$s is caused by y, then estimates of the coefficients ${\displaystyle a_{j}}$ are obtained. If the null hypothesis that ${\displaystyle a_{j}=0}$ is rejected, then the alternative hypothesis that ${\displaystyle a_{j}\neq 0}$ and equivalently that ${\displaystyle x_{j}}$ causes y cannot be rejected. On the other hand, if the null hypothesis that ${\displaystyle a_{j}=0}$ cannot be rejected, then equivalently the hypothesis of no causal effect of ${\displaystyle x_{j}}$ on y cannot be rejected. Here the notion of causality is one of contributory causality: If the true value ${\displaystyle a_{j}\neq 0}$, then a change in ${\displaystyle x_{j}}$ will result in a change in y unless some other causative variable(s), either included in the regression or implicit in the error term, change in such a way as to exactly offset its effect; thus a change in ${\displaystyle x_{j}}$ is not sufficient to change y. Likewise, a change in ${\displaystyle x_{j}}$ is not necessary to change y, because a change in y could be caused by something implicit in the error term (or by some other causative explanatory variable included in the model). 

Regression analysis controls for other relevant variables by including them as regressors (explanatory variables). This helps to avoid mistaken inference of causality due to the presence of a third, underlying, variable that influences both the potentially causative variable and the potentially caused variable: its effect on the potentially caused variable is captured by directly including it in the regression, so that effect will not be picked up as a spurious effect of the potentially causative variable of interest. In addition, the use of multivariate regression helps to avoid wrongly inferring that an indirect effect of, say x₁ (e.g., x₁ → x₂ → y) is a direct effect (x₁ → y). 

Just as an experimenter must be careful to employ an experimental design that controls for every confounding factor, so also must the user of multiple regression be careful to control for all confounding factors by including them among the regressors. If a confounding factor is omitted from the regression, its effect is captured in the error term by default, and if the resulting error term is correlated with one (or more) of the included regressors, then the estimated regression may be biased or inconsistent (see omitted variable bias). 

In addition to regression analysis, the data can be examined to determine if Granger causality exists. The presence of Granger causality indicates both that x precedes y, and that x contains unique information about y.

Other relationships

There are several other relationships defined in statistical analysis as follows.

• Direct relationship
• Mediating relationship
• Moderating relationship

All models are wrong

From Wikipedia, the free encyclopedia

 

"All models are wrong" is a common aphorism in statistics; it is often expanded as "All models are wrong, but some are useful". It is usually considered to be applicable to not only statistical models, but to scientific models generally. The aphorism is generally attributed to the statistician George Box, although the underlying concept predates Box's writings.

Quotations of George Box

The first record of Box saying "all models are wrong" is in a 1976 paper published in the Journal of the American Statistical Association. The 1976 paper contains the aphorism twice. The two sections of the paper that contain the aphorism are copied below.

2.3  Parsimony
Since all models are wrong the scientist cannot obtain a "correct" one by excessive elaboration. On the contrary following William of Occam he should seek an economical description of natural phenomena. Just as the ability to devise simple but evocative models is the signature of the great scientist so overelaboration and overparameterization is often the mark of mediocrity.
2.4  Worrying Selectively
Since all models are wrong the scientist must be alert to what is importantly wrong. It is inappropriate to be concerned about mice when there are tigers abroad.

Box repeated the aphorism in a paper that was published in the proceedings of a 1978 statistics workshop. The paper contains a section entitled "All models are wrong but some are useful". The section is copied below.

Now it would be very remarkable if any system existing in the real world could be exactly represented by any simple model. However, cunningly chosen parsimonious models often do provide remarkably useful approximations. For example, the law PV = RT relating pressure P, volume V and temperature T of an "ideal" gas via a constant R is not exactly true for any real gas, but it frequently provides a useful approximation and furthermore its structure is informative since it springs from a physical view of the behavior of gas molecules.
For such a model there is no need to ask the question "Is the model true?". If "truth" is to be the "whole truth" the answer must be "No". The only question of interest is "Is the model illuminating and useful?".

Box repeated the aphorism twice more in his 1987 book, Empirical Model-Building and Response Surfaces (which was co-authored with Norman Draper). The first repetition is on p. 74: "Remember that all models are wrong; the practical question is how wrong do they have to be to not be useful." The second repetition is on p. 424, which is excerpted below.

... all models are approximations. Essentially, all models are wrong, but some are useful. However, the approximate nature of the model must always be borne in mind ...

A second edition of the book was published in 2007, under the title Response Surfaces, Mixtures, and Ridge Analyses. The second edition also repeats the aphorism twice, in contexts identical with those of the first edition (on p. 63 and p. 414).

Box repeated the aphorism two more times in his 1997 book, Statistical Control: By Monitoring and Feedback Adjustment (which was co-authored with Alberto Luceño). The first repetition is on p. 6, which is excerpted below.

It has been said that "all models are wrong but some models are useful." In other words, any model is at best a useful fiction—there never was, or ever will be, an exactly normal distribution or an exact linear relationship. Nevertheless, enormous progress has been made by entertaining such fictions and using them as approximations.

The second repetition is on p. 9: "So since all models are wrong, it is very important to know what to worry about; or, to put it in another way, what models are likely to produce procedures that work in practice (where exact assumptions are never true)".

A second edition of the book was published in 2009, under the title Statistical Control By Monitoring and Adjustment (co-authored with Alberto Luceño and Maria del Carmen Paniagua-Quiñones). The second edition also repeats the aphorism two times. The first repetition is on p. 61, which is excerpted below.

All models are approximations. Assumptions, whether implied or clearly stated, are never exactly true. All models are wrong, but some models are useful. So the question you need to ask is not "Is the model true?" (it never is) but "Is the model good enough for this particular application?"

The second repetition is on p. 63; its context is essentially the same as that of the second repetition in the first edition.

Box's widely cited book Statistics for Experimenters (co-authored with William Hunter) does not include the aphorism in its first edition (published in 1978). The second edition (published in 2005; co-authored with William Hunter and J. Stuart Hunter) includes the aphorism three times: on p. 208, p. 384, and p. 440. On p. 440, the relevant sentence is this: "The most that can be expected from any model is that it can supply a useful approximation to reality: All models are wrong; some models are useful". 

In addition to stating the aphorism verbatim, Box sometimes stated the essence of the aphorism with different words. One example is from 1978, while Box was President of the American Statistical Association. At the annual meeting of the Association, Box delivered his Presidential Address, wherein he stated this: "Models, of course, are never true, but fortunately it is only necessary that they be useful".

Discussions

There have been varied discussions about the aphorism. A selection from those discussions is presented below. 

In 1983, the statisticians Peter McCullagh and John Nelder published their much-cited book on generalized linear models. The book includes a brief discussion of the aphorism (though without citing Box). A second edition of the book, published in 1989, contains a very similar discussion of the aphorism. The discussion from the first edition is as follows.

Modelling in science remains, partly at least, an art. Some principles do exist, however, to guide the modeller. The first is that all models are wrong; some, though, are better than others and we can search for the better ones. At the same time we must recognize that eternal truth is not within our grasp.

In 1995, the statistician Sir David Cox commented as follows.

... it does not seem helpful just to say that all models are wrong. The very word model implies simplification and idealization. The idea that complex physical, biological or sociological systems can be exactly described by a few formulae is patently absurd. The construction of idealized representations that capture important stable aspects of such systems is, however, a vital part of general scientific analysis and statistical models, especially substantive ones, do not seem essentially different from other kinds of model.

In 1996, an Applied Statistician's Creed was proposed. The Creed includes, in its core part, the aphorism.

In 2002, K.P. Burnham and D.R. Anderson published their much-cited book on statistical model selection. The book states the following.

A model is a simplification or approximation of reality and hence will not reflect all of reality. ... Box noted that "all models are wrong, but some are useful." While a model can never be "truth," a model might be ranked from very useful, to useful, to somewhat useful to, finally, essentially useless.

The statistician J. Michael Steele has commented on the aphorism as follows.

... there are wonderful models — like city maps....
If I say that a map is wrong, it means that a building is misnamed, or the direction of a one-way street is mislabeled. I never expected my map to recreate all of physical reality, and I only feel ripped off if my map does not correctly answer the questions that it claims to answer.
My maps of Philadelphia are useful. Moreover, except for a few that are out-of-date, they are not wrong.
So, you say, "Yes, a map can be thought of as a model, but surely it would be more precise to say that a map is a 'visually enhanced database.' Such databases can be correct. These are not the kinds of models that Box had in mind."
I agree. ...

In 2008, the statistician Andrew Gelman responded to that, saying in particular the following.

I take his general point, which is that a street map could be exactly correct, to the resolution of the map.
... The saying, "all models are wrong," is helpful because it is not completely obvious....
This is a simple point, and I can see how Steele can be irritated by people making a big point about it. But, the trouble is, many people don't realize that all models are wrong.

In 2013, the philosopher of science Peter Truran published an essay related to the aphorism. The essay notes, in particular, the following.

... seemingly incompatible models may be used to make predictions about the same phenomenon. ... For each model we may believe that its predictive power is an indication of its being at least approximately true. But if both models are successful in making predictions, and yet mutually inconsistent, how can they both be true? Let us consider a simple illustration. Two observers are looking at a physical object. One may report seeing a circular disc, and the other may report seeing a rectangle. Both will be correct, but one will be looking at the object (a cylindrical can) from above and the other will be observing from the side. The two models represent different aspects of the same reality.

Truran's essay further notes that Newton's theory of gravitation has been supplanted by Einstein's theory of relativity and yet Newton's theory remains generally "empirically adequate". Indeed, Newton's theory generally has excellent predictive power. Yet Newton's theory is not an approximation of Einstein's theory. For illustration, consider an apple falling down from a tree. Under Newton's theory, the apple falls because Earth exerts a force on the apple—what is called "the force of gravity". Under Einstein's theory, Earth does not exert any force on the apple. Hence, Newton's theory might be regarded as being, in some sense, completely wrong but extremely useful. (The usefulness of Newton's theory comes partly from being vastly simpler, both mathematically and computationally, than Einstein's theory.)

In 2014, the statistician David Hand made the following statement.

In general, when building statistical models, we must not forget that the aim is to understand something about the real world. Or predict, choose an action, make a decision, summarize evidence, and so on, but always about the real world, not an abstract mathematical world: our models are not the reality—a point well made by George Box in his oft-cited remark that "all models are wrong, but some are useful".

In 2016, P.J. Bickel and K.A. Doksum published the second volume of their book on mathematical statistics. The volume includes the quote from Box's Presidential Address, given above. It states that the quote is the best formulation of the "guiding principle of modern statistics".

Additionally, in 2011, a workshop on model selection was held in The Netherlands. The name of the workshop was "All models are wrong...".

Historical antecedents

Although the aphorism seems to have originated with George Box, the underlying concept goes back decades, perhaps centuries. Some exemplifications of that are given below.

In 1960, Georg Rasch said the following.

… no models are [true]—not even the Newtonian laws. When you construct a model you leave out all the details which you, with the knowledge at your disposal, consider inessential…. Models should not be true, but it is important that they are applicable, and whether they are applicable for any given purpose must of course be investigated. This also means that a model is never accepted finally, only on trial.

— Rasch, G. (1960), Probabilistic Models for Some Intelligence and Attainment Tests, Copenhagen: Danmarks Paedagogiske Institut, pp. 37–38; republished in 1980 by University of Chicago Press

In 1947, the mathematician John von Neumann said that "truth … is much too complicated to allow anything but approximations".

In 1942, the French philosopher-poet Paul Valéry said the following.

Ce qui est simple est toujours faux. Ce qui ne l’est pas est inutilisable.	What is simple is always wrong. What is not is unusable.
—Valéry, Paul (1942), Mauvaises pensées et autres, Paris: Éditions Gallimard

In 1939, the founder of statistical process control, Walter Shewhart, said the following.

… no model can ever be theoretically attainable that will completely and uniquely characterize the indefinitely expansible concept of a state of statistical control. What is perhaps even more important, on the basis of a finite portion of the sequence [X₁, X₂, X₃, …]—and we can never have more than a finite portion—we can not reasonably hope to construct a model that will represent exactly any specific characteristic of a particular state of control even though such a state actually exists. Here the situation is much like that in physical science where we find a model of a molecule; any model is always an incomplete though useful picture of the conceived physical thing called a molecule.

— Shewhart, W. A. (1939), Statistical Method From the Viewpoint of Quality Control, U.S. Department of Agriculture, p. 19

In 1923, a related idea was articulated by the artist Pablo Picasso.

We all know that art is not truth. Art is a lie that makes us realize truth, at least the truth that is given us to understand. The artist must know the manner whereby to convince others of the truthfulness of his lies.

— Picasso, Pablo (1923), "Picasso speaks", The Arts, 3: 315–326; reprinted in Barr, Alfred H., Jr. (1939), Picasso: Forty Years of his Art (PDF), Museum of Modern Art, pp. 9–12

Mathematical model

From Wikipedia, the free encyclopedia

A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, biology, earth science, chemistry) and engineering disciplines (such as computer science, electrical engineering), as well as in the social sciences (such as economics, psychology, sociology, political science).

A model may help to explain a system and to study the effects of different components, and to make predictions about behaviour.

Elements of a mathematical model

Mathematical models can take many forms, including dynamical systems, statistical models, differential equations, or game theoretic models. These and other types of models can overlap, with a given model involving a variety of abstract structures. In general, mathematical models may include logical models. In many cases, the quality of a scientific field depends on how well the mathematical models developed on the theoretical side agree with results of repeatable experiments. Lack of agreement between theoretical mathematical models and experimental measurements often leads to important advances as better theories are developed. 

In the physical sciences, a traditional mathematical model contains most of the following elements:

1. Governing equations
2. Supplementary sub-models
   1. Defining equations
   2. Constitutive equations
3. Assumptions and constraints
   1. Initial and boundary conditions
   2. Classical constraints and kinematic equations

Classifications

Mathematical models are usually composed of relationships and variables. Relationships can be described by operators, such as algebraic operators, functions, differential operators, etc. Variables are abstractions of system parameters of interest, that can be quantified. Several classification criteria can be used for mathematical models according to their structure:

• Linear vs. nonlinear: If all the operators in a mathematical model exhibit linearity, the resulting mathematical model is defined as linear. A model is considered to be nonlinear otherwise. The definition of linearity and nonlinearity is dependent on context, and linear models may have nonlinear expressions in them. For example, in a statistical linear model, it is assumed that a relationship is linear in the parameters, but it may be nonlinear in the predictor variables. Similarly, a differential equation is said to be linear if it can be written with linear differential operators, but it can still have nonlinear expressions in it. In a mathematical programming model, if the objective functions and constraints are represented entirely by linear equations, then the model is regarded as a linear model. If one or more of the objective functions or constraints are represented with a nonlinear equation, then the model is known as a nonlinear model.
  Nonlinearity, even in fairly simple systems, is often associated with phenomena such as chaos and irreversibility. Although there are exceptions, nonlinear systems and models tend to be more difficult to study than linear ones. A common approach to nonlinear problems is linearization, but this can be problematic if one is trying to study aspects such as irreversibility, which are strongly tied to nonlinearity.
• Static vs. dynamic: A dynamic model accounts for time-dependent changes in the state of the system, while a static (or steady-state) model calculates the system in equilibrium, and thus is time-invariant. Dynamic models typically are represented by differential equations or difference equations.
• Explicit vs. implicit: If all of the input parameters of the overall model are known, and the output parameters can be calculated by a finite series of computations, the model is said to be explicit. But sometimes it is the output parameters which are known, and the corresponding inputs must be solved for by an iterative procedure, such as Newton's method (if the model is linear) or Broyden's method (if non-linear). In such a case the model is said to be implicit. For example, a jet engine's physical properties such as turbine and nozzle throat areas can be explicitly calculated given a design thermodynamic cycle (air and fuel flow rates, pressures, and temperatures) at a specific flight condition and power setting, but the engine's operating cycles at other flight conditions and power settings cannot be explicitly calculated from the constant physical properties.
• Discrete vs. continuous: A discrete model treats objects as discrete, such as the particles in a molecular model or the states in a statistical model; while a continuous model represents the objects in a continuous manner, such as the velocity field of fluid in pipe flows, temperatures and stresses in a solid, and electric field that applies continuously over the entire model due to a point charge.
• Deterministic vs. probabilistic (stochastic): A deterministic model is one in which every set of variable states is uniquely determined by parameters in the model and by sets of previous states of these variables; therefore, a deterministic model always performs the same way for a given set of initial conditions. Conversely, in a stochastic model—usually called a "statistical model"—randomness is present, and variable states are not described by unique values, but rather by probability distributions.
• Deductive, inductive, or floating: A deductive model is a logical structure based on a theory. An inductive model arises from empirical findings and generalization from them. The floating model rests on neither theory nor observation, but is merely the invocation of expected structure. Application of mathematics in social sciences outside of economics has been criticized for unfounded models. Application of catastrophe theory in science has been characterized as a floating model.

Significance in the natural sciences

Mathematical models are of great importance in the natural sciences, particularly in physics. Physical theories are almost invariably expressed using mathematical models.

Throughout history, more and more accurate mathematical models have been developed. Newton's laws accurately describe many everyday phenomena, but at certain limits theory of relativity and quantum mechanics must be used. Though even these theories can't model or explain all phenomena themselves or together, such as black holes. It is possible to obtain the less accurate models in appropriate limits, for example relativistic mechanics reduces to Newtonian mechanics at speeds much less than the speed of light. Quantum mechanics reduces to classical physics when the quantum numbers are high. For example, the de Broglie wavelength of a tennis ball is insignificantly small, so classical physics is a good approximation to use in this case. 

It is common to use idealized models in physics to simplify things. Massless ropes, point particles, ideal gases and the particle in a box are among the many simplified models used in physics. The laws of physics are represented with simple equations such as Newton's laws, Maxwell's equations and the Schrödinger equation. These laws are a basis for making mathematical models of real situations. Many real situations are very complex and thus modeled approximate on a computer, a model that is computationally feasible to compute is made from the basic laws or from approximate models made from the basic laws. For example, molecules can be modeled by molecular orbital models that are approximate solutions to the Schrödinger equation. In engineering, physics models are often made by mathematical methods such as finite element analysis.

Different mathematical models use different geometries that are not necessarily accurate descriptions of the geometry of the universe. Euclidean geometry is much used in classical physics, while special relativity and general relativity are examples of theories that use geometries which are not Euclidean.

Some applications

Since prehistorical times simple models such as maps and diagrams have been used.

Often when engineers analyze a system to be controlled or optimized, they use a mathematical model. In analysis, engineers can build a descriptive model of the system as a hypothesis of how the system could work, or try to estimate how an unforeseeable event could affect the system. Similarly, in control of a system, engineers can try out different control approaches in simulations.

A mathematical model usually describes a system by a set of variables and a set of equations that establish relationships between the variables. Variables may be of many types; real or integer numbers, boolean values or strings, for example. The variables represent some properties of the system, for example, measured system outputs often in the form of signals, timing data, counters, and event occurrence (yes/no). The actual model is the set of functions that describe the relations between the different variables.

Building blocks

In business and engineering, mathematical models may be used to maximize a certain output. The system under consideration will require certain inputs. The system relating inputs to outputs depends on other variables too: decision variables, state variables, exogenous variables, and random variables.

Decision variables are sometimes known as independent variables. Exogenous variables are sometimes known as parameters or constants. The variables are not independent of each other as the state variables are dependent on the decision, input, random, and exogenous variables. Furthermore, the output variables are dependent on the state of the system (represented by the state variables).

Objectives and constraints of the system and its users can be represented as functions of the output variables or state variables. The objective functions will depend on the perspective of the model's user. Depending on the context, an objective function is also known as an index of performance, as it is some measure of interest to the user. Although there is no limit to the number of objective functions and constraints a model can have, using or optimizing the model becomes more involved (computationally) as the number increases. 

For example, economists often apply linear algebra when using input-output models. Complicated mathematical models that have many variables may be consolidated by use of vectors where one symbol represents several variables.

A priori information

To analyse something with a typical "black box approach", only the behavior of the stimulus/response will be accounted for, to infer the (unknown) box. The usual representation of this black box system is a data flow diagram centered in the box.

 

Mathematical modeling problems are often classified into black box or white box models, according to how much a priori information on the system is available. A black-box model is a system of which there is no a priori information available. A white-box model (also called glass box or clear box) is a system where all necessary information is available. Practically all systems are somewhere between the black-box and white-box models, so this concept is useful only as an intuitive guide for deciding which approach to take.

Usually it is preferable to use as much a priori information as possible to make the model more accurate. Therefore, the white-box models are usually considered easier, because if you have used the information correctly, then the model will behave correctly. Often the a priori information comes in forms of knowing the type of functions relating different variables. For example, if we make a model of how a medicine works in a human system, we know that usually the amount of medicine in the blood is an exponentially decaying function. But we are still left with several unknown parameters; how rapidly does the medicine amount decay, and what is the initial amount of medicine in blood? This example is therefore not a completely white-box model. These parameters have to be estimated through some means before one can use the model.

In black-box models one tries to estimate both the functional form of relations between variables and the numerical parameters in those functions. Using a priori information we could end up, for example, with a set of functions that probably could describe the system adequately. If there is no a priori information we would try to use functions as general as possible to cover all different models. An often used approach for black-box models are neural networks which usually do not make assumptions about incoming data. Alternatively the NARMAX (Nonlinear AutoRegressive Moving Average model with eXogenous inputs) algorithms which were developed as part of nonlinear system identification can be used to select the model terms, determine the model structure, and estimate the unknown parameters in the presence of correlated and nonlinear noise. The advantage of NARMAX models compared to neural networks is that NARMAX produces models that can be written down and related to the underlying process, whereas neural networks produce an approximation that is opaque.

Subjective information

Sometimes it is useful to incorporate subjective information into a mathematical model. This can be done based on intuition, experience, or expert opinion, or based on convenience of mathematical form. Bayesian statistics provides a theoretical framework for incorporating such subjectivity into a rigorous analysis: we specify a prior probability distribution (which can be subjective), and then update this distribution based on empirical data.

An example of when such approach would be necessary is a situation in which an experimenter bends a coin slightly and tosses it once, recording whether it comes up heads, and is then given the task of predicting the probability that the next flip comes up heads. After bending the coin, the true probability that the coin will come up heads is unknown; so the experimenter would need to make a decision (perhaps by looking at the shape of the coin) about what prior distribution to use. Incorporation of such subjective information might be important to get an accurate estimate of the probability.

Complexity

In general, model complexity involves a trade-off between simplicity and accuracy of the model. Occam's razor is a principle particularly relevant to modeling, its essential idea being that among models with roughly equal predictive power, the simplest one is the most desirable. While added complexity usually improves the realism of a model, it can make the model difficult to understand and analyze, and can also pose computational problems, including numerical instability. Thomas Kuhn argues that as science progresses, explanations tend to become more complex before a paradigm shift offers radical simplification.

For example, when modeling the flight of an aircraft, we could embed each mechanical part of the aircraft into our model and would thus acquire an almost white-box model of the system. However, the computational cost of adding such a huge amount of detail would effectively inhibit the usage of such a model. Additionally, the uncertainty would increase due to an overly complex system, because each separate part induces some amount of variance into the model. It is therefore usually appropriate to make some approximations to reduce the model to a sensible size. Engineers often can accept some approximations in order to get a more robust and simple model. For example, Newton's classical mechanics is an approximated model of the real world. Still, Newton's model is quite sufficient for most ordinary-life situations, that is, as long as particle speeds are well below the speed of light, and we study macro-particles only.

Training and tuning

Any model which is not pure white-box contains some parameters that can be used to fit the model to the system it is intended to describe. If the modeling is done by an artificial neural network or other machine learning, the optimization of parameters is called training, while the optimization of model hyperparameters is called tuning and often uses cross-validation. In more conventional modeling through explicitly given mathematical functions, parameters are often determined by curve fitting.

Model evaluation

A crucial part of the modeling process is the evaluation of whether or not a given mathematical model describes a system accurately. This question can be difficult to answer as it involves several different types of evaluation.

Fit to empirical data

Usually the easiest part of model evaluation is checking whether a model fits experimental measurements or other empirical data. In models with parameters, a common approach to test this fit is to split the data into two disjoint subsets: training data and verification data. The training data are used to estimate the model parameters. An accurate model will closely match the verification data even though these data were not used to set the model's parameters. This practice is referred to as cross-validation in statistics.

Defining a metric to measure distances between observed and predicted data is a useful tool of assessing model fit. In statistics, decision theory, and some economic models, a loss function plays a similar role.

While it is rather straightforward to test the appropriateness of parameters, it can be more difficult to test the validity of the general mathematical form of a model. In general, more mathematical tools have been developed to test the fit of statistical models than models involving differential equations. Tools from non-parametric statistics can sometimes be used to evaluate how well the data fit a known distribution or to come up with a general model that makes only minimal assumptions about the model's mathematical form.

Scope of the model

Assessing the scope of a model, that is, determining what situations the model is applicable to, can be less straightforward. If the model was constructed based on a set of data, one must determine for which systems or situations the known data is a "typical" set of data.

The question of whether the model describes well the properties of the system between data points is called interpolation, and the same question for events or data points outside the observed data is called extrapolation.

As an example of the typical limitations of the scope of a model, in evaluating Newtonian classical mechanics, we can note that Newton made his measurements without advanced equipment, so he could not measure properties of particles travelling at speeds close to the speed of light. Likewise, he did not measure the movements of molecules and other small particles, but macro particles only. It is then not surprising that his model does not extrapolate well into these domains, even though his model is quite sufficient for ordinary life physics.

Philosophical considerations

Many types of modeling implicitly involve claims about causality. This is usually (but not always) true of models involving differential equations. As the purpose of modeling is to increase our understanding of the world, the validity of a model rests not only on its fit to empirical observations, but also on its ability to extrapolate to situations or data beyond those originally described in the model. One can think of this as the differentiation between qualitative and quantitative predictions. One can also argue that a model is worthless unless it provides some insight which goes beyond what is already known from direct investigation of the phenomenon being studied.

An example of such criticism is the argument that the mathematical models of optimal foraging theory do not offer insight that goes beyond the common-sense conclusions of evolution and other basic principles of ecology.

Examples

• One of the popular examples in computer science is the mathematical models of various machines, an example is the deterministic finite automaton (DFA) which is defined as an abstract mathematical concept, but due to the deterministic nature of a DFA, it is implementable in hardware and software for solving various specific problems. For example, the following is a DFA M with a binary alphabet, which requires that the input contains an even number of 0s.

The state diagram for M

M = (Q, Σ, δ, q₀, F) where

• Q = {S₁, S₂},
• Σ = {0, 1},
• q₀ = S₁,
• F = {S₁}, and
• δ is defined by the following state transition table:

	0	1
S1	S2	S1
S2	S1	S2

The state S₁ represents that there has been an even number of 0s in the input so far, while S₂ signifies an odd number. A 1 in the input does not change the state of the automaton. When the input ends, the state will show whether the input contained an even number of 0s or not. If the input did contain an even number of 0s, M will finish in state S₁, an accepting state, so the input string will be accepted.

The language recognized by M is the regular language given by the regular expression 1*( 0 (1*) 0 (1*) )*, where "*" is the Kleene star, e.g., 1* denotes any non-negative number (possibly zero) of symbols "1".

• Many everyday activities carried out without a thought are uses of mathematical models. A geographical map projection of a region of the earth onto a small, plane surface is a model^[7] which can be used for many purposes such as planning travel.
• Another simple activity is predicting the position of a vehicle from its initial position, direction and speed of travel, using the equation that distance traveled is the product of time and speed. This is known as dead reckoning when used more formally. Mathematical modeling in this way does not necessarily require formal mathematics; animals have been shown to use dead reckoning.
• Population Growth. A simple (though approximate) model of population growth is the Malthusian growth model. A slightly more realistic and largely used population growth model is the logistic function, and its extensions.
• Model of a particle in a potential-field. In this model we consider a particle as being a point of mass which describes a trajectory in space which is modeled by a function giving its coordinates in space as a function of time. The potential field is given by a function ${\displaystyle V\!:\mathbb {R} ^{3}\!\rightarrow \mathbb {R} }$ and the trajectory, that is a function ${\displaystyle \mathbf {r} \!:\mathbb {R} \rightarrow \mathbb {R} ^{3}}$, is the solution of the differential equation:

${\displaystyle -{\frac {\mathrm {d} ^{2}\mathbf {r} (t)}{\mathrm {d} t^{2}}}m={\frac {\partial V[\mathbf {r} (t)]}{\partial x}}\mathbf {\hat {x}} +{\frac {\partial V[\mathbf {r} (t)]}{\partial y}}\mathbf {\hat {y}} +{\frac {\partial V[\mathbf {r} (t)]}{\partial z}}\mathbf {\hat {z}} ,}$

that can be written also as:

${\displaystyle m{\frac {\mathrm {d} ^{2}\mathbf {r} (t)}{\mathrm {d} t^{2}}}=-\nabla V[\mathbf {r} (t)].}$

Note this model assumes the particle is a point mass, which is certainly known to be false in many cases in which we use this model; for example, as a model of planetary motion.

• Model of rational behavior for a consumer. In this model we assume a consumer faces a choice of n commodities labeled 1,2,...,n each with a market price p₁, p₂,..., p_n. The consumer is assumed to have an ordinal utility function U (ordinal in the sense that only the sign of the differences between two utilities, and not the level of each utility, is meaningful), depending on the amounts of commodities x₁, x₂,..., x_n consumed. The model further assumes that the consumer has a budget M which is used to purchase a vector x₁, x₂,..., x_n in such a way as to maximize U(x₁, x₂,..., x_n). The problem of rational behavior in this model then becomes an optimization problem, that is:

${\displaystyle \max U(x_{1},x_{2},\ldots ,x_{n})}$

subject to:

${\displaystyle \sum _{i=1}^{n}p_{i}x_{i}\leq M.}$

${\displaystyle x_{i}\geq 0\;\;\;\forall i\in \{1,2,\ldots ,n\}}$

This model has been used in a wide variety of economic contexts, such as in general equilibrium theory to show existence and Pareto efficiency of economic equilibria.

• Neighbour-sensing model explains the mushroom formation from the initially chaotic fungal network.
• Computer science: models in Computer Networks, data models, surface model,...
• Mechanics: movement of rocket model,...