Regression toward the mean

From Wikipedia, the free encyclopedia

 

Galton's experimental setup (Fig.8)

 

In statistics, regression toward (or to) the mean is the phenomenon that arises if a random variable is extreme on its first measurement but closer to the mean or average on its second measurement and if it is extreme on its second measurement but closer to the average on its first. To avoid making incorrect inferences, regression toward the mean must be considered when designing scientific experiments and interpreting data. Historically, what is now called regression toward the mean has also been called reversion to the mean and reversion to mediocrity.

The conditions under which regression toward the mean occurs depend on the way the term is mathematically defined. The British polymath Sir Francis Galton first observed the phenomenon in the context of simple linear regression of data points. Galton developed the following model: pellets fall through a quincunx to form a normal distribution centered directly under their entrance point. These pellets might then be released down into a second gallery corresponding to a second measurement. Galton then asked the reverse question: "From where did these pellets come?"

The answer was not 'on average directly above'. Rather it was 'on average, more towards the middle', for the simple reason that there were more pellets above it towards the middle that could wander left than there were in the left extreme that could wander to the right, inwards.

As a less restrictive approach, regression towards the mean can be defined for any bivariate distribution with identical marginal distributions. Two such definitions exist. One definition accords closely with the common usage of the term "regression towards the mean". Not all such bivariate distributions show regression towards the mean under this definition. However, all such bivariate distributions show regression towards the mean under the other definition.

Jeremy Siegel uses the term "return to the mean" to describe a financial time series in which "returns can be very unstable in the short run but very stable in the long run." More quantitatively, it is one in which the standard deviation of average annual returns declines faster than the inverse of the holding period, implying that the process is not a random walk, but that periods of lower returns are systematically followed by compensating periods of higher returns, as is the case in many seasonal businesses, for example.

Conceptual background

Consider a simple example: a class of students takes a 100-item true/false test on a subject. Suppose that all students choose randomly on all questions. Then, each student's score would be a realization of one of a set of independent and identically distributed random variables, with an expected mean of 50. Naturally, some students will score substantially above 50 and some substantially below 50 just by chance. If one takes only the top scoring 10% of the students and gives them a second test on which they again choose randomly on all items, the mean score would again be expected to be close to 50. Thus the mean of these students would "regress" all the way back to the mean of all students who took the original test. No matter what a student scores on the original test, the best prediction of their score on the second test is 50.

If choosing answers to the test questions was not random – i.e. if there were no luck (good or bad) or random guessing involved in the answers supplied by the students – then all students would be expected to score the same on the second test as they scored on the original test, and there would be no regression toward the mean.

Most realistic situations fall between these two extremes: for example, one might consider exam scores as a combination of skill and luck. In this case, the subset of students scoring above average would be composed of those who were skilled and had not especially bad luck, together with those who were unskilled, but were extremely lucky. On a retest of this subset, the unskilled will be unlikely to repeat their lucky break, while the skilled will have a second chance to have bad luck. Hence, those who did well previously are unlikely to do quite as well in the second test even if the original cannot be replicated.

The following is an example of this second kind of regression toward the mean. A class of students takes two editions of the same test on two successive days. It has frequently been observed that the worst performers on the first day will tend to improve their scores on the second day, and the best performers on the first day will tend to do worse on the second day. The phenomenon occurs because student scores are determined in part by underlying ability and in part by chance. For the first test, some will be lucky, and score more than their ability, and some will be unlucky and score less than their ability. Some of the lucky students on the first test will be lucky again on the second test, but more of them will have (for them) average or below average scores. Therefore, a student who was lucky on the first test is more likely to have a worse score on the second test than a better score. Similarly, students who score less than the mean on the first test will tend to see their scores increase on the second test.

Other examples

If your favorite sports team won the championship last year, what does that mean for their chances for winning next season? To the extent this result is due to skill (the team is in good condition, with a top coach, etc.), their win signals that it is more likely they will win again next year. But the greater the extent this is due to luck (other teams embroiled in a drug scandal, favorable draw, draft picks turned out to be productive, etc.), the less likely it is they will win again next year.

If one medical trial suggests that a particular drug or treatment is outperforming all other treatments for a condition, then in a second trial it is more likely that the outperforming drug or treatment will perform closer to the mean.

If a business organisation has a highly profitable quarter, despite the underlying reasons for its performance being unchanged, it is likely to do less well the next quarter.

If the country's GDP jumps in one quarter it is likely not to do as well in the next.

Baseball players who hit well in their rookie season are likely to do worse their 2nd; the "Sophomore slump".

History

The concept of regression comes from genetics and was popularized by Sir Francis Galton during the late 19th century with the publication of Regression towards mediocrity in hereditary stature. Galton observed that extreme characteristics (e.g., height) in parents are not passed on completely to their offspring. Rather, the characteristics in the offspring regress towards a mediocre point (a point which has since been identified as the mean). By measuring the heights of hundreds of people, he was able to quantify regression to the mean, and estimate the size of the effect. Galton wrote that, "the average regression of the offspring is a constant fraction of their respective mid-parental deviations". This means that the difference between a child and its parents for some characteristic is proportional to its parents' deviation from typical people in the population. If its parents are each two inches taller than the averages for men and women, then, on average, the offspring will be shorter than its parents by some factor (which, today, we would call one minus the regression coefficient) times two inches. For height, Galton estimated this coefficient to be about 2/3: the height of an individual will measure around a midpoint that is two thirds of the parents' deviation from the population average. 

Galton coined the term "regression" to describe an observable fact in the inheritance of multi-factorial quantitative genetic traits: namely that the offspring of parents who lie at the tails of the distribution will tend to lie closer to the centre, the mean, of the distribution. He quantified this trend, and in doing so invented linear regression analysis, thus laying the groundwork for much of modern statistical modelling. Since then, the term "regression" has taken on a variety of meanings, and it may be used by modern statisticians to describe phenomena of sampling bias which have little to do with Galton's original observations in the field of genetics.

Though his mathematical analysis was correct, Galton's biological explanation for the regression phenomenon he observed is now known to be incorrect. He stated: "A child inherits partly from his parents, partly from his ancestors. Speaking generally, the further his genealogy goes back, the more numerous and varied will his ancestry become, until they cease to differ from any equally numerous sample taken at haphazard from the race at large." This is incorrect, since a child receives its genetic make-up exclusively from its parents. There is no generation-skipping in genetic material: any genetic material from earlier ancestors must have passed through the parents (though it may not have been expressed in them). The phenomenon is better understood if we assume that the inherited trait (e.g., height) is controlled by a large number of recessive genes. Exceptionally tall individuals must be homozygous for increased height mutations at a large proportion of these loci. But the loci which carry these mutations are not necessarily shared between two tall individuals, and if these individuals mate, their offspring will be on average homozygous for "tall" mutations on fewer loci than either of their parents. In addition, height is not entirely genetically determined, but also subject to environmental influences during development, which make offspring of exceptional parents even more likely to be closer to the average than their parents.

This population genetic phenomenon of regression to the mean is best thought of as a combination of a binomially distributed process of inheritance plus normally distributed environmental influences. In contrast, the term "regression to the mean" is now often used to describe the phenomenon by which an initial sampling bias may disappear as new, repeated, or larger samples display sample means that are closer to the true underlying population mean.

Importance

Regression toward the mean is a significant consideration in the design of experiments.

Take a hypothetical example of 1,000 individuals of a similar age who were examined and scored on the risk of experiencing a heart attack. Statistics could be used to measure the success of an intervention on the 50 who were rated at the greatest risk. The intervention could be a change in diet, exercise, or a drug treatment. Even if the interventions are worthless, the test group would be expected to show an improvement on their next physical exam, because of regression toward the mean. The best way to combat this effect is to divide the group randomly into a treatment group that receives the treatment, and a control group that does not. The treatment would then be judged effective only if the treatment group improves more than the control group. 

Alternatively, a group of disadvantaged children could be tested to identify the ones with most college potential. The top 1% could be identified and supplied with special enrichment courses, tutoring, counseling and computers. Even if the program is effective, their average scores may well be less when the test is repeated a year later. However, in these circumstances it may be considered unethical to have a control group of disadvantaged children whose special needs are ignored. A mathematical calculation for shrinkage can adjust for this effect, although it will not be as reliable as the control group method.

The effect can also be exploited for general inference and estimation. The hottest place in the country today is more likely to be cooler tomorrow than hotter, as compared to today. The best performing mutual fund over the last three years is more likely to see relative performance decline than improve over the next three years. The most successful Hollywood actor of this year is likely to have less gross than more gross for his or her next movie. The baseball player with the greatest batting average by the All-Star break is more likely to have a lower average than a higher average over the second half of the season.

Misunderstandings

The concept of regression toward the mean can be misused very easily. 

In the student test example above, it was assumed implicitly that what was being measured did not change between the two measurements. Suppose, however, that the course was pass/fail and students were required to score above 70 on both tests to pass. Then the students who scored under 70 the first time would have no incentive to do well, and might score worse on average the second time. The students just over 70, on the other hand, would have a strong incentive to study and concentrate while taking the test. In that case one might see movement away from 70, scores below it getting lower and scores above it getting higher. It is possible for changes between the measurement times to augment, offset or reverse the statistical tendency to regress toward the mean.

Statistical regression toward the mean is not a causal phenomenon. A student with the worst score on the test on the first day will not necessarily increase his score substantially on the second day due to the effect. On average, the worst scorers improve, but that is only true because the worst scorers are more likely to have been unlucky than lucky. To the extent that a score is determined randomly, or that a score has random variation or error, as opposed to being determined by the student's academic ability or being a "true value", the phenomenon will have an effect. A classic mistake in this regard was in education. The students that received praise for good work were noticed to do more poorly on the next measure, and the students who were punished for poor work were noticed to do better on the next measure. The educators decided to stop praising and keep punishing on this basis. Such a decision was a mistake, because regression toward the mean is not based on cause and effect, but rather on random error in a natural distribution around a mean.

Although extreme individual measurements regress toward the mean, the second sample of measurements will be no closer to the mean than the first. Consider the students again. Suppose the tendency of extreme individuals is to regress 10% of the way toward the mean of 80, so a student who scored 100 the first day is expected to score 98 the second day, and a student who scored 70 the first day is expected to score 71 the second day. Those expectations are closer to the mean than the first day scores. But the second day scores will vary around their expectations; some will be higher and some will be lower. In addition, individuals that measure very close to the mean should expect to move away from the mean. The effect is the exact reverse of regression toward the mean, and exactly offsets it. So for extreme individuals, we expect the second score to be closer to the mean than the first score, but for all individuals, we expect the distribution of distances from the mean to be the same on both sets of measurements.

Related to the point above, regression toward the mean works equally well in both directions. We expect the student with the highest test score on the second day to have done worse on the first day. And if we compare the best student on the first day to the best student on the second day, regardless of whether it is the same individual or not, there is a tendency to regress toward the mean going in either direction. We expect the best scores on both days to be equally far from the mean.

Regression fallacies

Many phenomena tend to be attributed to the wrong causes when regression to the mean is not taken into account. 

An extreme example is Horace Secrist's 1933 book The Triumph of Mediocrity in Business, in which the statistics professor collected mountains of data to prove that the profit rates of competitive businesses tend toward the average over time. In fact, there is no such effect; the variability of profit rates is almost constant over time. Secrist had only described the common regression toward the mean. One exasperated reviewer, Harold Hotelling, likened the book to "proving the multiplication table by arranging elephants in rows and columns, and then doing the same for numerous other kinds of animals".

The calculation and interpretation of "improvement scores" on standardized educational tests in Massachusetts probably provides another example of the regression fallacy. In 1999, schools were given improvement goals. For each school, the Department of Education tabulated the difference in the average score achieved by students in 1999 and in 2000. It was quickly noted that most of the worst-performing schools had met their goals, which the Department of Education took as confirmation of the soundness of their policies. However, it was also noted that many of the supposedly best schools in the Commonwealth, such as Brookline High School (with 18 National Merit Scholarship finalists) were declared to have failed. As in many cases involving statistics and public policy, the issue is debated, but "improvement scores" were not announced in subsequent years and the findings appear to be a case of regression to the mean.

The psychologist Daniel Kahneman, winner of the 2002 Nobel Memorial Prize in Economic Sciences, pointed out that regression to the mean might explain why rebukes can seem to improve performance, while praise seems to backfire.

“

I had the most satisfying Eureka experience of my career while attempting to teach flight instructors that praise is more effective than punishment for promoting skill-learning. When I had finished my enthusiastic speech, one of the most seasoned instructors in the audience raised his hand and made his own short speech, which began by conceding that positive reinforcement might be good for the birds, but went on to deny that it was optimal for flight cadets. He said, "On many occasions I have praised flight cadets for clean execution of some aerobatic maneuver, and in general when they try it again, they do worse. On the other hand, I have often screamed at cadets for bad execution, and in general they do better the next time. So please don't tell us that reinforcement works and punishment does not, because the opposite is the case." This was a joyous moment, in which I understood an important truth about the world: because we tend to reward others when they do well and punish them when they do badly, and because there is regression to the mean, it is part of the human condition that we are statistically punished for rewarding others and rewarded for punishing them. I immediately arranged a demonstration in which each participant tossed two coins at a target behind his back, without any feedback. We measured the distances from the target and could see that those who had done best the first time had mostly deteriorated on their second try, and vice versa. But I knew that this demonstration would not undo the effects of lifelong exposure to a perverse contingency.

”

To put Kahneman's story in simple terms, when one makes a severe mistake, their performance will later usually return to average level anyway. This will seem as an improvement and as "proof" of a belief that it is better to criticize than to praise (held especially by anyone who is willing to criticize at that "low" moment). In the contrary situation, when one happens to perform high above average, their performance will also tend to return to the average level later on; the change will be perceived as a deterioration and any initial praise following the first performance as a cause of that deterioration. Just because criticizing or praising precedes the regression toward the mean, the act of criticizing or of praising is falsely attributed causality. The regression fallacy is also explained in Rolf Dobelli's The Art of Thinking Clearly. 

UK law enforcement policies have encouraged the visible siting of static or mobile speed cameras at accident blackspots. This policy was justified by a perception that there is a corresponding reduction in serious road traffic accidents after a camera is set up. However, statisticians have pointed out that, although there is a net benefit in lives saved, failure to take into account the effects of regression to the mean results in the beneficial effects being overstated.

Statistical analysts have long recognized the effect of regression to the mean in sports; they even have a special name for it: the "sophomore slump". For example, Carmelo Anthony of the NBA's Denver Nuggets had an outstanding rookie season in 2004. It was so outstanding, in fact, that he could not possibly be expected to repeat it: in 2005, Anthony's numbers had dropped from his rookie season. The reasons for the "sophomore slump" abound, as sports are all about adjustment and counter-adjustment, but luck-based excellence as a rookie is as good a reason as any. Regression to the mean in sports performance may also be the reason for the apparent "Sports Illustrated cover jinx" and the "Madden Curse". John Hollinger has an alternate name for the phenomenon of regression to the mean: the "fluke rule", while Bill James calls it the "Plexiglas Principle".

Because popular lore has focused on regression toward the mean as an account of declining performance of athletes from one season to the next, it has usually overlooked the fact that such regression can also account for improved performance. For example, if one looks at the batting average of Major League Baseball players in one season, those whose batting average was above the league mean tend to regress downward toward the mean the following year, while those whose batting average was below the mean tend to progress upward toward the mean the following year.

Other statistical phenomena

Regression toward the mean simply says that, following an extreme random event, the next random event is likely to be less extreme. In no sense does the future event "compensate for" or "even out" the previous event, though this is assumed in the gambler's fallacy (and the variant law of averages). Similarly, the law of large numbers states that in the long term, the average will tend towards the expected value, but makes no statement about individual trials. For example, following a run of 10 heads on a flip of a fair coin (a rare, extreme event), regression to the mean states that the next run of heads will likely be less than 10, while the law of large numbers states that in the long term, this event will likely average out, and the average fraction of heads will tend to 1/2. By contrast, the gambler's fallacy incorrectly assumes that the coin is now "due" for a run of tails to balance out.

Definition for simple linear regression of data points

This is the definition of regression toward the mean that closely follows Sir Francis Galton's original usage.

Suppose there are n data points {y_i, x_i}, where i = 1, 2, …, n. We want to find the equation of the regression line, i.e. the straight line

${\displaystyle y=\alpha +\beta x,\,}$

which would provide a "best" fit for the data points. (Note that a straight line may not be the appropriate regression curve for the given data points.) Here the "best" will be understood as in the least-squares approach: such a line that minimizes the sum of squared residuals of the linear regression model. In other words, numbers α and β solve the following minimization problem:

Find ${\displaystyle \min _{\alpha ,\,\beta }Q(\alpha ,\beta )}$, where ${\displaystyle Q(\alpha ,\beta )=\sum _{i=1}^{n}{\hat {\varepsilon }}_{i}^{\,2}=\sum _{i=1}^{n}(y_{i}-\alpha -\beta x_{i})^{2}\ }$

Using calculus it can be shown that the values of α and β that minimize the objective function Q are

${\displaystyle {\begin{aligned}&{\hat {\beta }}={\frac {\sum _{i=1}^{n}(x_{i}-{\bar {x}})(y_{i}-{\bar {y}})}{\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}}={\frac {{\overline {xy}}-{\bar {x}}{\bar {y}}}{{\overline {x^{2}}}-{\bar {x}}^{2}}}={\frac {\operatorname {Cov} [x,y]}{\operatorname {Var} [x]}}=r_{xy}{\frac {s_{y}}{s_{x}}},\\&{\hat {\alpha }}={\bar {y}}-{\hat {\beta }}\,{\bar {x}},\end{aligned}}}$

where r_xy is the sample correlation coefficient between x and y, s_x is the standard deviation of x, and s_y is correspondingly the standard deviation of y. Horizontal bar over a variable means the sample average of that variable. For example:

 ${\displaystyle {\overline {xy}}={\tfrac {1}{n}}\textstyle \sum _{i=1}^{n}x_{i}y_{i}\ .}$

Substituting the above expressions for ${\displaystyle {\hat {\alpha }}}$ and ${\displaystyle {\hat {\beta }}}$ into ${\displaystyle y=\alpha +\beta x,\,}$ yields fitted values

${\displaystyle {\hat {y}}={\hat {\alpha }}+{\hat {\beta }}x,\,}$

which yields

${\displaystyle {\frac {{\hat {y}}-{\bar {y}}}{s_{y}}}=r_{xy}{\frac {x-{\bar {x}}}{s_{x}}}}$

This shows the role r_xy plays in the regression line of standardized data points.

If −1 < r_xy < 1, then we say that the data points exhibit regression toward the mean. In other words, if linear regression is the appropriate model for a set of data points whose sample correlation coefficient is not perfect, then there is regression toward the mean. The predicted (or fitted) standardized value of y is closer to its mean than the standardized value of x is to its mean.

Definitions for bivariate distribution with identical marginal distributions

Restrictive definition

Let X₁, X₂ be random variables with identical marginal distributions with mean μ. In this formalization, the bivariate distribution of X₁ and X₂ is said to exhibit regression toward the mean if, for every number c > μ, we have

μ ≤ E[X₂ | X₁ = c] < c,

with the reverse inequalities holding for c < μ.

The following is an informal description of the above definition. Consider a population of widgets. Each widget has two numbers, X₁ and X₂ (say, its left span (X₁ ) and right span (X₂)). Suppose that the probability distributions of X₁ and X₂ in the population are identical, and that the means of X₁ and X₂ are both μ. We now take a random widget from the population, and denote its X₁ value by c. (Note that c may be greater than, equal to, or smaller than μ.) We have no access to the value of this widget's X₂ yet. Let d denote the expected value of X₂ of this particular widget. (i.e. Let d denote the average value of X₂ of all widgets in the population with X₁=c.) If the following condition is true:

Whatever the value c is, d lies between μ and c (i.e. d is closer to μ than c is),

then we say that X₁ and X₂ show regression toward the mean. 

This definition accords closely with the current common usage, evolved from Galton's original usage, of the term "regression toward the mean." It is "restrictive" in the sense that not every bivariate distribution with identical marginal distributions exhibits regression toward the mean (under this definition).

Theorem

If a pair (X, Y) of random variables follows a bivariate normal distribution, then the conditional mean E(Y|X) is a linear function of X. The correlation coefficient r between X and Y, along with the marginal means and variances of X and Y, determines this linear relationship:

${\displaystyle {\frac {E(Y\mid X)-E[Y]}{\sigma _{y}}}=r{\frac {X-E[X]}{\sigma _{x}}},}$

where E[X] and E[Y] are the expected values of X and Y, respectively, and σ_x and σ_y are the standard deviations of X and Y, respectively. 

Hence the conditional expected value of Y, given that X is t standard deviations above its mean (and that includes the case where it's below its mean, when t < 0), is rt standard deviations above the mean of Y. Since |r| ≤ 1, Y is no farther from the mean than X is, as measured in the number of standard deviations.

Hence, if 0 ≤ r < 1, then (X, Y) shows regression toward the mean (by this definition).

General definition

The following definition of reversion toward the mean has been proposed by Samuels as an alternative to the more restrictive definition of regression toward the mean above.

Let X₁, X₂ be random variables with identical marginal distributions with mean μ. In this formalization, the bivariate distribution of X₁ and X₂ is said to exhibit reversion toward the mean if, for every number c, we have

μ ≤ E[X₂ | X₁ > c] < E[X₁ | X₁ > c], and

μ ≥ E[X₂ | X₁ < c] > E[X₁ | X₁ < c]

This definition is "general" in the sense that every bivariate distribution with identical marginal distributions exhibits reversion toward the mean.

Gambler's fallacy

From Wikipedia, the free encyclopedia

 

The gambler's fallacy, also known as the Monte Carlo fallacy or the fallacy of the maturity of chances, is the mistaken belief that if something happens more frequently than normal during a given period, it will happen less frequently in the future (or vice versa). In situations where the outcome being observed is truly random and consists of independent trials of a random process, this belief is false. The fallacy can arise in many situations, but is most strongly associated with gambling, where it is common among players.

The term "Monte Carlo fallacy" originates from the best known example of the phenomenon, which occurred in the Monte Carlo Casino in 1913.

Examples

Coin toss

Simulation of coin tosses: Each frame, a coin is flipped which is red on one side and blue on the other. The result of each flip is added as a coloured dot in the corresponding column. As the pie chart shows, the proportion of red versus blue approaches 50-50 (the law of large numbers). But the difference between red and blue does not systematically decrease to zero.

The gambler's fallacy can be illustrated by considering the repeated toss of a fair coin. The outcomes in different tosses are statistically independent and the probability of getting heads on a single toss is 1/2 (one in two). The probability of getting two heads in two tosses is 1/4 (one in four) and the probability of getting three heads in three tosses is 1/8 (one in eight). In general, if A_i is the event where toss i of a fair coin comes up heads, then:

${\displaystyle \Pr \left(\bigcap _{i=1}^{n}A_{i}\right)=\prod _{i=1}^{n}\Pr(A_{i})={1 \over 2^{n}}}$.

If after tossing four heads in a row, the next coin toss also came up heads, it would complete a run of five successive heads. Since the probability of a run of five successive heads is 1/32 (one in thirty-two), a person might believe that the next flip would be more likely to come up tails rather than heads again. This is incorrect and is an example of the gambler's fallacy. The event "5 heads in a row" and the event "first 4 heads, then a tails" are equally likely, each having probability 1/32. Since the first four tosses turn up heads, the probability that the next toss is a head is:

${\displaystyle \Pr \left(A_{5}|A_{1}\cap A_{2}\cap A_{3}\cap A_{4}\right)=\Pr \left(A_{5}\right)={\frac {1}{2}}}$.

While a run of five heads has a probability of 1/32 = 0.03125 (a little over 3%), the misunderstanding lies in not realizing that this is the case only before the first coin is tossed. After the first four tosses, the results are no longer unknown, so their probabilities are at that point equal to 1 (100%). The reasoning that it is more likely that a fifth toss is more likely to be tails because the previous four tosses were heads, with a run of luck in the past influencing the odds in the future, forms the basis of the fallacy.

Why the probability is 1/2 for a fair coin

If a fair coin is flipped 21 times, the probability of 21 heads is 1 in 2,097,152. The probability of flipping a head after having already flipped 20 heads in a row is 1/2. This is an application of Bayes' theorem.

This can also be shown without knowing that 20 heads have occurred, and without applying Bayes' theorem. Assuming a fair coin:

• The probability of 20 heads, then 1 tail is 0.5²⁰ × 0.5 = 0.5²¹
• The probability of 20 heads, then 1 head is 0.5²⁰ × 0.5 = 0.5²¹

The probability of getting 20 heads then 1 tail, and the probability of getting 20 heads then another head are both 1 in 2,097,152. When flipping a fair coin 21 times, the outcome is equally likely to be 21 heads as 20 heads and then 1 tail. These two outcomes are equally as likely as any of the other combinations that can be obtained from 21 flips of a coin. All of the 21-flip combinations will have probabilities equal to 0.5²¹, or 1 in 2,097,152. Assuming that a change in the probability will occur as a result of the outcome of prior flips is incorrect because every outcome of a 21-flip sequence is as likely as the other outcomes. In accordance with Bayes' theorem, the likely outcome of each flip is the probability of the fair coin, which is 1/2.

Other examples

The fallacy leads to the incorrect notion that previous failures will create an increased probability of success on subsequent attempts. For a fair 16-sided die, the probability of each outcome occurring is 1/16 (6.25%). If a win is defined as rolling a 1, the probability of a 1 occurring at least once in 16 rolls is:

${\displaystyle 1-\left[{\frac {15}{16}}\right]^{16}\,=\,64.39\%}$

The probability of a loss on the first roll is 15/16 (93.75%). According to the fallacy, the player should have a higher chance of winning after one loss has occurred. The probability of at least one win is now:

${\displaystyle 1-\left[{\frac {15}{16}}\right]^{15}\,=\,62.02\%}$

By losing one toss, the player's probability of winning drops by two percentage points. With 5 losses and 11 rolls remaining, the probability of winning drops to around 0.5 (50%). The probability of at least one win does not increase after a series of losses; indeed, the probability of success actually decreases, because there are fewer trials left in which to win. The probability of winning will eventually equal the probability of winning a single toss, which is 1/16 (6.25%) and occurs when only one toss is left.

Reverse position

After a consistent tendency towards tails, a gambler may also decide that tails has become a more likely outcome. This is a rational and Bayesian conclusion, bearing in mind the possibility that the coin may not be fair; it is not a fallacy. Believing the odds to favor tails, the gambler sees no reason to change to heads. However it is a fallacy that a sequence of trials carries a memory of past results which tend to favor or disfavor future outcomes.

The inverse gambler's fallacy described by Ian Hacking is a situation where a gambler entering a room and seeing a person rolling a double six on a pair of dice may erroneously conclude that the person must have been rolling the dice for quite a while, as they would be unlikely to get a double six on their first attempt.

Retrospective gambler's fallacy

Researchers have examined whether a similar bias exists for inferences about unknown past events based upon known subsequent events, calling this the "retrospective gambler's fallacy".

An example of a retrospective gambler's fallacy would be to observe multiple successive "heads" on a coin toss and conclude from this that the previously unknown flip was "tails". Real world examples of retrospective gambler's fallacy have been argued to exist in events such as the origin of the Universe. In his book Universes, John Leslie argues that "the presence of vastly many universes very different in their characters might be our best explanation for why at least one universe has a life-permitting character". Daniel M. Oppenheimer and Benoît Monin argue that "In other words, the 'best explanation' for a low-probability event is that it is only one in a multiple of trials, which is the core intuition of the reverse gambler's fallacy." Philosophical arguments are ongoing about whether such arguments are or are not a fallacy, arguing that the occurrence of our universe says nothing about the existence of other universes or trials of universes. Three studies involving Stanford University students tested the existence of a retrospective gamblers' fallacy. All three studies concluded that people have a gamblers' fallacy retrospectively as well as to future events. The authors of all three studies concluded their findings have significant "methodological implications" but may also have "important theoretical implications" that need investigation and research, saying "[a] thorough understanding of such reasoning processes requires that we not only examine how they influence our predictions of the future, but also our perceptions of the past."

Childbirth

In 1796, Pierre-Simon Laplace described in A Philosophical Essay on Probabilities the ways in which men calculated their probability of having sons: "I have seen men, ardently desirous of having a son, who could learn only with anxiety of the births of boys in the month when they expected to become fathers. Imagining that the ratio of these births to those of girls ought to be the same at the end of each month, they judged that the boys already born would render more probable the births next of girls." The expectant fathers feared that if more sons were born in the surrounding community, then they themselves would be more likely to have a daughter. This essay by Laplace is regarded as one of the earliest descriptions of the fallacy.

After having multiple children of the same sex, some parents may believe that they are due to have a child of the opposite sex. While the Trivers–Willard hypothesis predicts that birth sex is dependent on living conditions, stating that more male children are born in good living conditions, while more female children are born in poorer living conditions, the probability of having a child of either sex is still regarded as near 0.5 (50%).

Monte Carlo Casino

Perhaps the most famous example of the gambler's fallacy occurred in a game of roulette at the Monte Carlo Casino on August 18, 1913, when the ball fell in black 26 times in a row. This was an extremely uncommon occurrence: the probability of a sequence of either red or black occurring 26 times in a row is (18/37)^26-1 or around 1 in 66.6 million, assuming the mechanism is unbiased. Gamblers lost millions of francs betting against black, reasoning incorrectly that the streak was causing an imbalance in the randomness of the wheel, and that it had to be followed by a long streak of red.

Non-examples

Non-independent events

The gambler's fallacy does not apply in situations where the probability of different events is not independent. In such cases, the probability of future events can change based on the outcome of past events, such as the statistical permutation of events. An example is when cards are drawn from a deck without replacement. If an ace is drawn from a deck and not reinserted, the next draw is less likely to be an ace and more likely to be of another rank. The probability of drawing another ace, assuming that it was the first card drawn and that there are no jokers, has decreased from 4/52 (7.69%) to 3/51 (5.88%), while the probability for each other rank has increased from 4/52 (7.69%) to 4/51 (7.84%). This effect allows card counting systems to work in games such as blackjack.

Bias

In most illustrations of the gambler's fallacy and the reverse gambler's fallacy, the trial (e.g. flipping a coin) is assumed to be fair. In practice, this assumption may not hold. For example, if a coin is flipped 21 times, the probability of 21 heads with a fair coin is 1 in 2,097,152. Since this probability is so small, if it happens, it may well be that the coin is somehow biased towards landing on heads, or that it is being controlled by hidden magnets, or similar. In this case, the smart bet is "heads" because Bayesian inference from the empirical evidence — 21 heads in a row — suggests that the coin is likely to be biased toward heads. Bayesian inference can be used to show that when the long-run proportion of different outcomes is unknown but exchangeable (meaning that the random process from which the outcomes are generated may be biased but is equally likely to be biased in any direction) and that previous observations demonstrate the likely direction of the bias, the outcome which has occurred the most in the observed data is the most likely to occur again.

For example, if the a priori probability of a biased coin is say 1%, and assuming that such a biased coin would come down heads say 60% of the time, then after 21 heads the probability of a biased coin has increased to about 32%.

The opening scene of the play Rosencrantz and Guildenstern Are Dead by Tom Stoppard discusses these issues as one man continually flips heads and the other considers various possible explanations.

Changing probabilities

If external factors are allowed to change the probability of the events, the gambler's fallacy may not hold. For example, a change in the game rules might favour one player over the other, improving his or her win percentage. Similarly, an inexperienced player's success may decrease after opposing teams learn about and play against their weaknesses. This is another example of bias.

Psychology

Origins

The gambler's fallacy arises out of a belief in a law of small numbers, leading to the erroneous belief that small samples must be representative of the larger population. According to the fallacy, streaks must eventually even out in order to be representative. Amos Tversky and Daniel Kahneman first proposed that the gambler's fallacy is a cognitive bias produced by a psychological heuristic called the representativeness heuristic, which states that people evaluate the probability of a certain event by assessing how similar it is to events they have experienced before, and how similar the events surrounding those two processes are. According to this view, "after observing a long run of red on the roulette wheel, for example, most people erroneously believe that black will result in a more representative sequence than the occurrence of an additional red", so people expect that a short run of random outcomes should share properties of a longer run, specifically in that deviations from average should balance out. When people are asked to make up a random-looking sequence of coin tosses, they tend to make sequences where the proportion of heads to tails stays closer to 0.5 in any short segment than would be predicted by chance, a phenomenon known as insensitivity to sample size. Kahneman and Tversky interpret this to mean that people believe short sequences of random events should be representative of longer ones. The representativeness heuristic is also cited behind the related phenomenon of the clustering illusion, according to which people see streaks of random events as being non-random when such streaks are actually much more likely to occur in small samples than people expect.

The gambler's fallacy can also be attributed to the mistaken belief that gambling, or even chance itself, is a fair process that can correct itself in the event of streaks, known as the just-world hypothesis. Other researchers believe that belief in the fallacy may be the result of a mistaken belief in an internal locus of control. When a person believes that gambling outcomes are the result of their own skill, they may be more susceptible to the gambler's fallacy because they reject the idea that chance could overcome skill or talent.

Variations

Some researchers believe that it is possible to define two types of gambler's fallacy: type one and type two. Type one is the classic gambler's fallacy, where individuals believe that a particular outcome is due after a long streak of another outcome. Type two gambler's fallacy, as defined by Gideon Keren and Charles Lewis, occurs when a gambler underestimates how many observations are needed to detect a favorable outcome, such as watching a roulette wheel for a length of time and then betting on the numbers that appear most often. For events with a high degree of randomness, detecting a bias that will lead to a favorable outcome takes an impractically large amount of time and is very difficult, if not impossible, to do. The two types differ in that type one wrongly assumes that gambling conditions are fair and perfect, while type two assumes that the conditions are biased, and that this bias can be detected after a certain amount of time.

Another variety, known as the retrospective gambler's fallacy, occurs when individuals judge that a seemingly rare event must come from a longer sequence than a more common event does. The belief that an imaginary sequence of die rolls is more than three times as long when a set of three sixes is observed as opposed to when there are only two sixes. This effect can be observed in isolated instances, or even sequentially. Another example would involve hearing that a teenager has unprotected sex and becomes pregnant on a given night, and concluding that she has been engaging in unprotected sex for longer than if we hear she had unprotected sex but did not become pregnant, when the probability of becoming pregnant as a result of each intercourse is independent of the amount of prior intercourse.

Relationship to hot-hand fallacy

Another psychological perspective states that gambler's fallacy can be seen as the counterpart to basketball's hot-hand fallacy, in which people tend to predict the same outcome as the previous event - known as positive recency - resulting in a belief that a high scorer will continue to score. In the gambler's fallacy, people predict the opposite outcome of the previous event - negative recency - believing that since the roulette wheel has landed on black on the previous six occasions, it is due to land on red the next. Ayton and Fischer have theorized that people display positive recency for the hot-hand fallacy because the fallacy deals with human performance, and that people do not believe that an inanimate object can become "hot." Human performance is not perceived as random, and people are more likely to continue streaks when they believe that the process generating the results is nonrandom. When a person exhibits the gambler's fallacy, they are more likely to exhibit the hot-hand fallacy as well, suggesting that one construct is responsible for the two fallacies.

The difference between the two fallacies is also found in economic decision-making. A study by Huber, Kirchler, and Stockl in 2010 examined how the hot hand and the gambler's fallacy are exhibited in the financial market. The researchers gave their participants a choice: they could either bet on the outcome of a series of coin tosses, use an expert opinion to sway their decision, or choose a risk-free alternative instead for a smaller financial reward. Participants turned to the expert opinion to make their decision 24% of the time based on their past experience of success, which exemplifies the hot-hand. If the expert was correct, 78% of the participants chose the expert's opinion again, as opposed to 57% doing so when the expert was wrong. The participants also exhibited the gambler's fallacy, with their selection of either heads or tails decreasing after noticing a streak of either outcome. This experiment helped bolster Ayton and Fischer's theory that people put more faith in human performance than they do in seemingly random processes.

Neurophysiology

While the representativeness heuristic and other cognitive biases are the most commonly cited cause of the gambler's fallacy, research suggests that there may also be a neurological component. Functional magnetic resonance imaging has shown that after losing a bet or gamble, known as riskloss, the frontoparietal network of the brain is activated, resulting in more risk-taking behavior. In contrast, there is decreased activity in the amygdala, caudate, and ventral striatum after a riskloss. Activation in the amygdala is negatively correlated with gambler's fallacy, so that the more activity exhibited in the amygdala, the less likely an individual is to fall prey to the gambler's fallacy. These results suggest that gambler's fallacy relies more on the prefrontal cortex, which is responsible for executive, goal-directed processes, and less on the brain areas that control affective decision-making.

The desire to continue gambling or betting is controlled by the striatum, which supports a choice-outcome contingency learning method. The striatum processes the errors in prediction and the behavior changes accordingly. After a win, the positive behavior is reinforced and after a loss, the behavior is conditioned to be avoided. In individuals exhibiting the gambler's fallacy, this choice-outcome contingency method is impaired, and they continue to make risks after a series of losses.

Possible solutions

The gambler's fallacy is a deep-seated cognitive bias and can be very hard to overcome. Educating individuals about the nature of randomness has not always proven effective in reducing or eliminating any manifestation of the fallacy. Participants in a study by Beach and Swensson in 1967 were shown a shuffled deck of index cards with shapes on them, and were instructed to guess which shape would come next in a sequence. The experimental group of participants was informed about the nature and existence of the gambler's fallacy, and were explicitly instructed not to rely on run dependency to make their guesses. The control group was not given this information. The response styles of the two groups were similar, indicating that the experimental group still based their choices on the length of the run sequence. This led to the conclusion that instructing individuals about randomness is not sufficient in lessening the gambler's fallacy.

An individual's susceptibility to the gambler's fallacy may decrease with age. A study by Fischbein and Schnarch in 1997 administered a questionnaire to five groups: students in grades 5, 7, 9, 11, and college students specializing in teaching mathematics. None of the participants had received any prior education regarding probability. The question asked was: "Ronni flipped a coin three times and in all cases heads came up. Ronni intends to flip the coin again. What is the chance of getting heads the fourth time?" The results indicated that as the students got older, the less likely they were to answer with "smaller than the chance of getting tails", which would indicate a negative recency effect. 35% of the 5th graders, 35% of the 7th graders, and 20% of the 9th graders exhibited the negative recency effect. Only 10% of the 11th graders answered this way, and none of the college students did. Fischbein and Schnarch theorized that an individual's tendency to rely on the representativeness heuristic and other cognitive biases can be overcome with age.

Another possible solution comes from Roney and Trick, Gestalt psychologists who suggest that the fallacy may be eliminated as a result of grouping. When a future event such as a coin toss is described as part of a sequence, no matter how arbitrarily, a person will automatically consider the event as it relates to the past events, resulting in the gambler's fallacy. When a person considers every event as independent, the fallacy can be greatly reduced.

Roney and Trick told participants in their experiment that they were betting on either two blocks of six coin tosses, or on two blocks of seven coin tosses. The fourth, fifth, and sixth tosses all had the same outcome, either three heads or three tails. The seventh toss was grouped with either the end of one block, or the beginning of the next block. Participants exhibited the strongest gambler's fallacy when the seventh trial was part of the first block, directly after the sequence of three heads or tails. The researchers pointed out that the participants that did not show the gambler's fallacy showed less confidence in their bets and bet fewer times than the participants who picked with the gambler's fallacy. When the seventh trial was grouped with the second block, and was perceived as not being part of a streak, the gambler's fallacy did not occur. 

Roney and Trick argued that instead of teaching individuals about the nature of randomness, the fallacy could be avoided by training people to treat each event as if it is a beginning and not a continuation of previous events. They suggested that this would prevent people from gambling when they are losing, in the mistaken hope that their chances of winning are due to increase based on an interaction with previous events.

Users

Studies have found that asylum judges, loan officers, baseball umpires and lotto players employ the gambler's fallacy consistently in their decision-making.

Illusion of control

From Wikipedia, the free encyclopedia

 

The illusion of control is the tendency for people to overestimate their ability to control events; for example, it occurs when someone feels a sense of control over outcomes that they demonstrably do not influence. The effect was named by psychologist Ellen Langer and has been replicated in many different contexts. It is thought to influence gambling behavior and belief in the paranormal. Along with illusory superiority and optimism bias, the illusion of control is one of the positive illusions.

The illusion might arise because people lack direct introspective insight into whether they are in control of events. This has been called the introspection illusion. Instead they may judge their degree of control by a process that is often unreliable. As a result, they see themselves as responsible for events when there is little or no causal link. In one study, college students were in a virtual reality setting to treat a fear of heights using an elevator. Those who were told that they had control, yet had none, felt as though they had as much control as those who actually did have control over the elevator. Those who were led to believe they did not have control said they felt as though they had little control.

Psychological theorists have consistently emphasized the importance of perceptions of control over life events. One of the earliest instances of this is when Adler argued that people strive for proficiency in their lives. Heider later proposed that humans have a strong motive to control their environment and Wyatt Mann hypothesized a basic competence motive that people satisfy by exerting control. Wiener, an attribution theorist, modified his original theory of achievement motivation to include a controllability dimension. Kelley then argued that people's failure to detect noncontingencies may result in their attributing uncontrollable outcomes to personal causes. Nearer to the present, Taylor and Brown argued that positive illusions, including the illusion of control, foster mental health.

The illusion is more common in familiar situations, and in situations where the person knows the desired outcome. Feedback that emphasizes success rather than failure can increase the effect, while feedback that emphasizes failure can decrease or reverse the effect. The illusion is weaker for depressed individuals and is stronger when individuals have an emotional need to control the outcome. The illusion is strengthened by stressful and competitive situations, including financial trading. Although people are likely to overestimate their control when the situations are heavily chance-determined, they also tend to underestimate their control when they actually have it, which runs contrary to some theories of the illusion and its adaptiveness. People also showed a higher illusion of control when they were allowed to become familiar with a task through practice trials, make their choice before the event happens like with throwing dice, and when they can make their choice rather than have it made for them with the same odds. People are more likely to show control when they have more answers right at the beginning than at the end, even when the people had the same number of correct answers.

By proxy

At times, people attempt to gain control by transferring responsibility to more capable or “luckier” others to act for them. By forfeiting direct control, it is perceived to be a valid way of maximizing outcomes. This illusion of control by proxy is a significant theoretical extension of the traditional illusion of control model. People will of course give up control if another person is thought to have more knowledge or skill in areas such as medicine where actual skill and knowledge are involved. In cases like these it is entirely rational to give up responsibility to people such as doctors. However, when it comes to events of pure chance, allowing another to make decisions (or gamble) on one's behalf, because they are seen as luckier is not rational and would go against people's well-documented desire for control in uncontrollable situations. However, it does seem plausible since people generally believe that they can possess luck and employ it to advantage in games of chance, and it is not a far leap that others may also be seen as lucky and able to control uncontrollable events.

In one instance, a lottery pool at a company decides who picks the numbers and buys the tickets based on the wins and losses of each member. The member with the best record becomes the representative until they accumulate a certain number of losses and then a new representative is picked based on wins and losses. Even though no member is truly better than the other and it is all by chance, they still would rather have someone with seemingly more luck to have control over them.

In another real-world example, in the 2002 Olympics men's and women's hockey finals, Team Canada beat Team USA but it was later believed that the win was the result of the luck of a Canadian coin that was secretly placed under the ice before the game. The members of Team Canada were the only people who knew the coin had been placed there. The coin was later put in the Hockey Hall of Fame where there was an opening so people could touch it. People believed they could transfer luck from the coin to themselves by touching it, and thereby change their own luck.

Demonstration

The illusion of control is demonstrated by three converging lines of evidence: 1) laboratory experiments, 2) observed behavior in familiar games of chance such as lotteries, and 3) self-reports of real-world behavior.

One kind of laboratory demonstration involves two lights marked "Score" and "No Score". Subjects have to try to control which one lights up. In one version of this experiment, subjects could press either of two buttons. Another version had one button, which subjects decided on each trial to press or not. Subjects had a variable degree of control over the lights, or none at all, depending on how the buttons were connected. The experimenters made clear that there might be no relation between the subjects' actions and the lights. Subjects estimated how much control they had over the lights. These estimates bore no relation to how much control they actually had, but was related to how often the "Score" light lit up. Even when their choices made no difference at all, subjects confidently reported exerting some control over the lights.

Ellen Langer's research demonstrated that people were more likely to behave as if they could exercise control in a chance situation where "skill cues" were present. By skill cues, Langer meant properties of the situation more normally associated with the exercise of skill, in particular the exercise of choice, competition, familiarity with the stimulus and involvement in decisions. One simple form of this effect is found in casinos: when rolling dice in a craps game people tend to throw harder when they need high numbers and softer for low numbers.

In another experiment, subjects had to predict the outcome of thirty coin tosses. The feedback was rigged so that each subject was right exactly half the time, but the groups differed in where their "hits" occurred. Some were told that their early guesses were accurate. Others were told that their successes were distributed evenly through the thirty trials. Afterwards, they were surveyed about their performance. Subjects with early "hits" overestimated their total successes and had higher expectations of how they would perform on future guessing games. This result resembles the irrational primacy effect in which people give greater weight to information that occurs earlier in a series. Forty percent of the subjects believed their performance on this chance task would improve with practice, and twenty-five percent said that distraction would impair their performance.

Another of Langer's experiments—replicated by other researchers—involves a lottery. Subjects are either given tickets at random or allowed to choose their own. They can then trade their tickets for others with a higher chance of paying out. Subjects who had chosen their own ticket were more reluctant to part with it. Tickets bearing familiar symbols were less likely to be exchanged than others with unfamiliar symbols. Although these lotteries were random, subjects behaved as though their choice of ticket affected the outcome. Participants who chose their own numbers were less likely to trade their ticket even for one in a game with better odds.

Another way to investigate perceptions of control is to ask people about hypothetical situations, for example their likelihood of being involved in a motor vehicle accident. On average, drivers regard accidents as much less likely in "high-control" situations, such as when they are driving, than in "low-control" situations, such as when they are in the passenger seat. They also rate a high-control accident, such as driving into the car in front, as much less likely than a low-control accident such as being hit from behind by another driver.

Explanations

Ellen Langer, who first demonstrated the illusion of control, explained her findings in terms of a confusion between skill and chance situations. She proposed that people base their judgments of control on "skill cues". These are features of a situation that are usually associated with games of skill, such as competitiveness, familiarity and individual choice. When more of these skill cues are present, the illusion is stronger.

Suzanne Thompson and colleagues argued that Langer's explanation was inadequate to explain all the variations in the effect. As an alternative, they proposed that judgments about control are based on a procedure that they called the "control heuristic". This theory proposes that judgments of control to depend on two conditions; an intention to create the outcome, and a relationship between the action and outcome. In games of chance, these two conditions frequently go together. As well as an intention to win, there is an action, such as throwing a die or pulling a lever on a slot machine, which is immediately followed by an outcome. Even though the outcome is selected randomly, the control heuristic would result in the player feeling a degree of control over the outcome.

Self-regulation theory offers another explanation. To the extent that people are driven by internal goals concerned with the exercise of control over their environment, they will seek to reassert control in conditions of chaos, uncertainty or stress. One way of coping with a lack of real control is to falsely attribute oneself control of the situation.

The core self-evaluations (CSE) trait is a stable personality trait composed of locus of control, neuroticism, self-efficacy, and self-esteem. While those with high core self-evaluations are likely to believe that they control their own environment (i.e., internal locus of control), very high levels of CSE may lead to the illusion of control.

Benefits and costs to the individual

Taylor and Brown have argued that positive illusions, including the illusion of control, are adaptive as they motivate people to persist at tasks when they might otherwise give up. This position is supported by Albert Bandura's claim that "optimistic self-appraisals of capability, that are not unduly disparate from what is possible, can be advantageous, whereas veridical judgements can be self-limiting". His argument is essentially concerned with the adaptive effect of optimistic beliefs about control and performance in circumstances where control is possible, rather than perceived control in circumstances where outcomes do not depend on an individual's behavior.

Bandura has also suggested that:

"In activities where the margins of error are narrow and missteps can produce costly or injurious consequences, personal well-being is best served by highly accurate efficacy appraisal."

Taylor and Brown argue that positive illusions are adaptive, since there is evidence that they are more common in normally mentally healthy individuals than in depressed individuals. However, Pacini, Muir and Epstein have shown that this may be because depressed people overcompensate for a tendency toward maladaptive intuitive processing by exercising excessive rational control in trivial situations, and note that the difference with non-depressed people disappears in more consequential circumstances.

There is also empirical evidence that high self-efficacy can be maladaptive in some circumstances. In a scenario-based study, Whyte et al. showed that participants in whom they had induced high self-efficacy were significantly more likely to escalate commitment to a failing course of action. Knee and Zuckerman have challenged the definition of mental health used by Taylor and Brown and argue that lack of illusions is associated with a non-defensive personality oriented towards growth and learning and with low ego involvement in outcomes. They present evidence that self-determined individuals are less prone to these illusions. In the late 1970s, Abramson and Alloy demonstrated that depressed individuals held a more accurate view than their non-depressed counterparts in a test which measured illusion of control. This finding held true even when the depression was manipulated experimentally. However, when replicating the findings Msetfi et al. (2005, 2007) found that the overestimation of control in nondepressed people only showed up when the interval was long enough, implying that this is because they take more aspects of a situation into account than their depressed counterparts. Also, Dykman et al. (1989) showed that depressed people believe they have no control in situations where they actually do, so their perception is not more accurate overall. Allan et al. (2007) has proposed that the pessimistic bias of depressives resulted in "depressive realism" when asked about estimation of control, because depressed individuals are more likely to say no even if they have control.

A number of studies have found a link between a sense of control and health, especially in older people.

Fenton-O'Creevy et al. argue, as do Gollwittzer and Kinney, that while illusory beliefs about control may promote goal striving, they are not conducive to sound decision-making. Illusions of control may cause insensitivity to feedback, impede learning and predispose toward greater objective risk taking (since subjective risk will be reduced by illusion of control).

Applications

Psychologist Daniel Wegner argues that an illusion of control over external events underlies belief in psychokinesis, a supposed paranormal ability to move objects directly using the mind. As evidence, Wegner cites a series of experiments on magical thinking in which subjects were induced to think they had influenced external events. In one experiment, subjects watched a basketball player taking a series of free throws. When they were instructed to visualise him making his shots, they felt that they had contributed to his success.

One study examined traders working in the City of London's investment banks. They each watched a graph being plotted on a computer screen, similar to a real-time graph of a stock price or index. Using three computer keys, they had to raise the value as high as possible. They were warned that the value showed random variations, but that the keys might have some effect. In fact, the fluctuations were not affected by the keys. The traders' ratings of their success measured their susceptibility to the illusion of control. This score was then compared with each trader's performance. Those who were more prone to the illusion scored significantly lower on analysis, risk management and contribution to profits. They also earned significantly less.