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Example
David Hamilton and Robert Gifford (1976) conducted a series of experiments that demonstrated how stereotypic beliefs regarding minorities could derive from inaccurate ideas from majority group members. To test their hypothesis, Hamilton and Gifford had research participants read a series of 39 sentences, which were associated with either Group A or B. Abstract groups were used so no previous stereotypes would influence results. Twenty-six out of the 39 sentences were associated with Group A, making it the majority, and 13 were associated with Group B, making it the minority. Positive behaviors were represented in 27 sentences and negative behaviors were described in 12. The following table summarises the information given.

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 real life example of Illusory Correlations is discussed by Michael Smith and Geoffrey Alpert. They performed a study looking into the cause of a disproportionate number of Blacks and Hispanics being stopped, questioned and arrested by police officers. This form of profiling could be the result of an illusory correlation that minorities perform more criminal activity than Caucasians. Smith and Alpert found that stereotype formation had led to profiling, which increased police activity towards minorities. Smith and Alpert say that illusory correlations can lead to stereotype formation. They argue that since illusory correlations are caused by unconscious events, the disproportionate number of are not always the result of racist intentions. Research about the effects of stereotypes and behavior are inconsistent, with some showing a correlation and others do not. Due to the lack of supervision of police officers in routine traffic stops, the high levels of discretion given to the officer, and minimal required evidence, Smith and Alpert argue it is the perfect environment for unconscious racial bias to occur.

An illusory correlation was also found in an experiment performed by Viken et al. This study looked into the possibility of an illusory correlation between perceived happiness and body type. The study had women look at images of other women and had them rate the images on happiness, degree of overweight, and attractiveness. The study found that women thought that an increase in degree of overweight led to a decrease in perceived happiness, even though there was no correlation. The study also found that the more eating disorder symptoms presented by the subject, the illusory correlation became larger.

These real life examples show that illusory correlations exist outside of the original illusory correlation paradigm. With the knowledge of this, it is necessary to find out how and why these correlations are made. Once these are known it will become easier to figure out how to prevent illusory correlations from being formed.

Explanations
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. Representativeness is the degree of similarity between an individual and the most typical example of a category. This heuristic has been shown to underlie many illusory correlations: people may correlate sexually ambiguous interpretations of inkblots with homosexuality because those judgements resemble their idea of a typical homosexual person.

Illusory correlation can also be explained in terms of biases in hypothesis-testing behavior. People tend to test hypotheses in a one-sided way, searching for evidence consistent with their current hypothesis. A parallel effect occurs when people judge whether two events, such as illness and bad weather, are correlated. They rely heavily on the number of cases where the two go together: in this example, instances of both pain and bad weather. They pay relatively little attention to the other kinds of observation (of no pain and/or good weather).

A study done by Murphy et al. was done to see if increased learning would have any effect on illusory correlations. In the study they set up two fictional minority groups to see if there were any judgments on the groups based on illusory correlations. They then educated one group about illusory correlations, and found that with more education less judgments based on illusory correlations were made.

In an article published by Berndsen et al., the authors discussed two studies regarding why illusory correlations are made. In the first study, three groups of participants were formed. All groups were told that two populations exist in the world, Group A and Group B, with Group A being larger. Then, each group was given a list of statements about each group but were told that these statements were only a selection of the statements, because reading all of them would take too long. The first group was then given 24 positive statements about Group A, and 12 positive statements about Group B. The second group was given 24 positive statements and 12 negative statements about Group A, and no statements about group B.  The control group was given equal number of positive and negative statements about both groups. The subjects then rated each group.

This design led to a creation of an illusory correlation, with Group B being seen more negatively than Group A despite no evidence of this existing. The authors attributed this differently to each group. They said that in the first group (24 positive for Group A and twelve positive for Group B) the positive behaviors in Group B were interpreted as relatively negative to create a difference between Group A and Group B. In the second group (24 positive and 12 negative statements about Group A), the negative statements were perceived relatively positive compared to the control condition, making Group A be perceived in a generally positive way. The authors argued that these differences between the groups were due to the amount of thought processes required in each group. In the first group, the differences were made explicit, as information on both groups was provided. The second group would have to differentiate implicitly, as there was no information provided about Group B.

In the second study in their research, Berndsen et al. had only one group, and this group was given all information about the two groups: Group A had 16 positive statements and 8 negative, Group B had 8 positive and 4 negative statements. The different number of statements did not matter as the ration of positive to negative statements was the same in both groups. The participants in this study were asked to verbalize their thought process, in an attempt to see subjects were looking for differences in the groups and to see how reinterpretations of statements and impressions influenced the formation of illusory correlations.

The results from this study showed the importance of evaluating statements through reinterpretation and looking for differences between the two groups. It also showed that these two processes started early during the presentation of statements. The confirmation bias also plays a role here, as looking for confirming information led to an illusory correlation while looking for disconfirming information did not create an illusory correlation.

Formation of Illusory Correlations
There are many different theories that apply to the formation of illusory correlations, several of which are discussed below.

General Theory
In a series of experiments performed by Shavitt et al. investigated the broad conditions for forming illusory correlations. In their first experiment, they found that despite evaluatively equal descriptions, an illusory correlation was found that favored the majority group. They attributed this to a difference in the number of descriptions given about each group. They thought that because less information was given about the minority group participants would be more likely to make extreme judgments about the majority group, simply because respondents felt they knew more about the majority group.

In the second experiment performed, Shavitt et al. looked into whether the set-size differences attributed to the creation of illusory correlations in experiment 1 were the true cause of the creation. They found that the chances of an illusory correlation being created was affected by differences in set-sizes. Specifically, they found that larger illusory correlations were found when the differences in set-sizes were larger, and when the set-sizes were relatively equal in size, little to no illusory correlation was found.

In the third experiment, the effect of set-size differences was again the focus of study. However, instead of looking at an absolute difference between majority and minority groups, this study looked into if differences could be found when just the set-size of the minority group was altered. This experiment found the same results as experiment 2; that the greater the set-size difference, the more likley to form an illusory correlation.

Information Processing
Denrell and Le Mens discuss many different theories about the formation of illusory correlations. One set of theories they discuss is theories based on biased information processing. The traditional way of looking at this, as Hamilton and Gifford did, is that the observer has access to all information in the 2x2 diagram. The model that Denrell and Le Mens propose that the majority of the time observing this information is not always clear cut and is influenced by outside noise. They also say that the decision-maker will often times interact with individuals in the population one at a time, and each interaction will steer there next interaction. This, paired with the confirmation bias, will cause the decision maker to move away from certain parts of the population, causing illusory correlations to be created.

Sampling Effects
Another set of theories about the formation of illusory correlations are based on sampling. Small samples can lead to illusory correlations as they can poorly represent the entire population, and result in stronger correlations than truly exist. Non-proportional sampling can also lead to illusory correlations through misrepresentations of the population, causing overestimation of the correlation. Another sampling issue that can lead to illusory correlations is skewed base rates. These skewed base rates can lead to using irrelevant information in creating correlations, causing illusory correlations.

Explicit versus Implicit Attitudes
Two studies performed by Ratliff and Nosek looked into the differences between explicit and implicit attitudes in relation with illusory correlations. Essentially, Ratliff and Nosek were looking at whether or not explicit attitudes and explicit 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 got overwhelming pro-majority readings, one got pro-minority readings, and one got neutral readings. The groups that had pro-majority and pro-minority readings both 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 very similar to the first study, but instead of readings pictures of behaviors were shown, and the decision makers wrote a sentence describing the behavior. This study found the same pattern as the first study. The authors argue 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).

Self-Affirmation
In a study done by Munro and Stansbury, effects of self-affirmation on illusory correlations were examined. They found that self-affirmation led to inaccuracy in identifying correlations. In the study, it was found that participants that were not self-affirmed were more likley to correctly disconfirm an illusory correlation. However, not self-affirmed participants were still not very good at properly identifying exactly why the illusory correlation did not exist.

Working-Memory Capacity
In an experimental study done by Eder, Fiedler and Hamm-Eder, 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.

Does Paradigm Structure Lead to Illusory Correlations?
Berndsen et al. looked into the effects of the structure of testing for illusory correlations, and if this actually could lead to the formation of illusory correlation. For example, they thought that calling the groups Group A and Group B may be causing the participants to look for differences between the groups, resulting in the creation of illusory correlations. To test this, they set up an experiment where one group of participants were told the groups wer Group A and Group B, and the other participants were labeled 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, compared to students of '93 or '94.

Terror Management Theory
Lieberman applied the Terror Management Theory to illusory correlations, and hypothesized that individuals who had to deliberate about their own mortality would be more likely to create an illusory correlation than individuals who had not contemplated their mortality. To do this Lieberman had participants enter a room, fill out a survey about their personality and then perform a basic illusory correlation paradigm. The manipulation occurred in the personality survey, where participants either discussed what they thought would happen to them when they were dead, or to describe emotions that occur while watching television. Individuals who contemplated their mortality were found to create stronger illusory correlations than those who contemplated emotions while watching television.

Learning Effects on Illusory Correlations
Murphy et al. performed two experiments to examine the effects of learning on illusory correlations. To do this, they performed a similar task to the original illusory correlation paradigm of Hamilton and Gifford, but instead of giving only 39 trials, subjects were given 90 trials. Throughout these 90 trials participants were asked to rate their views on the two groups. As participants saw more trials, the illusory correlation originally increased, but eventually began to diminish.

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 quicker. This can result in illusory correlations. The Attention theory also argues that instead of one stereotype forming regarding the minority group, two different stereotypes, one for the majority and one for the minority, are formed.

Age
A study performed by Primi and Agnoli found that even children can create illusory correlations. In their experiment they had children in grades 1, 3, 5, and 7, and adults, look at the same illusory correlation paradigm. The study found that children did create significant illusory correlations, but they were weaker than the correlations created by adults. In a second study, instead of using groups of people, they used groups of shapes with different colors, and found that the ilusory correlation persisted. This shows that social stimuli are not necessary for creating these correlations.

Johnson and Jacobs performed two experiments to see how early in life individuals begin forming illusory correlations. The first experiment used a typical illusory correlation paradigm to see if negative attributes were associated with the minority group in second and fifth graders. The authors found that the second and fifth graders both formed illusory correlations. The second experiment was performed to check for a priori judgments made about the minority groups. This step is normally skipped for adults, but the authors worried that this may not be true for children. In order to do this, the study was set up so that fewer positive behaviors and more negative behaviors were in both the majority and the minority groups, essentially switching the normal illusory correlation paradigm. Johnson and Jacobs found that the miority groups were seen as more positive now, showing that a priori judgments were not made. These findings support those of Primi and Agnoli.

Conclusions
All of these theories add to the knowledge of how and why illusory correlations are formed. With this knowledge it is possible to start to actively avoid creating these false ideas. As shown by Primi and Johnson, these correlations begin forming at a very young age. The attention theory of learning shows that because majority group information is learned first, minority groups are based off of this and must differentiate to distinguish between the two. Terror management theory showed that once someone contemplates their own mortality they are more likely to form illusory correlations. Working-memory use can also alter the creation of illusory correlations, as more working memory is used more illusory correlations are formed. It has also been shown that the way in which individuals process information affects the number of illusory correlations formed. Another effect on the formation of illusory correlations is sampling, as non proportional samples and small samples greatly increase the number formed. All of these things interact in many different ways in the creation of illusory correlations.

There is some hope though. The research shows that increased learning can decrease the formation of illusory correlations. Also, self-affirmation has been proven to increase the ability to disprove illusory correlations. Through these two practices it is possible to reduce the number of illusory correlations made, and prove existing illusory correlations false.