Abstract
The ability to extract statistical contingencies (e.g., between cause and effect, between response and feedback) is commonly presupposed as a basic module of adaptive behavior. In reality, however, stimulus input rarely contains the complete sets of correlated attributes required to assess the actual contingencies. Instead, cognitive inferences often rely on a base-rate-driven pseudocontingency rule, which links the more (or less) frequent level of one variable to the more (or less) frequent level of the other variable. Empirical evidence shows that logically unwarranted pseudocontingency inferences override genuine contingencies across many research paradigms. Although pseudocontingencies can be severely misleading, they also provide a useful proxy that accurately predicts existing contingencies most of the time.
It has been shown that methodological tools can become theories in cognitive psychology (Gigerenzer, 1991). There is hardly any more compelling example for this wisdom than the statistical tool of a correlation—or contingency in case of dichotomous variables. The ability to assess correlations is widely recognized as a core module of adaptive behavior. Theories in diverse areas of cognitive psychology rely on contingencies between conditional and unconditional stimuli (Rescorla, 1969), between causes and effects (Cheng, 1997; Einhorn & Hogarth, 1986; White, 2009), or between heuristic cues and environmental states in rational decision making (Gigerenzer & Goldstein, 1996).
It is noteworthy that the cognitive process of correlation assessment is also modeled after the contingency tool. Subjective contingencies, ΔP, are supposed to be inferred from the joint frequencies a, b, c, and d given in the 2 × 2 cells of Figure 1, as differences between two proportions, ΔP = a/(a +b) − c/(c +d). Virtually all accounts of subjective contingency estimates assume unequal weighting or selective processing of a, b, c, and d (Fiedler, 2000; McKenzie, 1994; Wasserman, Dorner, & Kao, 1990; White, 2009). Different theories predict contingency inferences to be more sensitive to the joint presence (cell frequency a) than to the joint absence (d) of events (Wasserman et al., 1990) or to be biased toward the most expected (Chapman & Chapman, 1969) or most infrequent events (Hamilton & Gifford, 1976). That the cognitive process resembles the statistical model is generally taken for granted. How could cognitive contingency assessment not rely on the joint frequencies a, b, c, and d?
Notation for the frequency distribution of two dichotomous variables: exposure to an insect and contracting a disease across individuals living in an ecology. Although contingencies are computed from joint frequencies a, b, c, and d, pseudocontingencies are inferred from the marginal frequencies or base rates (a +b), (c +d), and (a +c), (b +d).
A positive answer to this rhetorical question can be found in the pseudocontingency heuristic (Fiedler, Freytag, & Meiser, 2009), which suggests that cognitive inferences are simpler than the statistical rationale. In brief, pseudocontingency inferences relate what is frequent in one variable to what is frequent in another variable. As the prefix pseudo- suggests, these pseudocontingency inferences appear unwarranted and strange. Closer inspection will soon reveal, however, that pseudocontingencies reflect not only a severe cognitive illusion but, as many illusions do, also an adaptive, fast, and frugal heuristic that often yields correct estimates.
To illustrate, assume that only the base rates of two variables are known (Table 1); most people living in an ecology have been exposed to an insect (75%), and most people have contracted an infectious disease (75%). Because nothing is known about the joint frequencies within the cells, the contingency might be positive (disease rate enhanced after insect exposure), zero (disease rate equal), or negative (disease rate reduced). Nevertheless, the alignment of two skewed base-rate distributions gives rise to a positive pseudocontingency: Insect and disease appear to be positively correlated. Inferred contingencies are positive when the two base-rate distributions are skewed in the same direction (both insect and disease base rate are high) but negative when skewed in opposite directions (one frequently present and one infrequently).
Certainly, such an inference is logically unwarranted. Jointly elevated insect and disease rates in the same ecology suggest some common cause, such as climatic conditions or a virus carried by the insect. However, such ecology-level causes must not be confused with the causes that determine the within-ecology correlation across individuals, which may indeed work in the opposite direction. For example, people who engage in open-air sports may be more likely to be exposed to insects but less likely to contract diseases.
These uncritical pseudocontingency inferences from congruent base-rate differences at the ecological level to within-ecology correlation are reminiscent of ecological fallacies in social science (Robinson, 1950): Enhanced proportions of Black and illiterate people in particular districts (relative to others) do not imply that illiterates are more likely to be Black than White within districts (Robinson, 1950). Likewise, higher proportions of men in professions with high leadership ability do not imply that individual male leaders are superior to female leaders (Eagly & Carli, 2003). 1
Severe Pseudocontingency Illusions
Positive and negative pseudocontingencies
In pseudocontingency experiments, base rates are manipulated either in the absence of or in conflict to actually existing contingency information. For example, on each trial of a stimulus series used by Fiedler and Freytag (2004), a patient was described by high or low scores on two personality tests, X and Y. The correlation between X and Y was zero; high Y scores were equally likely in patients with high and low X scores. However, high scores on both tests were generally more frequent than low scores (75% vs. 25%). When participants later predicted one score from given low or high scores on the other attribute, their predictions reflected a positive pseudocontingency; frequency estimates of high Y scores were higher for patients with high than low X scores. Inferred correlations between X and Y were positive when base-rate distributions were aligned (i.e., skewed in the same direction) but negative when base rates were misaligned, even though the correlation actually presented was zero.
Pseudocontingencies without correlation data
Another way to demonstrate pure pseudocontingencies involves concealing all correlation evidence. Thus, presenting X and Y scores separately in successive runs facilitates the assessment of base rates but conceals the joint reference of individual scores Xi and Yi to the same patient i. Yet Fiedler and Freytag (2004) found that correlations were readily inferred from aligned base rates; the frequent level of one test was predicted about 46% more often from the frequent level than from the infrequent level of the other test.
Demanding multi-cue tasks
The absence of a correlation (being either zero or undefined) is not a precondition. Pseudocontingencies override even transparently given opposing correlations, especially in complex task settings. Fiedler (2010) provided participants with descriptions of students on four dichotomous attributes (gender, subject matter, hobby, and university), the two levels of which appeared at base rates of 75% and 25%. Pairwise contingencies varied from r = .31 (positive values indicate that r values are congruent with pseudocontingency) to r = 0 to r = −.33. Subsequent percentage estimates for each attribute, given both levels of all other attributes, revealed regular pseudocontingencies. Whether the actual correlation was congruent, zero, or incongruent did not have the slighted effect (Fig. 2).
Mean percentage estimates for a selected value of one cue as a function of actually presented contingency and frequency level (data from Fiedler, 2010).
Very strong pseudocontingency effects were found by Fiedler and Freytag (2004) when two tests, X and Y, could take on low, medium, and high scores, yielding a more complex 3 × 3 contingency, and when two contrasting ecologies (with predominantly high versus low scores on both tests) enhanced the salience of ecological base rates. Although the correlation within ecologies was negative (−.42), subsequent predictions of one test score from given scores on the other exhibited a highly positive correlation (.85).
Generality of Pseudocontingency Effects
Accuracy motivation
What if more sensible and socially meaningful tasks increased the judges’ accuracy motivation? Such a motivating task setting is implemented in the simulated classroom paradigm (Fiedler, Freytag, & Unkelbach, 2007; Fiedler, Walther, Freytag, & Plessner, 2002), in which participants (often future teachers) assess the performance of a class of students represented on the computer screen regarding their ability (rate of correct responses) and motivation (rate of raising hands).
Although evaluations are very sensitive to actual performance rates, they reflected distinct pseudocontingency biases. Given a 70% base rate of correct responses in a high-performing class, the correctness of aligned students with a high motivation base rate (70%) was appraised as higher than that of equally correct low-motivation students. Conversely, given 30% correctness in a low- performing class, misaligned high-motivation students (70%) suffered from reduced correctness ratings. Likewise, when the same high-correctness rates held for smart boys and girls, the gender group with the higher motivation rate received higher ability ratings. This pseudocontingency illusion was independent of whether the prevalent gender group was stereotypically linked to high ability (girls in language) or low ability (girls in science; see Fiedler et al., 2007).
Pseudocontingencies in stereotype acquisition
When a large (majority) and a small group (minority) exhibit the same high rate of positive behavior, the minority will nevertheless be evaluated less positively (Hamilton & Gifford, 1976; Mullen & Johnson, 1990). This illusory-correlation effect is commonly attributed to enhanced memory for the rarest combinations of the infrequent valence (negative) with the infrequent group (minority). An alternative pseudocontingency-account points to the alignment of low base rates of minorities and negative behavior. Pitting both accounts against each other, Eder, Fiedler, and Hamm-Eder (2010) presented a series of mostly positive (or negative) behaviors without group reference. Participants who had been told that one group was more frequent were asked to guess the group reference of each behavior. Both online assignments of behaviors to groups and post-sequence impression ratings of the two groups reflected a marked pseudocontingency bias. The more prevalent valence was regularly associated with the more prevalent group, as in traditional illusory correlations.
Pseudocontingencies in operant learning and priming
Other pseudocontingency effects in prominent paradigms of cognitive psychology highlight the generality of the phenomenon. In an operant-learning study with performance-contingent payment by Kutzner, Freytag, Vogel, and Fiedler (2008), 240 trials started with a low-pitch tone and only 80 with a high-pitch tone. On both types of trials, one of two response keys was frequently rewarded (75%). Unsurprisingly, participants exhibited a bias toward the more frequently rewarded response key. However, this response bias was consistently more pronounced for the frequent trial type—a distinct pseudocontingency effect.
Pseudocontingency effects also moderated and reversed the common congruity advantage in evaluative priming (i.e., faster and more accurate evaluation of targets after primes of the same valence; Freytag, Bluemke, & Fiedler, 2011). When the more frequent valence was the same for primes and targets, a positive pseudocontingency between prime valence and target valence supported a congruity effect. In contrast, when the more frequent valence of primes and targets differed, a negative pseudocontingency led to faster and more accurate responses on incongruent trials.
Adaptive Pseudocontingency Functions
Thus, numerous findings from various paradigms testify to pseudocontingency effects causing logically unwarranted illusions. However, pseudocontingencies also afford an adaptive inference tool for which there is hardly any alternative.
Feasibility of correlation assessment
Information encountered in reality is more likely to support pseudocontingencies than correlations proper. Most knowledge about the social and physical world comes as aggregated base-rate information. School books, newspapers, or the Internet usually provide us with summary statistics about the climate in geographic regions, cultures of ethnic groups, political and economic trends in different countries, or images of brands, professions, or universities. We are rarely given high-resolution raw data with which to compute correlations across specific events or persons within ecologies.
Even when detailed information is occasionally available, encoding and memorizing countless individual observations is much less likely than assessing the few base rates required for pseudocontingency inferences. The standard textbook assumption of a multivariate data matrix containing multiple correlated measures Xi, Yi, and Zi for each individual i is rarely met. Real-life input is full of missing data. It is hard to reconstruct which Xi belongs to which Yi because time and context often separates observations of different attributes. Moreover, given multiple (k) variables (e.g., many different insects causing diseases), it is virtually impossible to manage the complex contingency matrix, which contains 2 k cells in the simplest case, and many more are needed to deal with partial correlations. Pseudocontingency-based inferences, in contrast, require only an assessment of the base-rate trends for k variables.
Level of interventions
Ecological base rates are not only more amenable to assessment than individual events but also represent a more feasible level for interventions. To prevent a disease, avoiding an ecology with a high disease (or insect) base rate is a more feasible strategy than trying to avoid specific carriers of an invisible virus. Likewise, in a priming experiment, a response strategy that exploits the predominant stimulus base rate is more likely to facilitate performance than attending to individual stimuli. More generally, category base rates often inform rational decisions with minimal opportunity costs (Huttenlocher, Hedges, & Vevea, 2000; Olivola & Todorov, 2010).
Pseudocontingencies actually predict correlations
An ultimate adaptive advantage of the pseudocontingency heuristic is its predictive validity. Pseudocontingencies afford a highly useful proxy for predicting existing correlations, because skewed base rates impose asymmetric restrictions on the range of possible correlations (Kareev, 1995; Thorndike, 1949). If 80% of people were in contact with the insect and all got diseases, alignment allows for a perfect positive correlation (1.00), but negative correlations cannot be more extreme than −.25. 2 Assuming independence (equally likely combinations of all variable values), the distribution of possible correlations in skewed worlds is clearly biased toward aligned correlations (see Fig. 3), as predicted by pseudocontingency.
Given two aligned distributions of dichotomous variables (with focal base rates p = q > .5), the maximal correlation (r max) = 1.00, whereas the minimal correlation (r min) shrinks with increasing skew from −1.00 to 0, creating a marked positivity bias in the expected average correlation (r average) (Kareev, 1995).
To be sure, no pseudocontingency inference is possible when base rates are equal. However, an intriguing property of the probabilistic world is that small samples of correlated X-Y pairs, even when drawn from a nonskewed world, are sufficiently skewed to produce pseudocontingencies; whether these are negative or positive tends to be consistent with the existing correlation in the population (Kutzner, Vogel, Freytag, & Fiedler, 2011). In other words, sampling error allows for pseudocontingency inferences even when the world is unskewed.
Concluding Remarks
As promised in the title of this article, then, pseudocontingency inferences are logically unwarranted but useful and smart. They override but also emulate and approximate real contingencies. Base-rate-driven inferences that follow the pseudocontingency rule have been shown to influence behavior across various paradigms. To be sure, the pseudocontingency domain is limited (to skewed samples), and genuine contingency assessment is sometimes possible. However, the extent to which the statistical metaphor of a contingency (which has governed so much cognitive research on causality, conditioning, and decision making) has to be revisited in the light of new evidence on pseudocontingency effects remains an open empirical question.
Recommended Reading
Eder, A. B., Fiedler, K., & Hamm-Eder, S. (2011). (See References). Shows that pseudocontingencies can underlie illusory correlations.
Fiedler, K. (2010). (See References). Demonstrates pseudocontingencies that completely override actually existing contingencies.
Fiedler, K., Freytag, P., & Meiser, T. (2009). (See References). Provides a comprehensive review of theoretical underpinnings and empirical evidence on pseudocontingencies.
Kutzner, F., Vogel, T., Freytag, P., & Fiedler, K. (2011). (See References). Reports simulation studies showing that pseudocontingencies correctly predict the sign of existing correlations most of the time.
Footnotes
Declaration of Conflicting Interests
The authors declared that they had no conflicts of interest with respect to their authorship or the publication of this article.
Funding
The research underlying this chapter was supported by a Koselleck Grant of the Deutsche Forschungsgemeinschaft awarded to the author (Fi 294/23-1)