What is denominator neglect?
Denominator neglect is the habit of judging a ratio by its top number. Two research lines name it differently and explain it differently, and one experiment shows what reduces it.
Answer to the question
What is denominator neglect?
Denominator neglect is giving too much weight to the number of cases in a ratio and too little to the total they sit within. It is also called the ratio bias. In a 1994 prize task many people chose a bowl with seven winning beans in a hundred over one with a winner in ten, though the second had better odds.
Key takeaways
- Denes-Raj and Epstein (1994): many participants chose a bowl with seven red beans in a hundred over one with one in ten, and admitted the odds favoured the other bowl.
- Barbey and Sloman (2007) argued that base-rate neglect arises from strategies that miss how categories nest, and reported a smaller effect when nested groups are readily represented.
- A 2007 commentary by Brainerd credited Reyna's denominator neglect model with anticipating that account, naming three parallels.
- Garcia-Retamero, Galesic and Gigerenzer (2010): in two experiments, icon arrays drew attention to the denominators and improved the accuracy of risk judgments.
- The ratio bias and denominator neglect are labels from two research lines for closely related patterns.
Two bowls stand on a table. One holds seven red beans among a hundred, the other a single red bean among ten, and a red bean wins a prize. Many people reach for the first bowl, though the second has the better odds. Judging a ratio by its top number, the count of winners, and giving too little weight to the size of the bowl is called denominator neglect. Researchers have given the pattern more than one name and more than one explanation, and the name decides what the effect gets compared with.
One Pattern, Several Names
The terms map onto the bowls directly. The numerator is the number of red beans and the denominator is the number of beans in the bowl. Seven in a hundred is seven per cent and one in ten is ten per cent, so the second bowl is the better bet by three percentage points. A reader who compares seven with one sees the first bowl as richer, and that comparison of numerators is the pattern the term names.
Denominator neglect appears in the literature under two labels. In the line of work begun in Seymour Epstein's laboratory it is usually called the ratio bias, and the bowls of beans are its standard test. In the work of Valerie Reyna, denominator neglect is the term for a wider claim about how people handle groups that sit inside other groups, and the bowls are one case among several.
One more label sits beside both. Base-rate neglect is the tendency to underweight how common a category is once a description of a case is in hand. Denominator neglect resembles it, since a base is again being underweighted, though here the base is stated inside the ratio itself.
Daniel Kahneman discusses the effect in Thinking, Fast and Slow, using two urns of marbles to show how a vivid count of winners outweighs the size of the urn. Denominator Neglect and the Bigger Bowl follows the bowl experiments through paid trials and judgments by professionals. This article asks a narrower question, which is what the term covers and how its two main explanations differ.
Odds That People Knew and Overrode
The jelly bean task is the clearest demonstration. Veronika Denes-Raj and Epstein reported in 1994 a prize task in which a participant could win one dollar by drawing a red bean without looking. One bowl held seven red beans among a hundred. The other held one red bean among ten. Many participants chose the first, which offered seven chances in a hundred against ten in a hundred. In that design the better odds sat with the bowl holding fewer winners, so a preference for the larger count worked against the player.
The participants were not unaware of the odds. They admitted that the odds favoured the other bowl, and still felt they had a better chance with more red beans. A weaker version of the pattern appeared when drawing a red bean meant losing a dollar. The authors predicted these results from the principles of cognitive-experiential self-theory, in which an experiential system runs alongside a rational one. Choices in the laboratory also correlated with heuristic responses to related vignettes.
An earlier paper from the same laboratory, by Lee A. Kirkpatrick and Epstein in 1992, had brought the self-theory to bear on subjective probability. In the 1994 task participants could say which bowl was better and choose the other.
When Classes Nest Inside Each Other
The second explanation looks at the structure of the problem and not at the person's two systems. In 2007 the journal Behavioral and Brain Sciences ran a target article by Aron Barbey and Steven Sloman on base-rate neglect, followed by open commentaries. Barbey and Sloman argued that a dual-process model fits the evidence best, with base-rate neglect arising from associative strategies that fail to capture how the categories in a problem nest within one another. They report that the effect is smaller when problems are framed so that people can readily represent the nested groups.
Two commentaries placed denominator neglect in that picture. Charles Brainerd wrote that the base-rate respect model of Barbey and Sloman was anticipated by Reyna's denominator neglect model, and he named three parallels: both rest on a general cognitive theory, both treat problem structure as a source of error, and both see nested set relations as the cause of base-rate neglect. Christopher Wolfe, in the same issue, set out the prediction that overlapping class-inclusion relations create processing interference, which leads people to act as if they ignore the marginal denominators in a two-by-two table.
Barbey and Sloman approached base-rate neglect from two angles, how people judge under uncertainty and what the judgments imply about human rationality, and they reviewed the competing accounts for how well each explained the data and how coherent it was.
Side by side, the three accounts place the source of the effect in different spots:
- The ratio bias line points to an experiential system that runs alongside a rational one.
- Reyna's model points to interference between nested class-inclusion relations.
- Barbey and Sloman's base-rate respect model points to associative strategies that miss how categories nest.
Wolfe's commentary extends the claim beyond beans. Ignoring marginal denominators, he argues, produces fallacies in base-rate problems and also affects estimates of conjunctive and disjunctive probabilities. Reyna and Brainerd gave a 2008 paper in Learning and Individual Differences the title Numeracy, ratio bias, and denominator neglect in judgments of risk and probability, which shows the three terms being treated as one topic.
Drawing the Whole Group
Both accounts raise the same practical question, which is how easy the denominator is to see. Rocio Garcia-Retamero, Mirta Galesic and Gerd Gigerenzer tested this directly in 2010, under the title Do Icon Arrays Help Reduce Denominator Neglect? Their two experiments gave participants a risk reduction figure for an intervention, with treated and untreated groups of different sizes. Participants tended to focus on the numerators, the number of events, and neglected the denominators, the total number of people in each group.
Adding icon arrays to the numbers, pictures in which each person in a group is drawn as one symbol, drew attention to the denominators and improved the accuracy of participants' risk judgments. The benefit was similar for younger and older adults. The authors conclude that icon arrays are an effective way to eliminate denominator neglect, and that the difficulty lies largely in how the problem is represented and not in the reasoner's mind.
That finding sits comfortably with the 2007 account, in which representing the nested groups reduces the effect. A familiar place where the denominator matters is fuel economy, which Why Do MPG Figures Mislead Car Buyers takes up, and how a probability is displayed also shapes how large it feels, a theme that a companion piece on small chances follows through the weights people give them. Whichever account is preferred, the experiment points the same way: the whole group is easiest to weigh when it is drawn in front of the reader.
Common questions
What is denominator neglect?
Denominator neglect is giving too much weight to the number of cases in a ratio and too little to the total they sit within. It is also called the ratio bias. In Denes-Raj and Epstein's 1994 task many people preferred a bowl with seven winning beans in a hundred to one with one winner in ten.
Is denominator neglect the same as ratio bias?
The two terms describe closely related patterns. Ratio bias is the label used in Epstein's line of work on the bowl task, while denominator neglect is the label used in Reyna's account of nested groups. Many authors use them interchangeably.
How can denominator neglect be reduced?
In a 2010 study by Garcia-Retamero, Galesic and Gigerenzer, adding icon arrays to numerical risk information drew attention to the denominators and improved the accuracy of participants' judgments, for younger and older adults alike.
Sources
Kahneman, D. (2011). Thinking, Fast and Slow. Seed book; vault hub Books/Thinking Fast and Slow.
Denes-Raj, V., & Epstein, S. (1994). “Conflict between intuitive and rational processing: When people behave against their better judgment.” Journal of Personality and Social Psychology, 66(5), 819-829. doi:10.1037/0022-3514.66.5.819 https://doi.org/10.1037/0022-3514.66.5.819
Kirkpatrick, L. A., & Epstein, S. (1992). “Cognitive-experiential self-theory and subjective probability: Further evidence for two conceptual systems.” Journal of Personality and Social Psychology, 63(4), 534-544.
Barbey, A. K., & Sloman, S. A. (2007). “Base-rate respect: From ecological rationality to dual processes.” Behavioral and Brain Sciences, 30(3), 241-254. doi:10.1017/S0140525X07001653 https://doi.org/10.1017/S0140525X07001653
Brainerd, C. J. (2007). “Kissing cousins but not identical twins: The denominator neglect and base-rate respect models.” Behavioral and Brain Sciences, 30(3), 257-258. doi:10.1017/S0140525X07001689 https://doi.org/10.1017/S0140525X07001689
Wolfe, C. R. (2007). “Adaptive redundancy, denominator neglect, and the base-rate fallacy.” Behavioral and Brain Sciences, 30(3), 286-287. doi:10.1017/S0140525X07001951 https://doi.org/10.1017/S0140525X07001951
Reyna, V. F., & Brainerd, C. J. (2008). “Numeracy, ratio bias, and denominator neglect in judgments of risk and probability.” Learning and Individual Differences, 18(1), 89-107. doi:10.1016/j.lindif.2007.03.011 https://doi.org/10.1016/j.lindif.2007.03.011
Garcia-Retamero, R., Galesic, M., & Gigerenzer, G. (2010). “Do icon arrays help reduce denominator neglect?” Medical Decision Making, 30(6), 672-684. doi:10.1177/0272989X10369000 https://doi.org/10.1177/0272989X10369000