Bias in proportion judgments: The cyclical power model.

Abstract

When participants make part-whole proportion judgments, systematic bias is commonly observed. In some studies, small proportions are overestimated and large proportions underestimated; in other studies, the reverse pattern occurs. Sometimes the bias pattern repeats cyclically with a higher frequency (e.g., overestimation of proportions less than .25 and between .5 and .75; underestimation otherwise). To account for the various bias patterns, a cyclical power model was derived from Stevens' power law. The model proposes that the amplitude of the bias pattern is determined by the Stevens exponent, beta (i.e., the stimulus continuum being judged), and that the frequency of the pattern is determined by a choice of intermediate reference points in the stimulus. When beta < 1, an over-then-under pattern is predicted; when beta > 1, the under-then-over pattern is predicted. Two experiments confirming the model's assumptions are described. A mixed-cycle version of the model is also proposed that predicts observed asymmetries in bias patterns when the set of reference points varies across trials.

Intuitive proportion judgment in number-line estimation: Converging evidence from multiple tasks

“…This mirrors the developmental trend reported for the NP task (Slusser et al, 2013). However, more one-cycle patterns are seen in the PN task as compared to the NP task, suggesting that different tasks invite different estimation strategies (Hollands & Dyre, 2000;Hollands et al, 2002). It is possible, for example, that the marked position on the PN task, or some other quality of the task, interferes with participants' ability to infer a central reference point.…”

“…The alternative theory, based on a psychophysical model of proportion estimation (Hollands & Dyre, 2000;Spence, 1990), posits that the proportional structure of the typical number-line task must be taken into account when attempting to understand task performance.…”

“…Spence (1990) showed that estimates of proportions (estimation of a part relative to a whole, or of a part relative to another part) should take the form of S-shaped or inverse S-shaped curves, depending on the particular value of β in question. When this model is applied to a typical 0-1000 number line task, This model was later generalized to account for cases in which the observer makes use of additional reference points (for example, estimating the proportion of a cylinder that is partially filled with liquid by judging the liquid's level relative to a halfway point, rather than relative to the entire height of the cylinder; Hollands & Dyre, 2000). This work showed that the use of such reference points 1) produces a pattern of estimates with multiple S-shaped or inverse S-shaped curves or "cycles" (hence the term "cyclical power model" of proportion estimation; see Figure with this model are observed in children's number-line estimates Barth et al, 2016;Rouder & Geary, 2014;Slusser et al, 2013), as they are in adults' estimates of proportions using various continua (Cohen & Blanc-Goldhammer, 2011;Hollands & Dyre, 2000).…”

Two interpretations of reports of knowledge of subpopulation sizes

“…part-whole proportion judgements, and models for the occurrence of estimation bias are beginning to appear (e.g. Hollands and Dyre, 2000).…”

What very small numbers mean.

“…Nevertheless, researchers often make an implicit assumption that people interpret these numerical formats equivalently (termed the assumption of numerical equivalence). The implicit assumption of numerical equivalence is prevalent in studies of psychophysics (e.g., Gescheider, 1988;Marks, 1974;Marks & Algom, 1998;Stevens, 1956Stevens, , 1986 Schwarz, & Bless, 1995), and probability estimation (e.g., Begg, 1974;Brooke & MacRae, 1977;Hollands & Dyre, 2000;Shuford, 1961;Spence, 1990;Teigen, 1973;Tversky & Fox, 1995;Varey, Mellers, & Birnbaum, 1990), to name just a few. The assumption of numerical equivalence, however, should not be made lightly because people's interpretation of numbers likely affects the data they produce, and thus the conclusions that researchers draw.…”