Associative symmetry generalizes to asymmetric pairs Christopher R. Madan (cmadan@ualberta.ca) Department of Psychology, University of Alberta Edmonton, AB T6G 2E9 Canada Mackenzie G. Glaholt (mackenzie@psych.utoronto.ca) Department of Psychology, University of Toronto at Mississauga Mississauga, ON L5L 1C6 Canada Jeremy B. Caplan (jcaplan@ualberta.ca) Department of Psychology & Centre for Neuroscience, University of Alberta Edmonton, AB T6G 2E9 Canada Abstract For pairs of words (A - B) in which both items are drawn from the same stimulus pool, cued recall accuracy in the for- ward direction (A → ?) is, on average, equal to accuracy in the backward direction (? ← B). This led to Gestalt psychol- ogists suggesting that pairs are learned as holistic units rather than two separate forward and backward associations (Asch & Ebenholtz, 1962). Kahana (2002) pointed out that instead, di- rect evidence of holistic learning would be a near-perfect corre- lation between forward and backward probes of the same pair, which is mathematically independent of mean performance. We report that even when pairs are asymmetric in mean per- formance measures (pairing low-frequency words with high- frequency words), the forward–backward correlation remains high. Our findings force a re-evaluation of prior findings of asymmetries in heterogeneous pairs: namely, human partici- pants can learn asymmetric pairs as holistic, non-directional associations. Keywords: paired-associate learning; word frequency; asso- ciative symmetry; verbal memory Introduction Paired-associate learning has a long tradition of study in ex- perimental psychology (Calkins, 1896). The nature of the association between paired items, A - B, has been hypoth- esized to be either made of separate, unidirectional associa- tions (Wolford, 1971), referred to as the Independent Asso- ciations Hypothesis or as a holistic unit (Asch & Ebenholtz, 1962; K¨ ohler, 1947), refered to as the Associative Symmetry Hypothesis. According to the Independent Associations Hy- pothesis, the association consists of two steps, where A → B is learned in a statistically independent step from A ← B. The consequence is that performance on forward probes of a pair, A - ?, are expected to be independent of backward probes of the same pair, ? - B. In contrast, Associative Symmetry Hy- pothesis implies that forward and backward probes comprise a single Gestalt (Figure 1) and are learned as a single step. Two methods have been proposed to test between the Inde- pendent Associations Hypothesis and the Associative Sym- metry Hypothesis. Firstly, comparing mean performance in the forward and backward directions, Asch and Eben- holtz (1962) suggested that symmetry supports the Associa- tive Symmetry Hypothesis whereas asymmetry (such as a forward-probe advantage) would support the Independent As- sociations Hypothesis. Secondly, by examining the correla- (a) (b) Figure 1. Visualizations of the Association Hypotheses. (a) Independent Association Hypothesis. (b) Associative Sym- metry Hypothesis. tion between forward and backward probe performance at the level of single pairs over successive tests (Kahana, 2002). Ka- hana (2002) pointed out that, symmetry of mean performance is irrelevant to the holistic nature of the pair and is not a direct test of the nature of the associative symmetry. While this is true from a mathematical perspective, it is not known whether they are separable in human behaviour. Here we attempt to empirically de-couple the two measures. If a pair is learned equally well in both directions, there should be equal probabilities of cued recall in both the for- ward and backward directions (Levy & Nevill, 1974; Wollen, Allison, & Lowry, 1969). If the pair is learned as a Gestalt unit, the forward and backward cued recall should have nearly perfect correlation. Even if performance on forward and backward probes is independent, trial-to-trial variance in per- formance will induce a correlation. The correlation of for- ward and backward test performance has been demonstrated to be high, supporting the Associative Symmetry Hypothe- sis (Caplan, 2004, 2005; Caplan, Glaholt, & McIntosh, 2006; Rizzuto & Kahana, 2000, 2001; Sommer, Rose, & B¨ uchel, 2007). However, it is important to note that in all these stud- ies, mean performance was nearly symmetric. To understand why the mean performance and correlation measures are distinct, consider the following example. Given the pairs SHROUD - RUMOUR and HELMET - MALICE, if SHROUD - ? and ? - MALICE are answered correctly by the