Symmetry-based methodology for decision-rule identification in same different experiments

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3 Same Different Decision Rules 03 A Unbiased Optimal B Differencing DF C Differencing DF d / 0 d / D Covert Classification CC E Classification CCs = RCs F Covert Classification CCa c c c c c c c c c c G Likelihood Ratio LR H Likelihood Ratio LR I Reversed Classification RCa c c c c Figure. Schematic representation of the decision space. White areas are labeled same and shaded areas different. The perceptual effects of the first and second stimuli vary along the x- and y-axes, respectively. The four circles in each panel are contours of equal probability density for the joint effects of stimulus pairs,,, and. Panel A illustrates how an unbiased optimal observer partitions the space. Panels B I illustrate the eight decision rules listed in Table and discussed in the text. The diagonal(s) show the axis (or axes) of symmetry for each decision boundary. c, c, and c denote covert-classification criteria. This is a sensitivity measure that characterizes only the stimulus pair, not the decision rule or the response bias. To produce a same or different response, the observer must partition the decision space. Figure A shows the strategy of an ideal, unbiased observer. The decision boundary consists of two perpendicular straight lines, each equidistant from the centers of the respective magnitude distributions. These lines coincide with the axes in our chosen coordinate system. The two shaded quadrants are labeled different ; the two white quadrants are labeled same. Table lists eight

4 04 Petrov decision rules organized in three families: differencing, covert classification, and likelihood ratio. Figures A I illustrate how each family partitions the decision space. Differencing strategies. The defining feature of all differencing strategies is that they form an explicit representation of the difference, δ 5 (x x ), of the stimulus magnitudes and that all subsequent processing is based on this difference. Magnitude pairs that differ by the same amount are treated as equivalent. This collapses the twodimensional space x, x to a single dimension δ. Geometrically, the differencing operation projects the plane onto the negative diagonal. The simplest differencing strategy was proposed by Sorkin (96). We refer to it as DF (differencing strategy with one criterion). It compares the absolute value of the difference with a criterion c. The observer responds same iff δ, c. Equivalently, DF can be formulated with a pair of criteria equidistant from the neutral point. The observer responds same iff c, δ, c. Projected back to the two-dimensional space, the criteria become lines parallel to and equidistant from the positive diagonal (Figure B). The band that contains the origin is labeled same ; the regions on either side are labeled different. Exchanging the stimuli in the two presentation intervals reverses the sign of δ. Under DF, however, this sign does not affect the final response. Thus, DF predicts identical response probabilities for stimulus pairs that differ only in the order of presentation. In particular, DF predicts p 5 p for heterogeneous pairs. Our experimental data violate this identity and thus rule out DF. The data are consistent with a more general strategy that we refer to as DF (differencing strategy with two criteria). It involves two arbitrary criteria that may not be equidistant from the neutral point. The observer responds same iff c, δ, c. Figure C plots an example in which p. p. DF is a special case of DF in which the two criteria satisfy the constraint c c 5 0. Differencing strategies have several things to recommend them. DF maximizes accuracy in two-alternative forced choice (AFC) designs, where the two stimuli are always different and the task is to indicate which one came first (Macmillan & Creelman, 005). DF is also optimal in roving same different designs, where absolute intensities vary but differences are kept constant (Dai et al., 996). A plausible neural representation of δ has been found in physiological recordings with monkeys perform- Label DF DF CC CCa CCs RCa RCs LR LR Table Abbreviated Labels of the Decision Strategies Discussed in the Text Decision Strategy Differencing strategy with one criterion Differencing strategy with two criteria Covert classification (CC) with one criterion CC with two asymmetric criteria and bias for same responses CC with two symmetric criteria and bias for same responses Reversed-classification with two asymmetric criteria Equivalent to CCs Likelihood-ratio strategy with one criterion Likelihood-ratio strategy with two criteria ing roving AFC tactile flutter discrimination: The firing rates of many neurons in secondary somatosensory cortex correlate with the stimulus difference (Romo, Hernández, Zainos, Lemus, & Brody, 00). On the other hand, the differencing strategy is suboptimal for fixed same different designs, which are our main focus here (Irwin et al., 999; Noreen, 98). For example, suppose d 5 3 and the criterion is chosen so that the false alarm rate is %. Then the hit rate is approximately 50% under the optimal strategy (LR) but only 3% under DF. Conversely, a 50% hit rate implies d under DF and d 5 3 under LR. To carry out a differencing strategy, the observer must maintain the first magnitude in working memory and then calculate the difference to the second magnitude. This can be effortful and error prone. To relieve the cognitive load, the observer may opt for a covert-classification strategy instead. Covert-classification strategies. The defining feature of all covert-classification strategies is that they classify the stimuli individually. Neither memorization nor subtraction of continuous magnitudes is necessary. The most straightforward covert-classification strategy was proposed by Pollack and Pisoni (97). We refer to it as CC (covert classification with one criterion). The first stimulus is classified as either A or B. Then the second stimulus is classified independently and with the same criterion. The observer responds same iff the two labels match. Figure D plots the resulting partitioning of the decision space. When the criterion is unbiased, the overt responses are unbiased and optimal (Figure A; Noreen, 98). When the criterion is biased, the overt responses are biased in favor of same. This prediction is in qualitative agreement with much same different data (Macmillan & Creelman, 005). Quantitatively, however, CC requires unrealistically high covert biases to account for the observed probabilities of responding same. A preference for same responses seems natural with hard-to-discriminate stimuli. On many trials, the stimuli are perceived as similar because neither of them can be (covertly) classified with confidence. This idea can be formalized by a decision rule with two criteria. The first stimulus is labeled A if its magnitude is below c, B if the magnitude is above c, and ambiguous otherwise. Then the second stimulus is classified independently and with the same criteria. The observer responds different iff the labels are unambiguous and different. All ambiguous stimuli produce a same response. When the two criteria satisfy the constraint c c 5 0, we have a covertclassification strategy with two symmetric criteria (CCs; Figure E). Without this constraint, we have a CC strategy with two arbitrary criteria (CCa; Figure F). Reversed-classification strategies the final members of this family formalize the following intuition. Suppose that the first magnitude exceeds some high criterion and is confidently labeled as B. To respond different, the observer demands that the second magnitude be classified with comparable confidence in the opposite category, demarcated by an equally stringent criterion of the opposite sign. Thus, if the first stimulus is classified as B when x. c, the second stimulus is classified as A when

5 Same Different Decision Rules 05 x, c. If the first stimulus is classified as A when x, c, the second stimulus is classified as B when x. c. With symmetric criteria (c 5 c ), the resulting rule coincides with the CCs rule. With asymmetric criteria, however, RCa and CCa partition the decision space differently and predict different patterns among the observed probabilities (Figures I and F). Likelihood-ratio strategies. The third family of decision rules involves likelihood ratios. They guarantee statistically optimal performance but require substantial computational power and detailed knowledge of the parameters of the perceptual distributions. This is not psychologically realistic, especially in the beginning of the experimental session. Thus, likelihood-ratio rules are interesting mainly as ideal-observer benchmarks. Let L d/s denote the likelihood ratio of different over same responses that is, the likelihood that x, x is drawn from or divided by the likelihood that it is drawn from or. Under uniform presentation frequencies and standard representational assumptions, this ratio is given in Equation 3 (Dai et al., 996; Noreen, 98). L d/s defines a saddle-shaped surface that opens up along the positive diagonal and down along the negative diagonal. The likelihood- ratio strategy with one criterion (LR) is to respond different iff L d/s. c. The decision boundary consists of two disjoint nonlinear curves. When c., there is an overall bias for same responses, and the boundary intersects the negative diagonal as is illustrated in Figure G. When c,, there is an overall bias for different responses, and the boundary intersects the positive diagonal. When c 5, there is no overall bias, and the boundary consists of two perpendicular straight lines (Figure A). ( ) = xd xd L x x e e d/ s, + ( x x ) d e. (3) + + Data sets in which p p are inconsistent with the LR strategy. The likelihood-ratio strategy with two criteria (LR) is designed to accommodate such cases. It postulates separate criteria for the two branches of the decision boundary. Figure H plots an example in which p. p. The decision process involves two consecutive comparisons. First, adopt criterion c when x, 0 and criterion c otherwise. Second, respond different iff L d/s exceeds the criterion adopted on the first step. The curvilinear contours of equal likelihood ratios can be approximated by angle-shaped piecewise-linear contours (Irwin & Hautus, 997). In our notation, the LR strategy is approximated by the RCs strategy and, equivalently, by the CCs strategy (cf. Figures G and E). The LR strategy is approximated by the RCa strategy introduced here for the first time (cf. Figures H and I). The CC strategy is optimal only in the special case of unbiased criterion (Figure A). Symmetries and Their Observable Consequences So, a participant in a same different experiment can adopt a variety of decision rules organized in three families. Our goal is to develop a framework for making observation- based inferences about these rules. The notion of axial symmetry plays a key role in this framework. Definition. A curve is symmetric with respect to an axis when the mirror image of each point on the curve also lies on the curve. Our main leverage comes from the fact that the decision boundaries of different strategies are symmetric with respect to different axes; the positive diagonal (through µ A, µ A and µ B, µ B ), the negative diagonal (through µ A, µ B and µ B, µ A ), or both. Figures A I are a visual guide to these symmetries. Each panel plots the axes of symmetry of the corresponding strategy. Given that the perceptual distributions are symmetric too, each decision rule gives rise to a characteristic pattern of equalities among the response probabilities. These patterns are summarized in Theorem, which is the main theoretical result in this article. Theorem. Under the representational assumptions listed in the previous section, the decision rules in Table make the following parameter-free predictions about the response probabilities defined in Equation : DF implies p 5 p and p 5 p DF implies p 5 p CC implies p 5 p and ( p.5) 5 (p.5)( p.5) CCs implies p 5 p and p 5 p and p \$ p CCa implies p 5 p and p \$ p p RCa implies p 5 p and p p \$ p LR implies p 5 p and p 5 p LR implies p 5 p when there is an overall bias for same responses and p 5 p when there is an overall bias for different responses. The proof is straightforward and is given in Appendix A. In a nutshell, symmetric decision boundaries demarcate symmetric response regions, the perceptual distributions have (mutually) symmetric probability density functions, and the integrals of symmetric densities over symmetric regions are equal. Theorem reveals the implicit regularities of each family. The differencing family is characterized by p 5 p. Any observer who follows any differencing strategy must show this pattern, regardless of their sensitivity or criteria. The reason is built into the decision rule itself: The difference, δ 5 (x x ), always has the same (zero) mean when x and x are drawn from distributions with identical means. Thus, the distinction between and is lost, because each of these pairs projects to the same δ. The covert-classification family, with the exception of RCa, is characterized by p 5 p. Again, the reason is built into the decision rule itself: The XOR rule that combines the covert-classification labels is order invariant. Specifically, the overt response for an A followed by a B matches that for a B followed by an A. Therefore, any strategy that applies the same covert-classification criteria on both presentation intervals predicts identical performance for the two heterogeneous stimulus pairs. (Homogeneous pairs break the XOR symmetry because of their nonoverlapping cross-classification possibilities.)

6 06 Petrov The reversed-classification rule with two asymmetric criteria (RCa, Figure I) is exceptional, because it applies different criteria on the two intervals. The characterization of the likelihood-ratio family depends on the overall response bias. Typically, observers favor same over different responses. In those cases, all LR strategies predict p 5 p. This is because cutting the saddle surface defined by Equation 3 above the neutral point produces decision boundaries that are symmetric with respect to the negative diagonal (Figure H). Strategy RCa, designed to approximate LR, also follows this pattern. In the complementary (and rare) case, the saddle is cut below the neutral point, and the axis of symmetry flips to the positive diagonal. The observable manifestations are a preponderance of different responses and p 5 p. Finally, all three families contain a member whose decision boundaries are symmetric with respect to both diagonals. Thus, strategies DF, CCs, and LR predict both p 5 p and p 5 p. Methodology These predictions can be used as empirical tests to rule out individual strategies or even whole families. The strength of Theorem is that it makes qualitative predictions about the contrasting families. When such a prediction is violated in a given data set, an entire family can be ruled out from further consideration. This process is referred to as strong inference (Platt, 964) or falsification (Popper, 963) in philosophy of science. It differs from model fitting, where families compete via their best- fitting representatives. In strong inference, families are ruled out when their qualitative predictions are falsified. The two methodologies are complementary. Strong inference is applied first to narrow the field. No parameter search is necessary. Model fitting, which requires search, can then be applied to the remaining candidates. We recommend that the analysis of every same different data set begin with two qualitative tests: Is p equal to p? Is p equal to p? This leads to the following four cases. Case I: p 5 p but p p When there are statistically significant differences between the probabilities to respond different to stimuli and, strategies CC, CCs, CCa, DF, and LR can be rejected. The likelihood-ratio strategy LR can be rejected too when there is an overall preponderance of different responses. Otherwise, LR is technically compatible with the data. It remains psychologically implausible, however, because it depends on unrealistic amounts of knowledge and processing power. This leaves only two main contenders: the differencing rule with two criteria (DF) and the reversed-classification rule with two asymmetric criteria (RCa). Unfortunately, they imply different d s. The RCa model approximates the optimal decision strategy (LR) and thus needs higher perceptual noise to account for the observed error levels. The DF model accounts for them through a combina- tion of perceptual noise and suboptimal decision making. Thus, d RCa is always less than d DF on the same data. Case II: p 5 p but p p When there are statistically significant differences between the probabilities to respond different to stimuli and, strategies DF, DF, RCa, CCs, LR, and LR can be rejected. This leaves two covert-classification strategies: CC and CCa. They predict identical d s, because the former is a special case of the latter. CC can be identified by its strong prediction ( p.5) 5 (p.5)( p.5), proven in Appendix B. Case III: p 5 p and p 5 p This is the case that is implicitly assumed throughout the literature (e.g., Dai et al., 996; Macmillan & Creelman, 005; Noreen, 98). There is a common hit rate, H 5 p 5 p, and a common false alarm rate, FA 5 p 5 p (Equation ). All asymmetric rules (DF, CC, CCa, and RCa) can be rejected. Two main contenders remain: the differencing strategy with one criterion (DF) and the covert- classification strategy with two symmetric criteria (CCs, which coincides with RCs). The CCs rule closely approximates the optimal LR rule (Irwin & Hautus, 997) and is much more psychologically plausible. As in Case I, the near-optimal rule implies lower d than the differencing rule: d CCs, d DF. The discrepancy is most pronounced when the probability of responding different is small (Macmillan & Creelman, 005, p. 4). Case IV: p p and p p When both symmetries are broken, all decision rules in Table must be rejected. The assumptions of independence, stimulus unidimensionality, and/or negligible memory noise are probably violated. There may be strong sequential and/ or configural effects that are not captured by the signal detection framework. The observers may be following a memory-based strategy (Cohen & Nosofsky, 000). Statistical Test The present methodology requires a test of statistical difference between two proportions. We recommend the chi-squared test 3 because of its versatility. Let n i denote the number of presentations of stimulus pair i during an experimental block, and let y i denote the observed number of different responses. The proportions ˆp i 5 y i /n i estimate the true response probabilities p i. The null hypothesis is that these probabilities are equal for the two stimulus pairs under consideration: p 5 p 5 p. The pooled proportion ˆp in Equation 4 allows a more stable estimate of the sampling variance. When the null hypothesis is correct, the X statistic in Equation 5 has a χ distribution on one degree of freedom (Collett, 003); X y pˆ + y =, n + n ( ) + ( ) ( ) n pˆ pˆ n pˆ pˆ = pˆ pˆ (4). (5)

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