Absolute Identification 1 Running Head : RELATIVE JUDGMENT MODEL Absolute Identification by Relative Judgment Neil Stewart

In unidimensional absolute identification tasks, participants identify stimuli that vary along a single dimension. Performance is surprisingly poor compared to discrimination of the same stimuli. Existing models assume that identification is achieved using long-term representations of absolute magnitudes. We propose an alternative relative judgment model (RJM) in which the elemental perceptual units are representations of the differences between current and previous stimuli. These differences are used, together with the previous feedback, to respond. Without using long-term representations of absolute magnitudes, the RJM accounts for (a) information transmission limits, (b) bowed serial position effects, and (c) sequential effects, where responses are biased towards immediately preceding stimuli but away from more distant stimuli (assimilation and contrast). Absolute Identification 3 Absolute Identification by Relative Judgment Miller (1956) drew attention to a curious phenomenon. People have great difficulty identifying stimuli from a set that varies along a single psychological continuum, even though their ability to discriminate pairs of stimuli from the set suggests that they should be very good at the identification task. This phenomenon can be seen across a wide range of stimulus attributes the frequency and loudness of tones, the strength of tastes and smells, the magnitude of lengths and areas, the hue and brightness of colors, and the intensity and numerousness of cutaneous stimulation suggesting some common and fundamental source of the limitation. In an absolute identification task, participants are required to identify, with a unique label, stimuli drawn from a set of items that vary along only a single continuum. Typically, stimuli are evenly psychologically spaced. A stimulus's label is normally its ordinal position within the set. Three key phenomena, which we review in more detail below, are observed. First, there is a severe limit in the information transmitted from stimulus to response (i.e., the size of the set for which members can be identified perfectly) even when adjacent stimuli are perfectly discriminable. Second, a bow effect is observed when identification accuracy is plotted against stimulus, with an advantage for the smallest and largest stimuli. Third, there are strong sequential effects, whereby the stimuli on previous trials exert a strong bias on the response to the current stimulus. Many theoretical accounts have been offered of one or more of these phenomena. Nearly all of these models have in common the assumption that in an absolute identification task a representation of the absolute magnitude of the current stimulus is compared to some long-term representations of the absolute magnitudes either of other stimuli from the set, particular anchor values, or particular criterial values. However, in a review for the centenary issue of Psychological Review, Shiffrin and Nosofsky (1994, p. 359) conclude that since Miller's (1956) classic article "...a fully unified account of the numerous range, edge, and Absolute Identification 4 sequential effects has not been achieved." Here, in contrast to existing models (excepting Laming, 1984), we offer a relative judgment model (RJM) of absolute identification. The RJM does not utilize long-term representations of absolute magnitudes. Instead, the difference between the current stimulus and the previous stimulus is used, in conjunction with the feedback from the previous trial, to generate a response. Thus, the magnitude of the current stimulus is judged relative to the magnitude of only the immediately preceding stimulus (hence the name RJM). In this article, we review existing models of absolute identification and show that none offers a complete account of the phenomena described above. We then show that the RJM offers a unified account of these phenomena, and present new experimental evidence that supports the model. We begin with a review of the key empirical results. Empirical Results in Absolute Identification Information Transmission Limit Using multivariate information transmission as a dependent variable (McGill, 1954), it is possible to measure the information transmitted1 in an absolute identification task. If performance in an absolute identification task were perfect, the information transmitted would grow as the number of stimuli is increased. For example, perfect identification of two equally probable stimuli carries 1 bit of information, identification of four stimuli carries 2 bits, of eight stimuli carries 3 bits, and so on. However, the information transmitted from stimulus to response (sometimes, channel capacity) in an absolute identification task seems to be limited to very few bits (see Table 1) corresponding to perfect identification of very few stimuli across a very wide range of stimulus attributes (see Garner, 1962; Laming, 1984; Miller, 1956 for reviews). Figure 1 shows information transmitted as a function of the number of stimuli in the set (with range of the stimuli held constant) for data from Garner (1953) and Pollack (1952). (In all figures that present data, data are collapsed across participants.) With a small number of stimuli, obviously less information must be transmitted, but as the number of stimuli increases, Absolute Identification 5 the information transmitted from stimulus to response does not continue to increase. Although an increase in the range of stimuli (number held constant), and hence the separation of the stimuli, will produce an initial increase in information transmitted, the increase is a negatively accelerated function of range, and quickly reaches an asymptote once adjacent stimuli are discriminable (Alluisi & Sidorsky, 1958; Braida & Durlach, 1972; Eriksen & Hake, 1955a; Pollack, 1952). Bow or Edge Effects in the Serial Position Curve When accuracy is plotted as a function of the rank of the stimulus within the stimulus set, a characteristic bow is observed in the resulting serial position curve (e.g., Kent & Lamberts, in press; Lacouture & Marley, 2004; Murdock, 1960; W. Siegel, 1972). Performance on stimuli at the ends of the range is better than performance on mid-range stimuli even though, when presented in isolation, any two adjacent stimuli may be perfectly discriminable. As for information transmission, once stimuli are pairwise perfectly discriminable, increased spacing of items leads, at best, to only slight improvements in accuracy (Braida & Durlach, 1972; Brown, Neath, & Chater, 2002; Gravetter & Lockhead, 1973; Hartman, 1954; Lacouture, 1997; Luce, Green, & Weber, 1976; Pollack, 1952). Figure 2 shows the very similar stimulus-response confusion matrices obtained by Brown et al. (2002) for absolute identification of tones varying in their frequency. Tones were geometrically spaced, with each tone a constant ratio higher in frequency than the immediately lower tone. (Following Weber's Law, geometric spacing is typically used to produce stimuli that are presumed to be equally psychologically spaced.) Each confusion matrix is for a different stimulus spacing (from 420 563 Hz in the narrow spacing condition to 363 652 Hz in the wide spacing condition). As Figure 2 shows, increasing the stimulus spacing had almost no effect on performance. Increasing the number of stimuli in an absolute identification task increases the size of the bow effect (Alluisi & Sidorsky, 1958; Durlach & Braida, 1969; Lacouture & Marley, Absolute Identification 6 1995; Pollack, 1953; W. Siegel, 1972; Weber, Green, & Luce, 1977). Figure 3 shows the serial position curves obtained by Lacouture and Marley (1995; see also Kent & Lamberts, in press; Lacouture, Li, & Marley, 1998) for different stimulus set sizes, with a larger bow effect for larger set sizes. Note that although Stimuli 5 and 6 can be nearly perfectly discriminated when they constitute the entire stimulus set, performance on these same stimuli drops considerably when they are identified within a larger stimulus set. Simply shifting all the stimuli along the dimension, so that each stimulus increased in value by a constant multiplicative factor, has no effect on the accuracy against stimulus magnitude curve (Lacouture, 1997). The bow effect remains even after extensive practice, although small improvements in accuracy are observed (Alluisi & Sidorsky, 1958; Hartman, 1954; Weber et al., 1977; but see Rouder, Morey, Cowan, & Pfaltz, 2004, for a larger practice effect). The bow effect is greatly reduced by correcting for the asymmetry of errors on extreme verses interior stimuli (Weber et al., 1977), suggesting that the restricted opportunity to make errors at the ends of the range is a major factor underlying the bow effect (see also Eriksen & Hake, 1957). The bow effect is not due to response bias (at least, not response bias alone). In data where end responses are not used more frequently than central responses, the effect is still observed (W. Siegel, 1972). In our data from Experiment 1, the bow is observed although there is a bias against responding with extreme categories. Sequential Effects We know of no absolute identification experiment where strong sequence effects (where the response to the current stimulus is shown to depend on previous stimuli and responses) are not found. Of course, when performance in an absolute identification task is perfect, then there are no sequential dependencies. Thus the existence of sequential dependencies is likely to provide a useful insight into processing in an absolute identification task. The most salient sequential effect is that the response given to the current stimulus is Absolute Identification 7 shown to be assimilated to the immediately preceding stimulus (Garner, 1953; Holland & Lockhead, 1968; Hu, 1997; Lacouture, 1997; Lockhead, 1984; Long, 1937; Luce, Nosofsky, Green, & Smith, 1982; Petrov & Anderson, in press; Purks, Callahan, Braida, & Durlach, 1980; Rouder et al., 2004; Staddon, King, & Lockhead, 1980; Stewart, 2001; Ward & Lockhead, 1970, 1971). In other words, participants are systematically biased to respond as if the current stimulus is nearer to the previous stimulus than it actually is. Figure 4 shows data from the feedback condition of Ward and Lockhead's (1970) absolute identification experiment. Stimuli were tones varying in loudness. The average error in responding on the current trial is plotted for each stimulus as a function of the stimulus on the previous trial. When the current stimulus is greater than the previous stimulus, the error is negative (i.e., the stimulus is underestimated); when the current stimulus is less than the previous stimulus, the error is positive. The five lines are approximately parallel, with positive slopes, demonstrating that assimilation takes place for all combinations of current and previous stimuli (Lockhead, 1984). Assimilation to preceding items is also observed in magnitude estimation tasks (e.g., Jesteadt, Luce, & Green, 1977), in matching tasks (Stevens, 1975, p. 275), and in relative intensity judgment tasks (Lockhead & King, 1983). The effect of stimuli further back in the sequence on the current response is the opposite, that is, there is a contrast effect (Holland & Lockhead, 1968; Lacouture, 1997; Ward & Lockhead, 1970, 1971). Assimilation to the previous trial and contrast to trials further back has been demonstrated within the same experiments, for the same participants. Figure 5 shows the average error on the current trial (averaged across all possible stimuli on the current trial) as a function of the stimulus k trials ago for data from Holland and Lockhead (1968), Lacouture (1997), and Ward and Lockhead (1970). As described above, assimilation is shown to the stimulus on the immediately preceding trial (k = 1). Stimuli on less recent trials (k > 1) exhibit contrast, as shown by the reversal in the sign of the error. The contrast effect is smaller than the assimilation, and the error dependency reduces to zero with increased Absolute Identification 8 numbers of intervening trials. In the experiments on sequence effects discussed so far stimulus, response and feedback are all highly correlated. Which of these is the basis for assimilation (and contrast)? We focus on this question for the remainder of this section. The sequence effects observed are dependent on the quality of stimulus presentation. Ward and Lockhead (1971) examined performance in a standard absolute identification experiment using line length. When task difficulty was increased, by reducing the luminance and duration of line length presentations, more assimilation was observed. In the difficult condition, accuracy was low and therefore the correlation between stimuli and responses was reduced. Assimilation was demonstrated only to the previous stimulus and not the previous response. This suggests that assimilation to the previous response is only normally observed because the response is correlated with the previous stimulus (see also Garner, 1953; McGill, 1957; Mori, 1998). Ward and Lockhead (1971) also observed assimilation to the previous trial's feedback but not to the previous response in a guessing task (although there was slight evidence of a small contrast effect to responses further back in the sequence). The guessing task was identical to an absolute identification experiment, except that the stimuli were omitted, and therefore the stimuli could not have been the cause of the assimilation observed. As the task was guessing, there was no correlation between the feedback and the responses. Thus the observation of sequential effects only for the previous feedback but not the previous response in a task where the two are not correlated, also suggests that previous responses are not the locus of sequential effects. Manipulating the Sequence in an Absolute Identification Task Manipulating the relative frequencies of the size of the differences between consecutive trials affects identification accuracy. In an absolute identification of loudness task, Luce et al. (1982) used four differently constrained sequences. In one condition, the sequence of trials Absolute Identification 9 was constrained so that the current stimulus was either identical to, one step softer than, or one step louder than the previous stimulus. This condition was called the small step (3) condition, because the current stimulus was chosen from one of three stimuli centered on the previous stimulus. In the small step (5) condition, the current stimulus was selected from five adjacent intensities centered on the previous stimulus. In the random condition, the sequence was random. In the large step condition, the current stimulus was at least four stimuli different from the previous stimulus. For all four sequence types, each intensity was equally frequent over the course of the whole experiment. From the identification confusion matrix a measure of the confusability, d í, i + 1, of each loudness i with the adjacent loudness i + 1, was obtained. This method of analysis allows comparison of identification performance free from contamination by constraints imposed by the control of the sequences in each condition.2 (The procedure for calculating d í, i + 1 is given in Appendix A.) When d í, i + 1 is plotted against stimulus magnitude, each condition shows a characteristic bow, with poorer performance for the middle of the range of signals (see the bottom panel in Figure 6). The key result is that the curves lie one above the other, such that tones are more confusable in the conditions where the step size is larger: In order of decreasing identification performance, the curves are small step (3), small step (5), random step, and large step. Smaller transitions seem to lead to higher accuracy (see also Hu, 1997, and Petzold and Haubensak, 2001, for similar findings). (The top panel of Figure 6 shows the corresponding bows in the accuracy serial position curves. Here the ordering of the large step and random conditions is reversed, with better performance in the large step condition because of the restricted possibility for making mistakes imposed by the restricted set of possible responses on each trial.) Further work (Nosofsky, 1983a), testing alternative hypotheses, is consistent with the idea that smaller transitions lead to higher accuracy. Existing Models of Absolute Identification There are many existing accounts of some of the phenomena seen in absolute Absolute Identification 10 identification data. The extant models can be divided into four main classes: (a) models where memories of recent stimuli are assimilated (Holland & Lockhead, 1968; Lockhead & King, 1983), (b) modified Thurstonian models (Braida et al., 1984; Durlach & Braida, 1969; Purks et al., 1980; Luce et al., 1976; Treisman, 1985), (c) limited response or processing capacity models (Lacouture & Marley, 1991, 1995, 2004; Laming, 1984, 1987; Marley & Cook, 1984, 1986), and (d) exemplar models (Brown et al., 2002; Kent & Lamberts, in press; Nosofsky, 1997; Petrov & Anderson, in press). Below, we briefly review each of these models and consider which of the phenomena outlined (limit in information transmitted, bow effects, and assimilation and contrast) are and are not accounted for by each model. Table 2 gives an overview of the scope of these models. Two themes emerge from this review. First, there are two different types of explanation as to why increasing the range of stimuli does not increase information transmitted. Some models assume a perceptual locus and others assume the limit lies in the response process. The second theme is that current models which assume that longterm representations of absolute magnitudes are the basis for absolute identification do not provide a full account of sequential effects. Assimilation models Holland and Lockhead (1968). In Holland and Lockhead's (1968) model, participants are assumed to generate a response by adding the judged distance between the current stimulus and the previous stimulus to the feedback from the previous trial. Assimilation and contrast are accounted for in terms of the contamination of the representations of the absolute magnitudes of stimuli. Specifically, the memory of the previous stimulus is assumed to be contaminated by the memories of earlier stimuli. Of the phenomena outlined above, Holland and Lockhead's (1968) model accounts for assimilation and contrast, but only on average. For example, consider a low magnitude stimulus on the previous trial. The stimuli on preceding trials are likely to have been larger in magnitude and thus, when the previous stimulus is confused with them, the representation of Absolute Identification 11 its magnitude will be an overestimate. This will cause the difference between the current and previous stimuli to be underestimated on average (as the current stimulus is also likely to be larger than the previous stimulus), and lead to the current response being biased towards the previous stimulus (i.e., assimilation). Contrast also follows: on average the current response is biased away from stimuli two or more trials ago because these stimuli are, on average, greater in magnitude than the (low magnitude) stimulus on the immediately preceding trial. However, a detailed examination reveals this account to be unsatisfactory. Typically, assimilation is observed for all combinations of current and previous stimuli (e.g., Ward & Lockhead, 1970; our Experiment 1). Holland and Lockhead's model predicts contrast in some cases. For example, consider the case in absolute identification of 10 stimuli with Stimulus 3 on the preceding trial followed by Stimulus 2 on the current trial. The confused representation of Stimulus 3 will be an overestimate (as earlier stimuli are likely to have been larger), and thus the difference between the current stimulus and the previous stimulus will be overestimated. This produces a contrast effect where assimilation is observed. In addition to this difficulty for the model, Holland and Lockhead give no account of the other phenomena listed in Table 2. Lockhead and King (1983). In Lockhead and King's (1983, see also Lockhead, 1984) model, two assumptions are made: (a) that successive stimuli assimilate in memory; (b) that people compare each new stimulus to a collection of stimulus memories to determine a response. No psychological mechanism is chosen to motivate these assumptions, "the focus here is on a simple equation to fit the data" (Lockhead, 1984, p. 44). The equation is Rn=S n a1 S n S n 1 a2 S n S n 2 a3 S n S n 3 ... where Rn is the response on trial n, Sn is the stimulus on trial n, a1 < 0 for assimilation to Sn 1, a2 > a3 > ... > 0 for decreasing contrast to less recent stimuli, and is a noise term. Although such a model can inevitably describe assimilation and contrast, no consideration is given to other absolute identification phenomena. In our application of Lockhead and King's model, we have found that such a model does not offer an account of the limit in information transmitted Absolute Identification 12 or the bow effect. Modified Thurstonian Models In a Thurstonian account, presentation of a stimulus results in perception of an absolute magnitude, represented as a noisy value on an internal sensory scale. Criteria, or bounds, divide this scale into response categories. The criteria provide a long-term frame of reference for absolute magnitudes. There are important multidimensional extensions of this idea (e.g., Ashby & Townsend, 1986). The source of variability in responding in the standard Thurstonian model is the noise in the representation of the stimulus on the internal sensory scale. A simple Thurstonian model can offer some account of the limit in information transmitted as the number of stimuli is increased with the range held constant (the "Range constant" columns in Table 2): As the number of stimuli is increased, the bounds will become closer together, and the fixed magnitude noise on the sensory scale means that a stimulus is more likely to be classified incorrectly. The bow effect can also be explained because there is a limited ability to make mistakes for stimuli at the edges of the range. For example, if the smallest magnitude stimulus is greatly underestimated, then it will still be correctly classified into the first category. The invariance of these phenomena as range is increased (the "N constant" columns in Table 2) and an account of sequential effects require further modification of the model. We evaluate three modifications below. Durlach and Braida (1969). Durlach and Braida (1969) modified the simple Thurstonian decision model outlined above to include an internal-noise model. Durlach and Braida propose that memory can operate in one of two modes. Here we discuss only the context-coding mode that applies to absolute identification. In the context-coding mode, the presented stimulus is compared to the general context of recent stimulus presentations. The context-coding mode adds an additional source of variability in responding (over and above the perceptual noise in the simple Thurstonian model) which results from "the inability of the Absolute Identification 13 subject to determine the context precisely and his inability to determine or represent the relation of the sensation to this context precisely" (p. 374). The standard deviation of the context-coding noise is assumed to be proportional to the range of the stimuli, though no psychological motivation is given for this assumption. The inclusion of a source of variability that grows with the stimulus range allows Durlach and Braida's (1969) preliminary theory of intensity resolution to account for the invariance of the absolute identification phenomena as range is increased (either by increasing the number of stimuli with the spacing held constant or increasing the spacing). To account for the bow effects, Braida et al. (1984) suggested that the general context is set by two anchors at either end of the range. A stimulus is compared to the general context by counting steps (which are some proportion of the distance between the anchors) using a noisy measurement unit. Thus, there is less variability for stimuli near one of the anchors, as a small number of steps is small, and thus the cumulative error is small.3 No mechanism is offered to account for sequential effects. However, Purks et al. (1980) suggest that the distributions that represent signals are unaffected by the location of the previous signal, but that the category boundaries are. By partitioning their data by the previous signal, and fitting a Thurstonian decision bound model to each partition, Purks et al. demonstrate that the separation between signal distributions was unaffected, but the locations of decision boundaries were, being shifted away from the previous signal. The next modification of the simple Thurstonian model extends this idea. Treisman (1985). Treisman (1985) used criterion-setting theory (Treisman & Williams, 1984) to maintain response criteria in a simple Thurstonian model. Two opposing short-term mechanisms act on the criteria on a trial-by-trial basis. A tracking mechanism, motivated by the assumption that objects in the real world tend to persist, moves criteria away from the currently perceived sensory effect, increasing the probability of a repetition of the previous response. A stabilizing mechanism acts to locate criteria nearer to the prevailing flux of Absolute Identification 14 sensory inputs, motivated by the assumption that criteria will be adjusted to maximize information transmitted. Tracking shifts are larger in magnitude than stabilizing shifts, but decay more quickly. Thus, Treisman's model predicts assimilation to the immediately preceding stimulus, when tracking shifts will dominate, but contrast to less recent stimuli, when stabilizing shifts will dominate. However, this account does not fully explain the sequential biases. Treisman (1985, p. 192) states that the magnitude of criteria movement decreases with the distance of the criteria from the stimulus. Therefore, the model would be expected to predict greater assimilation where previous and current stimuli are similar. However, assimilation is greater where stimuli differ more, rather than less (see, e.g., Figure 4). For the same reason, Treisman's model predicts that the error in responding should be greater if the previous and current stimuli are similar: Luce et al. (1982), Nosfosky (1983), Hu (1997), and Rouder et al. (2004) found the opposite result. The magnitude of the stabilizing shift is, unlike tracking shifts, not fixed, but instead proportional to the distance between the sensory input and the nearest criterion. The magnitude of stabilization is thus proportional to the inter-criterion distance and therefore also proportional to the range. This allows Treisman's (1985) model to explain why increasing the range of stimuli does not increase information transmitted. But it also means that as the range is increased, stabilization should come to dominate, predicting a change in the pattern of assimilation and contrast which is not observed (e.g., Experiment 1 below). Treisman's (1985) model does not predict the gradual and smooth U-shaped pattern of the bow effect (instead accuracy is approximately equal for all but the two most extreme stimuli). However, if criteria in the central region are more closely spaced, bow effects will be more U-shaped. Such a spacing would also lead to a response bias for extreme stimuli, but the opposite pattern a central tendency in responding is typically observed (see Experiment 1). Luce, Green, and Weber (1976). Luce et al. (1976) proposed a modified Thurstonian model, where an attention band roves over the stimulus range. Items falling in the band result Absolute Identification 15 in a less variable Thurstonian representation than those that do not. As the stimulus range is increased, the probability of stimuli falling inside the fixed width band is reduced, causing a reduction in identification performance. This allows the model to predict a limit in performance as stimulus range is increased. With the additional assumption that the attention band dwells at the edges of the range of stimuli, bow or edge effects are accounted for, although no motivation for this assumption is given. The further assumption, that attention tends to dwell on the location of the last stimuli, explains the finding that there is typically reduced variation in responding when the previous stimulus is similar to the current stimulus (e.g., Luce et al., 1982). However, the attention band model does not offer an account of the systematic bias in responding to the current stimulus by preceding stimuli (i.e., assimilation and contrast). Restricted Capacity Models Cook, Lacouture, and Marley have presented three models of absolute identification that account for limits on information transmitted and bow effects by assuming a limited capacity process in either memory or response processes (not perceptual processes). These models can account for limits in performance as stimulus range is increased because they do not assume that the limit in information transmitted is perceptual. Marley and Cook (1984, 1986), Karpiuk, Lacouture, and Marley (1997). In Marley and Cook's (1984, 1986) models, perception is assumed to be absolute, with the location of the stimulus represented accurately on a Thurstonian continuum. The exact location on the continuum is unavailable to the response process, and must be deduced by comparing the stimulus to the context in which it is presented. Marley and Cook assume that the context, which comprises a set of elements, must be rehearsed. Each element's activation is incremented each time a pulse arrives from a Poisson pulse process, before its activation continues to decay exponentially. There is a fixed total rehearsal capacity modeled by limiting the pulse rate across all elements. The location of the stimulus within the continuum of Absolute Identification 16 elements is derived by summing the activation of the elements between the stimulus and known anchors (cf. Braida et al., 1984). The anchors are assumed to be at or outside the location of the extreme stimuli. Marley and Cook (1984) show that, under these assumptions, the variability of the total activity of elements to one side of the stimulus increases with the number of elements. Marley and Cook (1984) demonstrated that the model can account for: (a) the asymptote in information transmission as stimulus range increases, with the number of stimuli held constant, or as the number of stimuli increases, with the spacing held constant; and (b) the bow effect. Karpiuk et al. (1997) extended the model to predict reaction time distributions. Marley and Cook provide no account of the sequential effects observed. Marley and Cook also point out that, if their model were extended to provide the necessary account by assuming the range of the rehearsal is determined by the immediately preceding context, it is not clear how it could explain assimilation and contrast without a further addition to the model. Lacouture and Marley (1991). Lacouture and Marley (1991) demonstrated that a simple network model could provide a reasonable account of the limit in information transmitted. The model was a three layer feed-forward network that learned by mean-variance back-propagation of error. Input vectors of adjacent stimuli overlapped. For example, if Stimulus 5 was presented, input unit 5 would be activated, but neighboring hidden units, 4 and 6, would also be activated to a lesser extent. The model predicts the limit in information transmitted when the number of hidden units is one, although the observed characteristic shape of the information transmitted against set size (see our Figure 1) is not well reproduced. The model does however produce a good fit to Braida and Durlach's (1972) data, where information transmitted was measured as stimulus range was varied (with set size held constant). Modeling of bow and sequence effects is not described. The model could not provide an account of sequence effects without substantial modification because, once learning has reached asymptote, the representation and processing of a stimulus is independent of Absolute Identification 17 immediately preceding stimuli. Lacouture and Marley (1995, 2004). Lacouture and Marley's (1995, 2004) mapping model is a feed-forward network with one single input unit, one single hidden unit, and an output unit for each response. The activation of the input unit represents the magnitude of the stimulus. Perception is assumed to be noisy and repeated presentation of the same stimulus does not always lead to the same activation. The hidden unit normalizes this activation using a lower and an upper anchor value so that the resulting activation falls within the range 0 to 1. Fixed magnitude noise which represents a noisy mapping process is then added, resulting in a limited channel capacity. With a large number of stimuli, the resulting set of possible mean hidden unit activations will be closer together than for a smaller set and, thus, the fixed magnitude noise will have a greater effect on performance for larger sets. The mapping of the hidden unit activation onto output units acts to partition the unit interval into response categories. For each output unit, activation is accumulated over the course of the trial, with the corresponding response being emitted once the accumulator reaches a given threshold (Lacouture & Marley, 1995). The assumption of repeated intra-trial sampling of the output units allows the model to predict response times as well as accuracy, providing an extension over previous models. Lacouture and Marley (2004) replace the accumulator and threshold with a leaky, competing accumulator (Usher & McClelland, 2001) to capture full correct response time distributions. The mapping model provides an account of the limit on information transmitted and of bow effect for different set sizes. By incorporating the (unmotivated) assumption that, after a response is made, there will be less variation in the output of that unit and those immediately adjacent to it on the next trial, the data from the sequence manipulation experiments (Luce et al., 1982) are also accounted for. However, this model suffers the same difficulty as the attention band model described above in accounting for sequence effects. Lacouture and Marley (2004) suggest three modifications to the model that might allow a future version of Absolute Identification 18 the model to account for sequential effects: (a) Instead of normalizing hidden unit activation by using two anchor values, previous stimulus values could be used. (b) Hidden unit activations may be contaminated with hidden unit activations from previous stimulus presentations. (c) The leaky competing accumulators may begin each trial with some residual activation carried over from previous trials. Laming's (1984, 1997) Relative Judgment Model Laming (1984) describes a model that accounts for the limit in information transmission and the effects of constraining possible jump sizes between successive stimuli (i.e., Luce et al., 1982). The crucial assumption in Laming's model is that all judgments are relative to the immediate preceding context (i.e., that it is the differences between successive stimuli that are used, not the absolute magnitudes of the stimuli). Further, Laming proposes that such relative judgments are limited. Specifically, Laming suggests that the current stimulus can be judged as 'much less than', 'less than', 'equal to', 'more than', or 'much more than' the previous stimulus. This judgment limit provides a limit in the information transmitted. Numerical estimates of the stimulus magnitudes are assigned such that they follow the same pattern. If the difference between stimuli is judged as 'less than', for example, then the number assigned to the estimate of the magnitude of the current stimulus is less than the estimate of the magnitude of the previous stimulus. The ordering of Luce et al.'s (1982) conditions is also explained by Laming's model. Laming shows that the variability in responding depends mainly on the mean squared jump sizes in the sequence and, as jump size predicts perfectly the ordering of performance in Luce et al.'s conditions, so does Laming's model. Laming (1984) does not offer an account of the bow in the serial position curve, or of assimilation and contrast. Indeed, Laming states that an additional principle will be required to provide an account of the sequential effects observed in magnitude estimation and absolute identification. He suggests taking into account the prior expectations of the distribution of stimulus magnitudes as a candidate principle. Absolute Identification 19 Exemplar Models Exemplar models (Medin & Schaffer, 1978; Nosofsky, 1986) assume a long-term memory for each stimulus's magnitude, together with the label associated with that stimulus. On presentation of a stimulus, the probability of a given response is given by similarity of the presented stimulus to the memory of the stimulus associated with that response divided by the summed similarity of the presented stimulus to each stimulus memory. Brown, Neath, and Chater (2002). Brown et al.'s (2002) model of scale-invariant memory, perception and learning (SIMPLE) has been applied to absolute identification data (as well as free, serial, and probed recall memory tasks). The model is an exemplar model of absolute identification, and is equivalent to the generalized context model (Nosofsky, 1986) in its application to absolute identification. Exemplar models of absolute identification provide a reasonable account of bow effects. Bow effects are accounted for because items at the end of the range have fewer similar neighbors to be confused with. However, exemplar models do not predict the gradual bowing that is typically observed: Instead, all items tend to show almost identical levels of performance, except for superior performance on the very edge items. It is possible to provide a better fit by biasing the responses associated with more extreme stimuli. However, this bias for extreme responses is at odds with the central tendency in responding that is typically observed. Further, as described above, the bow effect is still observed in data where each response is used equally often (W. Siegel, 1972) or where middle responses are used more often (see Experiment 1). Exemplar models face a further problem. Recall that increasing the spacing of stimuli does not remove the bow effect, and leads only to a slight improvement in accuracy. Exemplar models, however, predict a large improvement in accuracy, as items become more discriminable. Brown et al. (2002) introduce the assumption that discriminability is inversely related to stimulus range and show that, with this additional assumption, the bow effects are Absolute Identification 20 invariant under stimulus range. Exemplar models do not predict any curves in d' without further assumptions. In their simplest form, exemplar models offer no account of sequence effects. When adapted to predict sequence effects, typically by assuming more recent exemplars are more available in memory and/or weighted more heavily in the subsequent decision process (e.g., Nosofsky & Palmeri, 1997; see also Elliott & Anderson, 1995) the models do not correctly predict sequence effects observed in classification (Stewart and Brown, 2004; Stewart, Brown, & Chater, 2002). Increased weighting of more recent items will make a prediction similar to assimilation, as repetition of the previous response will be more likely. However, the criticism applied to the Thurstonian models above applies: An increased probability of repetition is not equivalent to assimilation. Further, this modification will provide no account of contrast. Nosofsky (1997). Nosofsky (1997) applied Nosofsky and Palmeri's (1997) exemplarbased random walk model, which is an extension of the generalized context model (Nosofsky, 1986), to predict responses and reaction times in absolute identification. Stimuli are represented by normal distributions on a psychological continuum, with stimuli at the edges of the range assumed to be less variable. In this way, an account of bow or edge effects is built into the model. On presentation of a test item, the model assumes memories race to be retrieved. The probability of a memory being retrieved at a given time is a function of the exemplar's similarity to the test item, and the exemplar's strength in memory. Once an item is retrieved, a counter for the associated category label is incremented and all others decremented. The remaining items then race again. When any counter falls too low, the response associated with the counter leaves the race. When a counter reaches a given threshold, the response associated with that counter is emitted, with the reaction time being a function of the sum of the times for each retrieval. No mechanism is outlined for prediction of sequence effects, and no explanation is offered on the invariance of the bow in serial position Absolute Identification 21 when stimulus range is altered. Petrov and Anderson (in press). Petrov and Anderson (in press) present a scaling model (ANCHOR) based on the ACT-R architecture (Anderson & Lebière, 1998) that they apply to absolute identification and category rating. The perception of the absolute magnitude of a stimulus is compared to anchors or exemplars stored in memory. Perception is assumed to be stochastic. The selection of exemplars is also stochastic, and depends upon the similarity between the exemplars and the target stimulus and also upon the frequency and recency with which each exemplar was previously used. Exemplars compete for selection. One exemplar is selected and the associated response is retrieved. If there is a discrepancy between the exemplar magnitude and the percept magnitude, then an adjustment is applied to the response to correct it either up or down. The system is adaptive and, after feedback, the location of the associated exemplar is assimilated towards the percept. Petrov and Anderson fitted the model to their own data from an absolute identification of nine stimuli. The model was able to fit simultaneously the information transmitted, central tendency in responding, assimilation (on average), and a small practice effect. The model did not predict bows in d', but was able to predict an accuracy advantage for end stimuli because of the limited opportunity for errors at the ends of the range. Petrov and Anderson did not model the effect of increasing the number of stimuli (with the range held constant) or the range of the stimuli (with the number held constant) and, thus, it is uncertain whether the model could account for the effects of these variables. However, the model does include noisy components that are independent of the spacing of the stimuli, and so it may well be able to account for the effects without modification. Petrov and Anderson did not examine assimilation in detail. In Figure 7, we show the predictions of ANCHOR for the effect on the current response (a) of the previous stimulus and current stimulus (top panel) and (b) for the effect of the stimulus at different lags (bottom panel). We used the parameters that Petrov and Anderon (in press, pp. 23-24 and p. 35) report Absolute Identification 22 as best fitting their data and ran 200 simulations of 450 trials. Although ANCHOR does predict assimilation on average, it does not predict the detailed pattern that is normally observed (e.g., Figure 4 and Experiment 1) in which assimilation increases as the difference between the previous and current stimuli increases. In ANCHOR, sequential effects are caused by exemplars being weighted by their recency of use. Thus, ANCHOR fails to predict the more detailed pattern of assimilation for the same reason as Treisman (1985) and Luce et al.'s (1976) models: Predicting that the response associated with the previous stimulus is more likely to be repeated is not the same as predicting that the current response will be biased towards the previous stimulus. Further, the model does not predict contrast to stimuli at lags of 2 or greater, and instead predicts assimilation to stimuli at these lags. Kent and Lamberts (in press). Kent and Lamberts (in press) present an application of Lambert's (2000) extended generalized context model (EGCM) to absolute identification. The EGCM differs from the GCM in assuming that the amount of information about a stimulus magnitude increases over the time course of the stimulus presentation as stimulus elements are sampled. The probability that sampling is halted and a response is given increases as more elements are sampled. The EGCM was able to predict bow in accuracy and mean correct reaction times, as well as the complete reaction time distributions for individual stimuli. By allowing more generalization for larger set sizes and less sampling for larger set sizes, the EGCM could also predict changes in accuracy and reaction time as set size varied. An alternative, more parsimonious modification, in which the information contributed by each additional sample was a decreasing function of the number of samples, also allowed the model to account for set-size effects. Kent and Lamberts did not model the effect of changing the stimulus spacing. However, without altering the discriminability (cf. Brown et al., 2002), the EGCM predicts that performance will increase greatly with increased spacing (limits in information transmitted have also not been modeled). The model does not predict any sequential effects. Absolute Identification 23 Motivation of the Relative Judgment Model Having reviewed the key empirical phenomena and the existing models of absolute identification, we next lay out the motivation for the RJM. We have made two main choices in developing the RJM. First, we assume that the locus of the limit in performance is not perceptual but judgmental. Second, we assume that judgment is relative and not absolute. We give our motivation for these assumptions below. The Locus of the Effects is Not Perceptual As we have reviewed above, as the range of the stimuli is increased, performance quickly reaches asymptote (Braida & Durlach, 1972; Brown et al., 2002; Eriksen & Hake, 1955a; Gravetter & Lockhead, 1973; Hartman, 1954; Luce et al., 1976; Pollack, 1952). Further, stimuli that can be identified perfectly when presented in isolation are poorly identified when presented within a larger set (Lacouture & Marley, 1995; see also Nosofsky, 1983a; Pollack, 1953). Typically, the variability in magnitude estimates is approximately two orders of magnitude greater than variability in threshold discriminations of the same stimuli (Laming, 1997; see also Miller, 1956; Shiffrin & Nosofsky, 1994). Theorists have taken two different approaches in accounting for these effects. One approach is to assume that the locus of the limit in performance is perceptual, and that when the range of stimuli is increased, there is a large increase in perceptual noise that keeps the information transmitted at the same level. For example, Luce et al. (1982) assume that perceptual noise increases with stimulus range because of a limit in the range over which attention can be focussed. Braida and Durlach (1972) and Gravetter and Lockhead (1973) assume that the noise in the location of the criteria in a Thurstonian model increases with stimulus range. (Although Braida and Durlach's and Gravetter and Lockhead's assumption concerns noise in the location of criteria rather than percepts, it is still noise on a perceptual scale with perceptual units.) Instead, in the RJM, we assume that what limits performance is noise in the processes of mapping a continuous valued estimate of the response onto response categories. Lacouture and Marley's (1995, 2004) Absolute Identification 24 mapping model makes the same assumption. They assume that stimulus magnitudes are scaled onto a hidden unit activation which ranges over the unit interval and that constant variance noise (completely independent of the stimulus range) is responsible for the limit in capacity. In the RJM, in assuming that mapping rather than perceptual noise is responsible for the limit in capacity, we do not require any additional assumptions to explain the lack of an improvement in performance when stimulus range is increased. One reason for this approach is parsimony. As described above, the limit in information transmitted is approximately constant across a wide range of stimulus types (see Table 1; Miller, 1956; Garner, 1962; Laming, 1984). The differences in the exact amount of information that can be transferred are perhaps less important that the fact that the limit in channel capacity seems to be generally so low. Miller (1956, p. 86) concludes, "There seems to be some limit built into us either by learning or by the design of our nervous system, a limit that keeps our channel capacities in this general range." The fact that similarly low limits in channel capacity are found across such a wide range of stimulus attributes suggests that there is a common cause to this limitation, especially as the same bow and sequential effects are observed across the same wide range. Of course, it could be that this cause is duplicated across the different sensory apparatus used in each task. But a more parsimonious explanation is that the cause resides not in the perceptual system, but in the judgment system responsible for producing responses. Relative Rather than Absolute Judgment A limitation in all of the above models (excepting Lockhead and King's, 1983, descriptive model) is the difficulty in predicting the ubiquitous pattern of assimilation and contrast (see, e.g., Figure 5 and Experiment 1). For example, a variety of modifications of the Thurstonian model have not proved adequate: Allowing the location of the criteria to be updated from trial to trial (Treisman, 1985) and allowing a resolution improving attention band to shadow stimuli (Luce et al., 1976) have both failed. Similarly, an adequate account of Absolute Identification 25 sequential effects has also eluded exemplar models, where the weighting of recent exemplars and the updating of their locations from trial to trial has also failed (Petrov & Anderson, in press). Here, we propose that these models find accommodating sequential effects difficult because they are based on the assumption that long-term absolute magnitude information is the basis for absolute identification performance. In the Thurstonian models, the position of the criteria provide long-term absolute magnitude information. In the Lacouture and Marley's (1991, 1995, 2004) connectionist model, long-term absolute magnitude information about the most extreme stimuli is used in rescaling each stimulus magnitude. In the exemplar models, the memory for the magnitude of each exemplar provides long-term absolute magnitude information. It may be that some future modification of these models would allow them to fully predict the pattern of sequential effects, but in the RJM we show how these sequential effects follow naturally from a relative judgment account. J. A. Siegel and W. Siegel (1972) review evidence that long-term representation of attributes such as pitch and loudness may be very poor: Memory for pitch, as measured in a same-different judgment task, decays very rapidly with the duration of tone or unfilled interval between the standard and comparison tones (Bachem, 1954; Harris, 1952; Kinchla & Smyzer, 1967; Koester, 1945, as cited in Massaro, 1970, and Wickelgren, 1966; König, 1957; Tanner, 1961; Wickelgren, 1966, 1969; Wolfe, 1886, as cited in Massaro, 1970, and Wickelgren, 1966). Massaro (1970) found that, if the intervening tone was similar to the standard, this disrupted judgment further. In their review article, J. A. Siegel and W. Siegel conclude that the limit in absolute identification performance "is not limited by stimulus information, but rather by subjects' inability to maintain multiple representations of sensory stimuli in memory" (p. 313). If a long-term representation of the absolute magnitude of a stimulus cannot be maintained successfully across only a single intervening stimulus, or even an unfilled interval, in a trial of a same-different judgment task (where the intervening stimulus can be ignored), then it is very unlikely that long-term representations of absolute magnitude can be maintained Absolute Identification 26 across the (on average) larger number of intervening trials in an absolute identification experiment. In the RJM, we instead suggest that, in the absence of stable and accurate long-term representations of the absolute magnitudes of stimuli, participants instead rely upon a relative comparison of the current stimulus to the previous stimulus. This relative difference is then used in conjunction with the feedback from the previous trial to generate a response (cf. Holland & Lockhead, 1968). Our intuition, which we test below, was that a model where responding on the current trial depends on information from the preceding trial might offer a simple account of sequential effects. We are not the first to suggest that psychophysical judgment might be relative. In reviews of the psychophysical literature, Helson (1964) and Laming (1997) both suggest that psychophysical judgment is relative. Lockhead (1992; 2004) also reaches this conclusion, although he suggests that, because single attributes cannot be abstracted from the object in which they occur, it is entire objects, rather than their constituent attributes, that are judged relative to one another. In absolute identification, where objects (stimuli) vary on only a single attribute, these alternatives are equivalent. Stewart and Brown (2004) have found evidence in perceptual categorization that supports the idea that the response on the current trial is generated by comparing the current stimulus to the preceding stimulus. They examined sequential effects in a binary categorization of tones varying in frequency, where low frequency tones belonged to one category and high frequency tones belonged to the other. If participants could maintain even only a single longterm absolute magnitude (the category boundary), then this categorization task should be trivial, as stimuli could simply be compared to this reference point and categorized accordingly. Instead, Stewart and Brown found strong sequential effects, consistent with participants making an ordinal comparison between the current stimulus and the preceding stimulus. Accuracy was only high when comparison to an immediately preceding stimulus determined the categorization. For example, if a stimulus was lower in magnitude than the Absolute Identification 27 preceding stimulus, and the preceding stimulus was from the low category, then the current stimulus was correctly categorized as a member of the low category. These data are consistent with the idea that the categorization of the previous stimulus, together with a judgment of the difference between the current stimulus and the previous stimulus, inform the current categorization decision. Other data are difficult to explain with an absolute account. If judgment were absolute, then the effect of the previous stimulus on the current response should be viewed as a biasing of absolute judgment. Attenuation of sequential effects should therefore lead to an improvement in identification performance. Stewart and Chater (2003) found that a manipulation which attenuated sequential effects instead reduced identification accuracy. Stewart and Chater had participants perform an absolute identification of eight loudnesses. However, each loudness was randomly presented as either a pure, sinusoidal tone or a white noise hiss. When consecutive stimuli were of different types (a hiss followed by tone or a tone followed by hiss) there was a significantly smaller correlation between the previous stimulus and the current response compared to when stimuli were of the same types (two consecutive hisses or two consecutive tones). Accuracy was also significantly lower when consecutive stimuli were of different types compared to when consecutive stimuli were of the same type. This result is the opposite to what would be expected if absolute judgments are being made: Reducing the biasing caused by the previous stimulus should have increased accuracy. However, this result is expected if the loudness of current stimulus is judged relative to the previous stimulus: A switch in the stimulus type will make the comparison of loudnesses more difficult, reducing the accuracy on the current trial. The idea that long-term representations of absolute magnitudes are not available may well be too strong. There are some data that are problematic for this view. Ward and Lockhead (1970) and Ward (1987) ran several psychophysical tasks requiring either absolute or relative judgment (absolute identification, category judgment, estimation of the ratio of Absolute Identification 28 successive magnitudes, absolute magnitude estimation and cross modality matching). On different days, they varied the loudness of the entire stimulus set. Whether or not participants were performing relative or absolute judgment tasks, the judgments on each day were systematically biased towards the stimulus-response mapping from the previous day. This suggests that some representation of the absolute magnitudes of stimuli persists over an interval of at least one day. Thus, it may be that long-term absolute magnitude information is available in absolute identification, but that its representation is rather poor or "fuzzy" (Ward, 1987, p. 226) and not sufficient to support absolute identification. Alternatively, the information may be available, but (for some unknown reason) not used. Consistent with this possibility, long-term absolute magnitude information seems to be weighted more heavily when instructions suggest using a long-term frame of reference (DeCarlo & Cross, 1990; DeCarlo, 1994) or when inter-trial intervals were large (DeCarlo, 1992). (Stewart and Brown, 2004, give a more detailed discussion of these data.) Our core claim that absolute identification is achieved by relative judgment is consistent with either the possibility that long-term representations of absolute magnitudes are poor or that the long-term representations are (for some unknown reason) unused. Summary In summary, two shortcomings of existing models have motivated the RJM. Models which assume the locus of the limit in information transmitted is perceptual fail to predict (or require modification to predict) that channel capacity remains severely limited even for very large stimulus spacings. In the RJM, the limit in channel capacity is not perceptual. Models which use long-term representations of absolute magnitudes do not capture the sequential effects adequately. In the RJM, as the name suggests, judgment is instead relative to the immediately preceding stimulus. Next we give a detailed specification of the RJM. Mathematical Specification of the Relative Judgment Model In what follows, we refer to the current trial in an experiment as trial n, the previous Absolute Identification 29 trials as trial n 1, and the kth most recent trial as trial n k. The physical magnitude of the stimulus on trial n is denoted Xn, the rank of the stimulus within the set Sn, the response Rn, the feedback Fn, and the error in responding En = Rn -Sn. The elemental unit admitted to the decision process is assumed to be the difference between Sn and Sn 1. In other words, what is admitted to the decision process on trial n is not some representation of the magnitude of Sn, but a representation of the difference between Sn and Sn 1. This difference, Dn, n 1, is given by the logarithm of the ratio of the physical magnitudes. Dn , n 1=A ln X n X n 1 (1) where A is a constant that depends on the sensory dimension. The use of the ratio follows from Weber's Law. A rearrangement of Equation (1) gives Dn ,n 1=A ln X n A ln X n 1 . If Fechner's logarithmic law relating physical magnitude to the subjective, psychological percept holds, then Dn, n 1 is the arithmetic difference between psychological magnitudes. If stimuli are geometrically spaced with spacing r (i.e., each stimulus is a constant ratio r larger in physical magnitude than the next highest in magnitude), as is nearly always the case in absolute identification experiments, then Dn , n 1=A ln r S n S n 1 . (2) This difference Dn, n 1 is assumed to be contaminated by residual representations of earlier differences Dn 1, n 2, Dn 2, n 3, ... . Equivalently, elements of the representation of Dn, n 1 are assumed to be confused with elements of the representations of Dn 1, n 2, Dn 2, n 3, ... (cf. Estes, 1950). The result of this confusion or contamination is labeled Dn ,n 1 C . Dn ,n 1 C = i=0 n 2 i Dn i , n i 1 (3) The coefficients are constrained to be in the range 0 1. The coefficient for the current difference 0 is fixed at 1. Further, the coefficients are constrained to be monotonically Absolute Identification 30 decreasing (i.e., i > i + 1), so that more recent differences are more likely to be confused with the current difference. The idea that representations may be confused is quite ubiquitous in psychology. What is unique in the RJM is the assumption that it is stimulus differences that are confused, and this follows from our initial assumption that it is stimulus differences rather than absolute magnitudes that are elemental. That is, Dn ,n 1 C can be considered the result of a confusion of stimulus differences in exactly the same way as any other representations might