Testing the representation of time in reference memory in the bisection and the generalization task: The utility of a developmental approach

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1 PQJE TECHSET COMPOSITION LTD, SALISBURY, U.K. 6/16/2006 THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY 0000, 00 (0), 1 17 Testing the representation of time in reference memory in the bisection and the generalization task: The utility of a developmental approach Maria de Lurdes Delgado and Sylvie Droit-Volet Blaise Pascal University, Clermont-Ferrand, France This study examined the effect of the variability of representation of durations in reference memory on temporal discrimination performance in children aged 5 and 8 years as well as in adults using a bisection (Experiment 1) and a generalization task (Experiment 2). In each task, the participants were familiarized before the blocks of tested trials with either the same referent duration values (fixed condition) or a distribution of referent duration values, with a mean equal to the referent durations used in the fixed condition and a.20 coefficient of variation (variable condition). The results showed that the sensitivity to duration was lower in the variable than in the fixed condition in the children and, to a lesser extent, in the adults. The modelling of the data indicated that this effect was due to the increase in the variability of the representation of durations in reference memory, but also to changes in the decisional processes. The memory of past event durations plays an important role in the judgement of current event durations. Comparing the presented duration with those stored in reference memory allows us to judge whether the former is similar, shorter, or longer than usual. The importance of the role of memory representations of duration in temporal judgement was suggested in scalar timing theory (Gibbon, 1977) and its associated information-processing models (Church, 1984; Gibbon & Church, 1984; Gibbon, Church, & Meck, 1984), which were first proposed for animal timing and have since been successfully applied to timing in human adults (e.g., Allan & Gibbon, 1991; Wearden, 1991, 1992). According to this theory, the raw material for time judgement comes from a pacemaker accumulator clock mechanism. More precisely, the subjective duration depends on the number of pulses emitted by a pacemaker and accumulated in a counter during the presented stimulus. However, this theory assumes that the final temporal judgement is governed by the comparison of the presented duration with the representation of important durations stored in reference memory, such as those associated with reinforcement in animal studies or identified as referent durations in human studies. Correspondence should be addressed to Sylvie Droit-Volet, Laboratoire de Psychologie Sociale et Cognitive, CNRS (UMR 6024), Université Blaise Pascal, 34 avenue Carnot, Clermont-Ferrand Cedex, France. droit@srvpsy.univ-bpclermont.fr # 0000 The Experimental Psychology Society 1 DOI: /

2 DE LURDES DELGADO AND DROIT-VOLET Some principles of scalar timing theory have been tested in human adults with different temporal discrimination tasks (for a review see Meck, 2003). However, as stated by Allan (2002), the bisection task has become the preferred task for the study of reference memory in humans because it maximizes the dependence of temporal judgements on the memory of referent durations. In the prototypical version of the bisection task, the participants are familiarized with a short (S) and a long (L) referent duration either at the beginning (e.g., Lustig & Meck, 2001; Wearden & Ferrara, 1995, 1996) or at intervals throughout the blocks of test trials (e.g., Allan & Gibbon, 1991; Penney, Allan, Meck, & Gibbon, 1998). They are then presented with comparison durations that are equal to the referent durations or of intermediate value and are required to judge whether these durations are more similar to the short or to the long reference duration. In humans, as of the age of 3 years, the temporal bisection task yields psychophysical functions that are ogival in form, with the proportion of long responses (t identified as more similar to L than to S) increasing with the value of t. However, the slope of these bisection functions appears to be flatter in younger children, 8 years old being the age at which the curve slope approaches that found in adults (e.g., Droit- Q1 Volet & Clément, in press; Droit-Volet, Tourret, & Wearden, 2004; Droit-Volet & Wearden, 2001, 2002; McCormack, Brown, Maylor, Darby, & Green, 1999). The scalar-timing-based models used to simulate data in the bisection task have provided an excellent description of the temporal behaviour obtained in human adults as well as in animals (e.g., Allan & Gibbon, 1991; Gibbon, 1981; Penney, Gibbon, & Meck, 2000; Wearden, 1991). In these models, the referent durations are stored in memory in the form of Gaussian distributions, rather than single values, with means equal to the value of the referent duration and some coefficient of variation, c. On any test trial, the duration t is thus compared with samples taken from this memory representation, S Q2 and L. Sampling from these distributions therefore produces trial-by-trial variance. In general, the greater the coefficient of variation of these memory distributions, the flatter the psychophysical function will be. The variability of referent memory is thus the main sensitivity parameter controlling the slope of the psychophysical functions. In other words, the temporal reference memory is the main source of variability in temporal bisection judgement. Using the scalar timing models of bisection, Droit-Volet and Wearden (2001) thus found that the young children s flatter bisection functions are due to a greater variability in the representation of the referent durations in memory. Recently, the validity of the memory representation of referent durations as a major source of variance in temporal behaviour has been questioned. Rodriguez-Girones and Kacelnik (1995, 1998, 2001) used a variant of the bisection task (roving referent bisection) in which no temporal reference memory could be formed and called on for temporal judgement. The participants had to indicate whether a third duration (test duration) was more similar to one or the other of the two previously presented referent durations. The referent durations were varied from trial to trial in order to prevent subjects from using referent memory. In this new task, bisection took place as in the prototypical version. In line with this finding, Wearden and Ferrara (1995, 1996) obtained similar psychophysical functions when the participants either were or were not (partition bisection) familiarized with referent durations. More recently, McCormack, Wearden, Smith, and Brock Q3 (2005), using a generalization task, and Droit- Volet and Rattat (2006), using a bisection task, Q4 decided to eliminate the reference memory, and they found that the age-related differences in temporal judgements did not disappear, with the psychophysical functions again being flatter in the younger children. Given this body of results, the authors concluded that the comparison of the presented duration with the referents is not necessary for temporal bisection to occur. Since the reference memory was unavailable in these bisection conditions, the dominant source of variability in temporal judgement, and its age-related changes, 2 THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 0000, 00 (0)

3 REPRESENTATION OF TIME IN REFERENCE MEMORY would appear not to lie in the reference memory, but rather in the perceptual processes (McCormack et al., 1995, p. 3). In short, temporal bisection can occur without referent durations in memory. However, this does not exclude the possibility that when the referent durations are explicitly identified, and a direct comparison with the stimulus duration is required, the content of the referent memory could, to some extent, affect the slope of the psychophysical functions. However, the exact nature of the content of the reference memory is not clear in the scalar timing models and has been the object of little research to date (Jones & Wearden, 2003). More specifically, in these models, the variability of the memory representation of referent durations results not from noise in perceived time, the presented duration being processed accurately, but from noise in the memory storage process itself (Meck, 1983, 1996; Meck & Angell, 1992). The duration in the accumulator is multiplied by a memory storage constant k, which varies as a Gaussian distribution with a mean of 1.0 and a coefficient of variation. It is this resulting value that is entered in the memory distribution (for a review see Church, 1997; Rattat & Droit-Volet, 2005b). In animal studies, Meck (1983) has demonstrated that we can alter this storage component through the administration of drugs without altering the other components of temporal information processing. Nevertheless, the question is: How is this noise in temporal reference memory produced? We assume that the variability in the memory representation of referent durations does not result solely from variance in the memory storage process itself as suggested in the scalar timing theories, but also from noise prior to memorization that is, in the encoding of time. Indeed, the content of the reference memory comes from what was initially encoded that is, the evaluation of each presented instance of the referent duration. In bisection, the variability in the reference memory would be relatively low in adults, because the quantity of errors in their encoding of each duration would be reduced. In contrast, imprecision in the evaluation of each presented instance of the referent duration would be greater in young children, thus producing a higher level of variability in the temporal representation in memory. This would affect their bisection judgements by flattening the bisection curve. The aim of the present study was thus to examine the effect of noise in the memory representation of the referent durations on temporal performance in a bisection task involving children aged 5 and 8 years old and adults by directly introducing variance in the presentation examples of the short and the long referent duration. In one condition (fixed referent memory), the participants were thus presented with the same short or long referent durations. In the other (varied referent memory), they were presented with a set of values around the short and the long referent duration. These values were chosen such that the coefficient of variation of their distributions was higher in the varied than in the fixed condition, and their means were equal to the referent durations used in this fixed condition. We assumed that if each perceived example of the referent durations contributes to the memory representation of the referent durations in the form of distributions with means and some coefficient of variation, the psychophysical function slope would be flatter in the varied than in the fixed condition, thus indicating a lower sensitivity to time. Furthermore, the fixed varied memory difference would be greater in the younger children. The modelling of the data with a scalar timing model should indicate that this lower temporal sensitivity in the varied condition is due to a greater coefficient of variation of the representation of durations in reference memory. EXPERIMENT 1 Method Participants The sample consisted of 80 participants: 24 five-year-olds (13 females and 11 males, mean age ¼ 4.8 years, SD ¼ 0.41); 28 eight-year-olds (14 females and 14 males, mean age ¼ 7.9 years, SD ¼ 0.35); and 28 adults (21 females and THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 0000, 00 (0) 3

4 DE LURDES DELGADO AND DROIT-VOLET 7 males, mean age ¼ years, SD ¼ 1.39). The children came from nursery and primary schools, and the adults were first-year psychology students from Blaise Pascal University, in Auvergne, France. Materials The participants were tested individually in a quiet room in their school (in the case of the children) or in the university psychology department (adults). The experiment was run on a PowerMacintosh computer, which controlled stimulus presentation and recorded the data via PsyScope. The stimulus, whose duration was varied, was a 4.5-cm diameter blue circle presented in the centre of the computer screen located at about cm and cm from the children s and the adults eyes, respectively. The participants responded to the stimulus by pressing the K or the D key on the computer keyboard. Procedure In each age group, half the participants were randomly assigned to the fixed referent memory condition and the other half to the variable referent memory condition. The procedure was similar in these two conditions, except for the value of the duration presented on each trial during the familiarization phase with the referent durations, and was given before each test block. In this familiarization phase, the participants were alternately presented with the short and the long referent five times each. In the fixed condition, the stimulus duration was always 1 s for the short and 7 s for the long referent duration. In the variable condition, the stimulus duration was randomly chosen on each trial from the five duration values 0.750, 0.875, 1.000, 1.125, and s for the short and the five duration values 5.250, 6.250, 7.000, 7.750, and s for the long referent, with a mean equal to the referent value used in the fixed condition and a coefficient of variation of.20. In the fixed as well as in the variable condition, the experimenter said on each trial: Look, it s the short/long circle. It stays on for a short/long time. Each familiarization phase was immediately followed by a test block. There were four test blocks of 21 trials, 3 trials for each of the seven comparison durations: 1, 2, 3, 4, 5, 6, 7 s. The total number of trials was thus 84 trials. The trials were presented in a random order within each block. The intertrial interval was also randomly chosen between 500 ms and 2 s. In the test blocks, the participant pressed one key ( D ) after a comparison duration judged to be similar to the short referent duration and the other key ( K ) after one judged similar to the long referent, the key press order being counterbalanced. The participants were instructed not to count and were required to repeat aloud blablabla... as fast as possible in order to preclude vocal and subvocal counting (Gallistel & Gelman, 2000). The experimenter monitored the continuity of this oral activity. Results Data analysis Figure 1 shows the mean proportion of long responses plotted against comparison durations in the fixed and the variable referent condition. The upper panel shows data from the 5-year-olds, the centre panel data from the 8-year-olds, and the lower panel data from the adults. An examination of Figure 1 suggests that the psychophysical functions were flatter in the variable than in the fixed condition, but to a greater extent in the younger children. An analysis of variance (ANOVA) 1 was run on the proportion of long responses with two between-subjects factors (age and referent memory) and one within-subject factor (comparison duration). The ANOVA found a significant effect of comparison duration, F(6, 444) ¼ , p ¼.0001, and a marginally significant effect of age, F(2, 74) ¼ 2.90, p ¼.06. The main effect of referent memory was not significant, F(1, 74) ¼ 0.45, p ¼.51. However, there was a Q5 1 Previous analyses revealed neither a significant main effect nor any interaction involving the sex and button order factors. Thus, these factors were not included in the statistical analyses. 4 THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 0000, 00 (0)

5 REPRESENTATION OF TIME IN REFERENCE MEMORY Q6 Figure 1. Mean proportion of long responses plotted against stimulus duration for the fixed and the variable referent memory condition. The upper panel shows data from the 5-year-olds, the centre panel from the 8-year-olds, and lower panel data from the adults. significant Age Comparison duration interaction, F(12, 444) ¼ 6.85, p ¼.0001, indicating that the steepness of the bisection functions increased with age. Furthermore, the Referent Memory Comparison Duration interaction, F(6, 444) ¼ 2.30, p ¼.03, and the Referent Memory Age Comparison Duration interaction, F(12, 444) ¼ 2.75, p ¼.001, were significant, with the Referent Memory Age interaction close to significance, F(2, 74) ¼ 2.89, p ¼.06. We therefore decided to run an ANOVA on the proportion of long responses for each age group taken separately. In both the 5 and the 8-year-olds, there was a significant interaction between the referent memory and the comparison duration, F(6, 132) ¼ 3.84, p ¼.001, F(6, 156) ¼ 2.64, p ¼.02, respectively, which subsumed a significant main effect of comparison duration, F(6, 132) ¼ 52.83, F(6, 156) ¼ , all p ¼.0001, with no significant main effect of referent memory, F(1, 22) ¼ 2.85, F(1, 26) ¼ 2.44, all p..05. Unlike in the children, there was only a main effect of comparison duration in the adults, F(6, 156) ¼ , p ¼ In the adults, neither the effect of referent memory, F(1, 26) ¼ 1.10, p ¼.31, nor the Referent Memory Comparison Duration interaction, F(6, 156) ¼ 0.70, p ¼.65, were significant. These results suggest that increasing the variability of the memory representation of the referent durations, while holding the means similar between the two memory conditions, flattened the bisection functions in the children, with the effect being more pronounced in the 5-year-olds than in the 8-year-olds, but not significantly in the adults. In order to further investigate similarities and differences between the psychological functions in the different referent memory conditions, we calculated two new indexes for each participant: a Weber ratio and a bisection point. The Weber ratio is a measure of the steepness of the psychophysical function, which is in turn an index of temporal sensitivity. The higher the Weber ratio value is, the lower the temporal sensitivity. It is the difference limen (half the difference between the comparison duration giving rise to 75 and 25% long responses) divided by the bisection point. The bisection point is the point of subjective equality that is to say, an index of the localization of the bisection criterion, the boundary below which the subjects mostly respond short and above which they mostly respond long. It is thus THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 0000, 00 (0) 5

6 DE LURDES DELGADO AND DROIT-VOLET the stimulus duration giving rise to 50% long responses. The Weber ratio and the bisection point were calculated on the basis of the slope and intercept obtained from the linear regression performed on the steepest part of the psychophysical function for each individual (for the method used, see Church & Deluty, 1977; Droit-Volet & Wearden, 2001; Wearden, 1991). It was impossible to calculate an individual Weber ratio value for four of the 5-year-olds in the variable condition because their temporal performance was totally impaired by the increase in variability in the presentation of the referent durations. We therefore decided to exclude these participants from the subsequent statistical analyses of the Weber ratio and the bisection point (Table 1). The overall two-way ANOVA run on the individual Weber ratios revealed a main effect of age, F(2, 70) ¼ 33.11, p ¼ However, there was also a significant effect of referent memory, F(1, 70) ¼ 15.71, p ¼.0001, as well as a significant interaction between age and referent memory, F(2, 70) ¼ 3.61, p ¼.03. The one-way ANOVA carried out on the Weber ratio in the fixed and the variable conditions taken separately showed a significant effect of age in each condition, F(2, 37) ¼ 20.58, F(2, 35) ¼ 15.75, respectively, all p ¼ As indicated by the Sheffe post hoc tests, in both the fixed and the variable referent memory condition, the Weber ratio was higher in the 5-year-olds than in either the 8-year-olds or the adults (all p..01), whereas it was similar in these two older age groups (p..05). This is entirely consistent with Table 1. Weber ratio and bisection point in the fixed and the variable referent memory condition for the three age groups Weber ratio Bisection point Mean SD Mean SD 5-year-olds Fixed Variable year-olds Fixed Variable Adults Fixed Variable the data found in previous studies using bisection tasks, which have revealed an improvement in time sensitivity between 5 and 8 years of age (Droit-Volet, 2003; Droit-Volet et al., 2004; Droit-Volet & Wearden, 2001; McCormack et al., 1999). In addition, the statistical analysis revealed that the Weber ratio value was higher in the variable than in the fixed referent condition for both the 5- and the 8-year-olds, t(18) ¼ 2.70, p ¼.02, t(26) ¼ 2.39, p ¼.02, respectively, but not for the adults, t(26) ¼ 0.76, p ¼.46. In line with the statistical analyses of the proportion of long responses, these data suggest that increasing the variability in the memory representation of the referent durations decreased the children s sensitivity to duration in the bisection task. However, unlike the children, the adults maintained a high sensitivity to duration whatever the referent memory condition. The overall ANOVA run on the individual bisection points also revealed a significant interaction between age and referent memory, F(2, 70) ¼ 4.76, p ¼.01, whereas the main effect of age, F(2, 70) ¼ 2.48, p ¼.09, and of referent memory, F(1, 70) ¼ 0.12, p ¼.74, were not significant. The post hoc statistical analyses indicated that this significant interaction was mainly due to the 5-year-olds who produced a significantly lower bisection point in the variable than in the fixed referent condition, t(18) ¼ 2.10, p ¼.05. This difference between the fixed and the variable condition failed to reach significance in the 8-year-olds, t(26) ¼ 1.93, p ¼.07, and the adults, t(26) ¼ 1.26, p ¼.23, respectively. Thus, for the adults and the 8-year-olds, the bisection point was located closer to the arithmetic mean of the short and the long referent (i.e., 4 s) than to the geometric mean (2.65), and this finding did not change significantly with the increase in the variability of the memory representation of the referent durations. In contrast, in the 5-year-olds, the bisection clearly fell beyond the arithmetic mean of the short and the long referent as found in previous studies (Droit-Volet & Wearden, 2001). However, it moved toward the geometric mean as the variability in the presentation of the reference duration values increased. 6 THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 0000, 00 (0)

7 REPRESENTATION OF TIME IN REFERENCE MEMORY Modelling In order to identity the mechanism underlying the effect of our experimental manipulation of reference memory on bisection performance, we decided to use the developmental version of Wearden s modified difference model (1991), which has provided a good fit with bisection data obtained from human adults in a large number of studies. Like most bisection models (see Introduction), this model suggests that the referent durations presented in the familiarization phase (i.e., the short, s, and the long, l ) are stored in reference memory in the form of Gaussian distributions, with means equal to the values of the short and long referent and some coefficient of variation, c. The stimulus duration, t, is thus compared with samples drawn from the reference distributions (i.e., s and l ). Consequently, the higher the coefficient of variation, c, the fuzzier the memory representation of the referent duration is. As illustrated in Figure 2a, increasing the coefficient of variation c, while the other parameters are held constant, flattens the bisection curve but does so to a greater extent for the longer than for the shorter stimulus durations. The coefficient of the remembered time is thus a sensitivity parameter controlling the slope of the psychophysical function. Added to this memory parameter, there are two other parameters: b, a bias toward long responses, and p, a probability of random responses. The bisection model assumes that the participants classify a comparison stimulus, t, as more similar to the short or the long referent by calculating two Q6 Figure 2. Data derived from simulations using the developmental version of the modified difference model (MDM) discussed in the text, when a parameter value is varying while the other parameter are kept constant. THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 0000, 00 (0) 7

8 DE LURDES DELGADO AND DROIT-VOLET differences: D(s, t) and D(l, t), the absolute difference between t and a sample taken from the memory of the short and the long referent duration, s, l, which differ from trial to trial and are drawn from Gaussian distributions. If the difference between D(s, t) and D(l, t) is less than a threshold value, b, the model responds long. That is to say, the model responds long when faced with ambiguous cases that is, when it cannot tell whether t is closer to the short or the long referent duration. In contrast, when this difference is greater than b, and thus clearly differentiated, the model responds short if D(s, t), D(l, t) and long if D(s, t). D(l, t); b is thus a sort of bias toward responding long (for more details, see Droit-Volet & Wearden, 2001). Increasing the value of b thus shifts the proportion of long responses for intermediate stimulus durations to the left that is, toward the geometric mean of the two anchor durations without altering the slope of the psychophysical function. However, as we can see in Figure 2b, this shift remains small even when the bias value is large (i.e., from 0 to 0.40) and was not observed for the shortest and longest comparison durations. The last parameter, p, is the proportion of responses emitted at random on each trial that is, without reference to the stimulus duration values. Figure 2c shows that increasing p flattens the bisection curve just as increasing c does. However, the increase in the proportion of long responses is greater for the short comparison durations. Droit-Volet and Wearden (2001) have demonstrated that the removal of the parameter p impaired the fit between the data and the model in the youngest children. In the present study, in order to capture any possible distortion of the representation of the reference duration, we added a fourth parameter, k that is, a memory distortion parameter which serves as a multiplier of the remembered short and long referent duration. Thus, if k is 1.0 then the referent values are remembered correctly. If k, 1, the referent values are remembered as shorter than they are in reality, and if k. 1 then they are remembered as longer than they are. The effect of a decrease in the value of k on psychological functions is shown in Figure 2d. Obviously, decreasing k shifts the proportion of long responses to the left for short and intermediate stimulus durations. This model, implemented in a computer program written in Visual Basic 6 (Microsoft Corporation), was run for 1,000 trials, and c, b, p, and k were varied over a wide range in order to obtain the best fitting simulation for the data in terms of mean absolute deviation (MAD) that is, the sum of the absolute deviation between the data and the data derived from the modelling divided by 7. Our model fits the data well with a MAD smaller than.05 (Table 2, Figure 3). Furthermore, it replicates some findings obtained in previous studies in children, notably a decrease in the c parameter value with increasing age, thus indicating that the coefficient of variation of the memory representation of the referent durations decreases with age (for a discussion, see Droit-Volet, Delgado, & Rattat, 2006). However, and most interestingly here, the model shows that the value of the k parameter was quite similar in the two referent memory conditions. Thus, increasing the variability in the Table 2. Parameter values for the fits of the developmental version of the MDM with the data obtained in the bisection task in the fixed and variable referent memory conditions in the 5-year-olds, the 8-year-olds and the adults c b p k MAD 5-year-olds Fixed Variable year-olds Fixed Variable Adults Fixed Variable Note: MDM is the modified difference model used by Wearden (1991) to model data obtained in adults in the bisection task; c is the coefficient of variation of the memory representation of the short and long referent; b is the bias toward long responses; p is the probability of random responding; and k is a distortion of the memory representation of the short and long referent. MAD is the mean absolute deviation, which is the sum of absolute differences between the data points and the fitted function divided by 7 (i.e. the number of data points). 8 THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 0000, 00 (0)

9 REPRESENTATION OF TIME IN REFERENCE MEMORY Q6 the 8-year-olds, although this variation was greatest in the youngest children. In comparison with the children, the difference in c values between the two memory conditions appeared to be relatively small in the adults. Furthermore, our model suggests that the manipulation of the reference memory affected not only the parameter c, but also other parameters. Indeed, in the 5-year-olds, both the p and the b parameter values were greater in the variable than in the fixed condition. This suggests that when the representation of durations in reference memory is fuzzier, the young children find it more difficult to produce a temporal decision. Faced with more ambiguous cases, they produce more random responses and more bias toward long responses. In contrast, in the 8-year-olds and the adults, the b parameter value tended to be smaller in the variable than in the fixed condition, thus suggesting a change in the decisional criterion as discussed below. Figure 3. Temporal bisection data from Experiment 1 in the fixed and variable referent memory condition (line) and values predicted by the best fitting modified difference model (MDM; circle) for the three age groups. presentation of examples of the referent durations did not alter their mean value in memory, regardless of the age groups. In contrast, the coefficient of variation of the memory representation of the referent durations, c, was greater in the variable than in the fixed condition in both the 5- and Discussion The results of the present study show that increasing the variability in the examples of the referent durations flattened the bisection functions and increased the Weber ratio in the children, and more particularly in the 5-year-olds. Our bisection model explains this in terms of an increase in the coefficient of variation of the memory representation of referent durations. This, therefore, provides clear evidence that, in temporal bisection, the children did not store single values in reference memory, but instead distributions of values together that allow obtaining of means and some coefficient of variation. Consequently, each example of the two referent durations, or at least the majority of these examples, would seem to be encoded and transferred into reference memory. This provides clear support for the suggestion that temporal reference memory results from the encoding of important times (Meck, 1983; Meck & Angell, 1992; Meck, Church, & Olton, 1984). However, the comparison between age groups allowed us to show that the effect of our THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 0000, 00 (0) 9

10 DE LURDES DELGADO AND DROIT-VOLET experimental manipulation of reference memory on temporal bisection behaviour was different in the children and the adults. Indeed, unlike in the children, in the adults an increased variability in the presentation of the examples of the reference durations did not significantly affect the features of the psychophysical functions. In another temporal discrimination task, Jones and Wearden (2003) varied the number of presentations of standard durations from 1 to 5 presentations and also failed to find an effect of the number of presentations on temporal performance in human adults. They concluded that the different examples of the standard duration are not stored separately in reference memory and then sampled, since otherwise a fuzzier memory representation would have been obtained with 5 presentations. According to these authors, this challenges the idea that reference memory contains more than one value. Allan and Gerhardt (2001) suggested that, in a temporal bisection task, adults do not use comparisons with the referent durations, but a unique duration criterion. However, the number of presentations of the reference durations used by Jones and Wearden (2003) (and in our experiment) might have been too low to produce sufficient noise in the reference memories of human adults to result in a modification of temporal judgement. Studies of animals have used long training sessions with the referent durations. However, our study showed that, unlike in the adults, a greater number of presentations of the referent durations was not necessary in the young children for the effect of variability of reference memory on temporal judgement to occur. This is probably due to their memory referent representation that was already fuzzy in the fixed memory condition. Thus, the comparison of the age groups has allowed to us to gain a better understanding of the fundamental nature of the content of the reference memory in humans in the bisection task. To confirm and extend our results, we decided to run the same experiment with another temporal task that is, the temporal generalization task. In this task, the participants are not required to categorize the stimulus durations as a function of their similarity with the short or the long referent duration, but to judge the similarity between the stimulus durations and a unique referent duration. In this condition, a direct comparison between the stimulus durations and the referent duration appears to be obligatory. Consequently, the effect on temporal judgement of the increasing variability of the reference durations in the examples should be observed both in the adults and in the children, although to a lesser extent in the former. EXPERIMENT 2 Method Participants The sample consisted of 86 new participants: 30 five-year-olds (17 girls and 13 boys; mean age ¼ 5.70 years, SD ¼ 0.76); 30 eight-year-olds (19 girls and 11 boys; mean age ¼ 8.45 years, SD ¼ 0.21), and 26 students (26 women; mean age ¼ years, SD ¼ 0.85). The children were recruited from nursery and primary schools other than those involved in Experiment 1, and the adults were first-year psychology students from Blaise Pascal University. Materials and procedure The materials were the same as those in Experiment 1, and the procedure was similar. In each age group, the participants were randomly assigned to one of the two referent memory conditions that is, fixed or variable. In each condition, they were presented with the referent duration five times (familiarization phase) before each test block. In the fixed referent condition, the referent duration was always 4 s while in the variable condition it was randomly chosen on each trial from the five following values: 3, 3.5, 4, 4.5, 5. The mean of the referent durations presented in the familiarization phase was 4 s and the coefficient of variation.20. In each condition, the experimenter said: Look, it s your circle. It stays on for a certain time. After each familiarization phase, the participants saw a test block. There were four test blocks of 21 trials each, 3 trials for each of the seven comparison durations 1, 2, 3, 4, 5, 6, and 10 THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 0000, 00 (0)

11 REPRESENTATION OF TIME IN REFERENCE MEMORY 7 s. The participant s task was to press one key if they judged the comparison duration to be similar to the referent duration (yes response) and the other key if they judged that it was not similar to the referent duration (no response). The press button order was counterbalanced. The comparison durations were presented in random order within each block, and the intertrial interval value was also randomly chosen between 500 ms and 2 s. As in Experiment 1, the experimenter told the participants not to count and controlled the continuity of the verbal activity used to prevent counting. Results Data analysis Figure 4 shows the mean proportion of yes responses plotted against the comparison duration for the fixed and the variable referent memory condition in the 5-year-olds (top panel), the 8-year-olds (centre panel), and the adults (bottom panel). The ANOVA run on the proportion of yes responses with age, referent memory condition, and comparison duration as subject factors found a significant effect of age, F(2, 80) ¼ 20.63, p ¼.0001, and of comparison duration, F(6, 480) ¼ , p ¼ The interaction between age and comparison duration was also significant, F(12, 480) ¼ 6.25, p ¼ In each age group, we calculated the mean proportion of yes responses for the shortest (the three comparison duration values shorter than the reference duration value) and the longest comparison duration values (the three comparison duration values longer than the reference duration value). The effect of age appeared to be significant for both the shortest, F(2, 83) ¼ 19.29, p ¼.001, and the longest durations, F(2, 83) ¼ 5.97, p ¼.004. This is due to the proportion of yes responses being greater in the 5-year-olds than in the 8-year-olds and the adults for the shortest durations (Sheffe post hoc, all p,.05) and greater than in the adults for the longest durations ( p,.05). The 8-year-olds also produced more yes responses than the adults did for the shortest as well as for the longest durations ( p,.05). These various results therefore indicate Figure 4. Proportion of yes responses plotted against stimulus duration in the fixed and the variable referent memory condition for the 5-year-olds (upper panel), the 8-year-olds (centre panel), and the adults (lower panel). that the steepness of the generalization gradient increased with age. The overall ANOVA on the proportion of yes responses also showed a significant main effect of referent memory, F(1, 80) ¼ 10.29, p ¼.002, as Q6 THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 0000, 00 (0) 11

12 DE LURDES DELGADO AND DROIT-VOLET well as a significant interaction between the referent memory and the comparison duration, F(6, 480) ¼ 4.70, p ¼ This revealed that the shape of the generalization gradient was flatter in the variable than in the fixed referent condition, although the varied fixed difference was particularly more important for the longer stimulus durations. The difference in the mean proportion of yes responses between the shortest and the longest comparison durations appeared to be greater than zero for the fixed, t(42) ¼ 2.48, p ¼.02, and the varied condition, t(42) ¼ 7.11, p ¼.0001, thus indicating a right asymmetry in the generalization gradients in each memory condition. However, for the longest durations, the mean proportion of yes responses appeared to be greater in the varied than in the fixed condition, t(84) ¼ 3.17, p ¼.002. This suggested that the varied condition shifted the generalization gradients toward the right. The overall ANOVA found no significant interaction between referent memory and age, F(2, 80) ¼ 0.68, p ¼.53, nor between age, referent memory, and comparison duration, F(12, 480) ¼ 1.44, p ¼.15. This suggests that the effect of the variability in the examples of the reference durations on the generalization gradients did not significantly change with age. Modelling and discussion The model that we used was the generalization model originally used by Church and Gibbon (1982) with animals and later modified by Wearden (1992) for adults and more recently for children (Droit-Volet, 2002; Droit-Volet, Clément, & Wearden, 2001; Droit-Volet & Q7 Izaute, 2004). In this model, the referent duration is represented as a Gaussian distribution with a mean s and a coefficient of variation, c. For each trial, a value s was randomly sampled from this distribution. This model thus assumes that the participants respond yes when j(s t)/tj, b. The first parameter in this model is c. As far as bisection is concerned, c is the coefficient of variation of the memory representation of s. The threshold b was also represented as a Gaussian distribution with a mean b and the referent deviation held constant at 0.5b. Varying this last parameter produces no great improvement. The value b is a sample taken on each trial from the distribution of the threshold. The participants respond yes when t, the duration which has just been presented, is sufficiently close to s, with the decision being controlled by the threshold b. The second parameter in the model is thus b. The third parameter in the model is p, the proportion of responses given at random. On each trial, the model would be equally likely to make yes or no responses regardless of the stimulus duration. The fourth parameter is k, a memory distortion parameter, which is multiplied s over a range of values below 1.0 or above 1.0. If k were 1.0, the referent duration would be remembered correctly. If k were. 1.0, it would be remembered as longer than it really was, and if k were, 1.0, it would be remembered as shorter than it was. Figure 5 illustrates some proprieties of this model. In Panel a, the parameter c was varied while the other parameters were held constant. Increasing c make the reference memory fuzzier and flattens the generalization gradients, but few yes responses occur at the shortest stimulus durations. Panel b shows the effect of increasing b, while the other parameters were also held constant. Increasing b makes the decision as to whether to respond yes less conservative, and the overall proportion of yes responses increases, while the general shape of the gradient remain constant. Increasing p (Panel c) increases the proportion of yes responses occurring at each stimulus duration, even for the shortest stimulus durations. Increasing k shifts the gradient toward the right while decreasing k shifts it toward the left. This model was implemented in a computer program written in Visual Basic. As in Experiment 2, the different parameters were varied until we obtained the best fitting simulation in terms of mean absolute deviation that is, the sum of the absolute differences between the predictions from the simulation and the data (Figure 6). The model fits our data very well. Table 3 shows the parameter values obtained in this way. Our model captured the same Q8 Q9 12 THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 0000, 00 (0)

13 REPRESENTATION OF TIME IN REFERENCE MEMORY Q6 Figure 5. Data derived from simulations using the developmental version of the modified Church and Gibbon (MCG) model discussed in the text, when a parameter value is varying while the other parameter are kept constant. developmental trends as those found in previous generalization studies: (a) the age-related increase of the c value, which indicates a fuzzier memory representation of the referent duration in the younger children, and (b) a higher proportion of random responses (Droit-Volet, 2002; Droit-Volet et al., 2001; Droit-Volet & Izaute, Q7 2004). It also allows us to identify the mechanisms underlying the fixed varied differences in the generalization gradients. First, it indicates that the flatter generalization gradient in the varied condition is due to a greater coefficient of variation of the temporal representation in reference memory. Furthermore, our model suggests that introducing noise in the presentation of examples of the reference duration affects not only the content of the reference memory, but also the decisional processes, and particularly so in children. Indeed, the value of the b parameter was greater in the varied than in the fixed condition. This indicated that the participants became less conservative when the representation of durations in memory was fuzzier. This would explain why the temporal generalization gradients were more skewed to the right in the varied than in the fixed memory condition. Unlike the c and b parameters, the p (probability of random responses) and k parameters did not differ between the fixed and the variable memory condition. GENERAL DISCUSSION Experiment 1, which used a bisection task, and Experiment 2, which used a generalization task, THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 0000, 00 (0) 13

14 DE LURDES DELGADO AND DROIT-VOLET Table 3. Parameter values for the fits of the developmental version of the MCG model with the data obtained in the generalization task in the fixed and variable referent memory conditions in the 5-year-olds, the 8-year-olds and the adults c b p k MAD 5-year-olds Fixed Variable year-olds Fixed Variable Adults Fixed Variable Q6 Note: MCG is the modified Church and Gibbon model used bywearden (1992) to model data obtained in adults in the generalization task. c ¼ coefficient of variation of the memory representation of the standard. b ¼ decisional threshold. p ¼ probability of random responses. k ¼ distortion of the memory representation of the standard. MAD ¼ mean absolute difference between the data points and the fitted function divided by 7 (i.e. the number of data points). Figure 6. Temporal generalization data from Experiment 2 in the fixed and variable referent memory condition (line) and values predicted by the best fitting modified Church and Gibbon (MCG) model (circle) for the three age groups. provided consistent data showing that introducing noise in the presentation of examples of the referent durations altered our participants temporal judgements by decreasing their sensitivity to time, even though this alteration was systematically observed in the children and only in the generalization task in the adults. This provides evidence that each example of the referent duration was coded and stored in memory, thus forming a distribution of values with a mean and a coefficient of variation. As previously suggested, this supports the idea that temporal reference memory contains a set of important durations (e.g., Meck, 1996; Meck, Church, & Olton, 1984). These results also reveal that the main sources of noise in the memory reference do not reside solely in the storage processes themselves, as the scalar timing models suppose, but also in the initial imprecision in the perception of time, which is greater in the children than in the adults. In their model, McCormack et al. introduced a noise parameter representing an amount of noise added to the perceived duration. Their model showed that the proportion of noise in the perceived time was effectively higher in children than in adults. Consequently, the fuzzier memory representation of referent durations in children, initially identified by Droit-Volet and Wearden (2001) in the developmental version of the scalar timing models, would, to a large extent, be due to an initial noise in time encoding. However, it remains possible that certain memory Q10 14 THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 0000, 00 (0)

15 REPRESENTATION OF TIME IN REFERENCE MEMORY problems specific to children may also be involved (Droit-Volet et al., 2006). Using an interfering task and deferred temporal tests in bisection, Rattat and Droit-Volet (2005a, 2005b) showed that the forgetting of reference durations was more important in the younger children. That increased the variability of their memory representation of the reference durations and flattened their bisection curves. Young children s errors in the encoding of each stimulus duration would therefore be introduced into reference memory. This in turn would produce imprecision in the judgement of the similarity of t to the referent duration. Recently, McCormack et al. (2005), using a generalization Q4 task, and Droit-Volet and Rattat (2006), using a bisection task, have shown that the age-related differences in temporal judgements do not disappear when temporal reference memory is disabled. This suggests that the main source of developmental differences in sensitivity to time does not lie in reference memory, otherwise children would be expected to obtain, at least, the same performance without reference memory. Our study, in which the content of reference memory was manipulated, provided additional data supporting this suggestion. Indeed, it demonstrated that the variability in the examples of the referent durations made their memory representations fuzzier, thus reducing sensitivity to duration as indicated by the flatter psychophysical functions in the varied than in the fixed condition. Our experiments, and more particularly Experiment 1 with the bisection task, also revealed a gap between the temporal behaviour of adults and that of children. Our interpretation is that this is mainly due to the development of selfregulatory functions, which are dependent on the acquisition of metacognitive knowledge and skills (for a review see Kuhl & Kraska, 1996). Young children are known to experience difficulties in identifying and using new strategies and therefore follow the instructions given by the experimenter. In our experiments, they thus compared the test durations with the referent durations even if their representation in memory was fuzzy. Unlike young children, human adults are able to use strategies other than those presented by the experimenter. According to Allan and Gerhardt (2001), in a bisection task, adults do not use the decision rule based on direct comparisons with the referent durations, but a unique criterion established on the basis of the values of the two referent durations. One possibility is that, unlike the young children, the adults were able to calculate the mean of the presented referent durations since there was no difference in the psychophysical functions between the fixed and the varied condition. Another possibility is that the adults did not use the referent durations in the bisection task but instead transformed it into a partition task in which they classified each presented duration into two categories namely, a short or a long one. This interpretation is consistent with Wearden and Ferrara s (1995, 1996) findings of similar data in the prototypical and the partition task in which no standard durations were presented. Whatever the case may be, the results of our study did not allow us to come to any conclusion about the type of strategies used by human adults in bisection, and these will have to be investigated in future studies. Our model only suggests that the b parameter changes in the condition in which the representation of reference durations in memory is fuzzier. Little research has been devoted to the role of decisional strategies in temporal judgement. Furthermore, in the models of temporal information processing, the decisional processes have always been considered independently of the other processes (Wearden & Grindrod, 2002) that is, independently of the perceived durations or the quality of the memory representation of durations. In contrast, our experiments have provided original data suggesting that, in children, and probably also in adults, the value of decisional parameters changes as a function of the representation (more or less fuzzy) of the referent durations in memory. Indeed, the children became less conservative when their temporal representation in memory became fuzzier. The participants awareness of the quality of the content of temporal reference memory would thus affect the decision and the strategy used in a temporal discrimination task. THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 0000, 00 (0) 15