Effects of attention and external stimuli on duration estimation under a prospective paradigm 1

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1 Japanese Psychological Research 2000, Volume 42, No. 3, Effects of attention and external stimuli on duration estimation under a prospective paradigm 1 YOSHIKO KOJIMA and FUMIKO MATSUDA 2 Department of Psychology, Hiroshima University, Kagamiyama, Higashi-Hiroshima , Japan Abstract: The study examined whether there are two independent cognitive factors affecting duration estimation. In two experiments, we manipulated simultaneously and independently two variables, namely, the level of attention to the lapse of time and the quantity of perceived changes, and examined their effects on duration estimation under a prospective paradigm. The duration was estimated to be longer when subjects attended to the lapse of time than when they attended to tasks during the target interval (Experiments 1 and 2). The characteristics of external stimuli irrelevant to the tasks, namely, the rate of presentation of sounds (Experiment 1) and the velocity of moving dots (Experiment 2), affected duration estimation, even though the attention level was little changed by these stimuli. These findings suggest that there are at least two independent cognitive factors that affect duration estimation. Key words: duration estimation, level of attention, quantity of perceived changes, prospective paradigm. There are many models and theories of duration estimation. Most are based on either internal clocks or internal tempos (e.g., Fetterman & Killeen, 1990; Hoagland, 1981; Treisman, Faulkner, & Naish, 1992) or cognitive processes. Models included in the latter seem to be further categorized into two types (Arlin, 1986a; Zakay, 1989). One is based on a quantity of something in the cognitive processing of stimuli presented during the estimated interval, such as the number of perceived changes (e.g., Block & Reed, 1978; Fraisse, 1967; Poynter, 1983), the size of storage in memory (Ornstein, 1969), or the load on shortterm memory (Fortin & Breton, 1995; Fortin & Rousseau, 1987; Fortin, Rousseau, Bourque, & Kirouac, 1993). Models in this category all predict that the estimated duration increases with this quantity. The other category is based on level of attention to the lapse of time (e.g., Hicks, Miller, Gaes, & Bierman, 1977; Underwood & Swain, 1973). This type of model predicts that the estimated duration becomes longer as more attention is paid to the lapse of time. Does the fact that there are two such categories mean that there are two types of cognitive factor that affect duration estimation? Advocates of these models have usually insisted, albeit sometimes implicitly, that one of the two types of factor can fully explain the effects on duration estimation. For example, 1 This study was supported by a Grant-in-Aid for Scientific Research, Ministry of Education, Science, Culture, and Sports, No (F. Matsuda). 2 We wish to express our gratitude to M. Yuzawa (Hiroshima University) and N. Matsumi (Hiroshima University) for helpful suggestions in designing the experiments. We are also indebted to M. Miyatani (Hiroshima University) for his great help with writing the computer programs used in the present experiments Japanese Psychological Association. Published by Blackwell Publishers Ltd, 108 Cowley Road, Oxford OX4 1JF, UK and 350 Main Street, Malden, MA 02148, USA.

2 Effects of attention and stimuli on duration estimation under a prospective paradigm 145 Fraisse (1967) emphasized that a difference in attitude, motivation, or task difficulty caused a difference in attention to the lapse of time, and thereby a difference in the number of perceived changes, and thereby a difference in estimated duration. That is, according to Fraisse, attention to the lapse of time affects duration estimation only by changing the number of perceived changes. The purpose of the present study was to clarify whether there are two kinds of cognitive factor that independently affect duration estimation, or whether they are actually embodied in one factor. If the existence of independent cognitive factors is verified, constructing a model incorporating multiple factors will become more important than insisting on the validity of one of the models mentioned above. In order to solve the problem, experiments using factorial combinations of quantity and attention are needed. There have in fact been a few such experiments. Arlin (1986a, 1986b, 1989), whose subjects were all children, carried out experiments in which the level of attention to the lapse of time was manipulated by varying task difficulty, while the quantity of perceived changes was manipulated by altering the number of tasks given within the test period. Arlin found that both factors affected duration estimation, that is, the greater the quantity, the longer was the estimated time, and the less difficult the task (assuming greater attention to the lapse of time), the longer was the estimated time. However, the change in the number of the tasks itself may have affected attention to the lapse of time. That is, in his experiments the two factors may have been confounded, and this possibility was not examined. Zakay, Nitzan, and Glicksohn (1983) also examined the effects of the two factors. Attention to the lapse of time was changed by manipulating task difficulty in their experiment, as in Arlin s. However, the quantity of perceived changes was manipulated by altering the rate of external rhythmic stimulation, and this was not related to the tasks performed during the target interval. They also found that both factors affected duration estimation in the same directions as reported by Arlin (1986a, 1986b, 1989). However, as the mean numbers of responses varied significantly with the task difficulty in their experiment, the task difficulty could have affected duration estimation as a quantity factor, as well as the external tempo. From the results of these previous studies, it is clear that the most important experimental points are to manipulate the two factors simultaneously and independently, and to ascertain that this is so. We used a prospective paradigm (subjects knew in advance that duration estimation would be required), because it was known that the effect of attention is stronger under this paradigm than under a retrospective paradigm (where subjects do not know in advance that duration estimation will be required) (Block & Zakay, 1997; Zakay, 1993a). Experiment 1 In Experiment 1, we tried directly to change the level of attention to the lapse of time with instructions from the experimenter regarding attentional allocation, after Casini and Macar (1997), Macar, Grondin, and Casini (1994), and Mckay (1977). We did not use task difficulty for this purpose, although it has been a very common method, because it appears to affect the quantity factor as much as the attentional factor. The quantity, the number of perceived changes, was manipulated by varying the presentation rate of sounds, after Zakay et al. (1983). Since the sounds were not relevant to the tasks performed during the target interval, they were assumed to affect subjects attention level very little, and to give little load on shortterm memory. Methods Apparatus. An AV tachistoscope (Iwatsu ISEL, IS-710A) was used to present the stimuli and sounds, as well as to record response times; it was controlled by a personal computer (NEC, PC-9801VX). The distance between the CRT display (21 inches; Iwatsu ISEL, IS-720A) connected to the tachistoscope and the subject was about 1.5 m.

3 146 Y. Kojima and F. Matsuda Subjects. Subjects were 38 undergraduate students (12 males and 26 females). Their mean age was 19.8 years. Method of duration estimation. Durations were estimated by the reproduction method. The standard duration was marked off by two stimuli presented at the middle of the display. Each stimulus, consisting of six aligned asterisks (******), was presented for 500 ms. The subject was asked to push a button next to one of his or her hands after the offset of the second asterisk stimulus when the time elapsed was perceived to be the same as the standard duration. The period from the offset of the second asterisk stimulus to the pushing of the button was defined as the reproduced (estimated) duration. Task during the standard duration. Each subject carried out a classification task during the standard duration. Subjects were asked to classify successive stimuli, presented on the display, as meaningful or not. They pushed one of two buttons next to each hand according to their judgment of meaningfulness (the left and right buttons were pushed by the left and right hands, respectively). The button assigned to meaningful was the left one for half the subjects and the right for the others. The stimuli for the classification task were selected as follows. First, 60 nouns in Japanese comprising three hiragana letters (syllables), which had no homonyms, were selected from the list made by the National Language Research Institute (1991), and these were then transcribed as Roman letters. All of these were meaningful (they were designated as MFs). Second, for each of these 60 nouns, two of the three syllables were interchanged, such that the resulting spellings were meaningless (designated as MLs). Finally, four undergraduate students were asked to classify each of these 120 spellings according to whether they were meaningful or not. Based on their response times and number of errors, 48 MFs and 48 MLs were selected for experimental trials (e.g., MF: SENAKA; ML: KANASE). The residual 12 MFs and 12 MLs were used in training and dummy trials. Each MF and ML was presented only once to each subject in the experimental trials. The number of MFs and MLs were the same in a trial, and they were presented in the middle area (3.5 cm 10.5 cm) of the display, one by one in random order. Each spelling was shown for 1500 ms, at 500-ms intervals. In each trial, the first spelling in the classification task was presented 500 ms after the offset of the first asterisk stimulus, and the second asterisk stimulus was presented 500 ms after the offset of the last spelling. Conditions during the standard duration. Two conditions, the attention level and the presentation rate of sounds, were manipulated during the standard duration. We tried to change the level of attention to the lapse of time by using different instructions: During the classification task, allocate your attention to the lapse of time as much as you can. It is not important to respond fast and correctly in the classification task. During the classification task, allocate your attention to the classification task as much as you can in order to do it fast and correctly. It does not matter that your duration estimation may not be precise. Intermittent 500-Hz tones, each lasting 70 ms, were presented during the standard duration. The high rate of presentation of sounds was 10.0 tones/s and the low rate 2.5 tones/s. By combining the two attention levels and the two presentation rates of sounds, four experimental conditions were generated. Each subject participated in all four conditions. Procedure. Each subject was examined individually. At the beginning of the experiment, the subject was asked to remove his or her watch, and not to count during duration estimation. First, three training trials of duration estimation, with the classification task, were carried out, where the standard durations were 13 s, 9 s, and 17 s. In the third trial, intermittent sounds were presented at a rate of 2.0 tones/s during the standard duration. Just before this trial, the subject was told that he or she would hear some intermittent sounds, but should ignore them because they had nothing to do

4 Effects of attention and stimuli on duration estimation under a prospective paradigm 147 with either the duration estimation task or the classification task. After the training trials, 16 trials, consisting of 12 experimental and 4 dummy trials, were carried out. They were divided into four blocks, each containing four trials. Two of the blocks were after the subject had been told to attend to the lapse of time, and the other two after the other instruction. The order of the four blocks was Block X, Block Y, Block Y, and Block X. About half the subjects were instructed to attend to the lapse of time in Block X, and the others in Block Y (the other instruction was given before the other blocks). The presentation rate of the sounds was randomized across these 16 trials, although there were always eight of each. In all experimental trials, the standard duration was 17 s. They contained each of the four conditions, three times. In order not to let a subject notice that standard durations were always the same length, four dummy trials were added, one in each block. In two of the dummy trials the standard duration was 13 s, and in the other two it was 21 s. At the beginning of each block, the subject received one of the two instructions regarding attentional allocation, and just before the third trial in each block the same instruction was repeated. When all trials of duration estimation had ended, every subject was asked to what degree he or she had allocated attention to the lapse of time, in order to ascertain whether the instructions had been followed. After that, a recognition task was carried out. In this task, the subject was asked to recognize one by one whether spellings (now written on paper) had been presented or not in the classification task. Each sheet of paper had one of the 96 spellings used in the experimental trials of duration estimation, or one of 48 distracters (24 meaningful, 24 meaningless). If more attention had been allocated to the classification task than to the lapse of time, we would expect more spellings to be correctly recognized. Results and discussion Validity of the manipulations. In order to ascertain the effectiveness of the instructions about attention, total numbers of spellings recognized correctly in the recognition task were calculated for the four conditions (correctly rejected spellings were not analyzed) (Table 1). A two-way analysis of variance (ANOVA) (attention level rate of presentation of sounds) showed that only the main effect of attention level had a tendency to significance, F(1, 37) = 3.23, p.10. Total numbers of spellings classified correctly and mean response times in the classification task were calculated for the four conditions (Table 1). The ANOVAs showed that only the main effects of attention level were significant in both measures, F(1, 37) = 4.12, p.05; F(1, 37) = 30.88, p.001, respectively. These findings suggested that when instructed to pay less attention to the lapse of time, more Table 1. Mean (SD) total numbers of spellings recognized correctly and classified correctly, and mean (SD) response times of classification under the four conditions in Experiment 1 Attention level to the lapse of time Measurement Rate of sounds High Low Spellings recognized correctly a High 14.9 (3.5) 15.6 (3.1) Low 15.2 (2.7) 15.9 (2.4) Spellings classified correctly a High 19.7 (2.9) 20.8 (2.3) Low 19.8 (2.9) 20.7 (2.5) Response times (s) in classification task High 1.45 (0.24) 1.32 (0.23) Low 1.44 (0.24) 1.31 (0.22) a The maximum possible was 24.

5 148 Y. Kojima and F. Matsuda attention was indeed allocated to the classification task (and vice versa). Moreover, all subjects reported that they consciously allocated more attention to the lapse of time when instructed to do so. Therefore, it was reasonable to conclude that the level of attention to the lapse of time could be manipulated by such instructions. Since the ANOVAs above did not show either significant effects of the rate of presentation of the sounds or significant interactions, and the means were almost equal between the two conditions of the presentation rate of sounds, it was also reasonable to conclude that this factor had little effect on attention level, and therefore that the two factors were manipulated almost independently. Reproduced durations. Mean (SD) reproduced durations under the four conditions are shown in Table 2. A two-way ANOVA (attention level rate of presentation of sounds) showed significant main effects for both attention level and rate of presentation of sounds, F(1, 37) = 49.92, p.001; F(1, 37) = 10.51, p.005, respectively, though the interaction between them was not significant, F(1, 37) = 0.23, p.10. That is, the reproduced durations were longer when subjects were told to attend to the lapse of time than otherwise, and they were shorter when there was a higher rate than a lower rate of presentation of sounds. The findings clearly show that the two cognitive factors independently affected duration estimation. The present findings regarding the effect of attending to the lapse of time are in complete agreement with the prediction from the attention model (e.g., Hicks et al., 1977; Underwood & Swain, 1973), while those regarding the effect of the presentation rate of sounds are not Table 2. Mean (SD) reproduced durations (s) under the four conditions in Experiment 1 Attention level to the lapse of time Rate of sounds High Low High 11.8 (3.1) 10.0 (3.2) Low 12.6 (2.9) 11.0 (3.0) in accord with the prediction from the models based on the quantity of perceived changes (e.g., Block & Reed, 1978; Fraisse, 1967; Poynter, 1983). In previous studies, however, the effects of the presentation rate of intermittent stimuli have proved inconsistent across experiments with adult subjects, while in children there have been clear-cut results, that is, the higher the presentation rate, the longer the estimated duration (see Matsuda & Matsuda, 1976). It seems that if adult subjects are introspective enough, they may modify the quantity of perceived changes according to the interval duration they perceive, as suggested by Matsuda (1989a). Thus, the effects of the quantity may not be simple in the present experiment. To make doubly sure of the existence of two cognitive factors independently affecting duration estimation, we carried out another experiment, using another kind of external stimulus and another method of duration estimation, to minimize the possibility of such introspective modifications. Experiment 2 The purpose of Experiment 2 was to investigate again the effects of attention level to the lapse of time and quantity of perceived changes, but with the following two improvements on Experiment 1. First, in Experiment 2 we used moving dots to manipulate the quantity of perceived changes, after Tayama and Aiba (1982), because the salient attributes of these stimuli vary in the same direction of magnitude with respect to the quantity of perceived changes. That is, the faster the velocity of the dots, the larger is the number of times they are presented. In fact, previous investigations with adult subjects have consistently shown that faster-moving dots produce a longer estimated duration (Brown, 1995; Tayama & Aiba, 1982; Tayama, Nakamura, & Aiba, 1987). Second, in Experiment 2 the verbal estimation method was used. The method used in Experiment 1 might have allowed subjects to consider the effects of the stimuli on his or her cognition of the length of time elapsed

6 Effects of attention and stimuli on duration estimation under a prospective paradigm 149 while reproducing the standard duration, and this might complicate the effects of the stimuli. Nonetheless, it has been shown that effects of stimuli presented during the target interval on the length of the reproduced duration are comparable with those on the length of the verbal estimated duration (see Matsuda, 1985, 1996a; Zakay, 1993b). If the existence of two cognitive factors that independently affect duration estimation can be verified using two different methods of duration estimation, the results will be more conclusive. Methods Apparatus. The stimuli were presented using a personal computer (NEC, PC-9801BX). The distance between the CRT display (15 inches; NEC, PC-KD882) and the subject was about 55 cm. Subjects. Subjects were six graduate and six undergraduate students (six males and six females). Their mean age was 25.3 years. Method of duration estimation. The standard duration was marked off by two stimuli presented on the middle of the display. Each stimulus consisted of five aligned asterisks and was presented for 500 ms. After the presentation of the second asterisk stimulus, subjects were asked to write their estimated length of the standard duration, to the nearest a tenth of a second. Each estimated duration was written on a separate sheet of paper, so that the subject could not look again at previously estimated values. Task during the standard duration. As mentioned above, in Experiment 2 we used moving dots to manipulate the quantity of perceived changes. Therefore, tasks with visual stimuli during the standard duration, such as the classification task in Experiment 1, were inappropriate. For this reason, the following word association task was used. The subject was asked to associate words with a stimulus word (denoted the S-word), presented just before the beginning of the standard duration. Then, immediately after the verbal time estimation, the subject was asked whether or not his or her association words had included a particular word (designated as the A-word). The answer was written on the same sheet as the verbal estimation. The S-words were selected after consulting Japanese words in Suzuki and Kawase (1981) and some Japanese dictionaries. A-words were selected as follows. One graduate student and two undergraduate students were asked to associate words with each S-word. A word that was associated with a particular S-word by only one of them, and which the other two agreed was an appropriate association word, was chosen as an A-word. Thirty-six pairs of S-words and A-words were used in experimental trials, 7 in training trials, and 12 in dummy trials. Each of the paired words consisted of three to five hiragana letters. An example of the pairs is Kabuki (a traditional Japanese theater style) as an S-word and Naraku (a part of the typical Kabuki stage setting) as an A-word. These pairs were used in the trials of duration estimation in a random order. Each pair was used only once on each subject. At the beginning of a trial, an S-word was presented on the middle of the display for 700 ms, and the first asterisk stimulus was presented for 500 ms after the offset of the S-word. Next, the moving dots were presented 500 ms after the offset of the asterisk stimulus. Looking at the moving dots, the subject tried to associate words with the S-word. The second asterisk stimulus was presented 500 ms after the offset of the dots, and then the subject estimated the standard duration. Finally, the A-word was presented for 700 ms and the subject answered whether it had been one of his or her association words. Conditions during standard duration. There were thus two conditions, the attention level and the velocity of dots. As in Experiment 1, the attention level to the lapse of time was manipulated by verbal instructions: During the presentation of moving dots, allocate your attention to the lapse of time as much as you can. You need not care if you do not do well in the word association task. During the presentation of moving dots, allocate your attention to the word association task as much as you can and try to suggest as many words as possible. It does not matter if

7 150 Y. Kojima and F. Matsuda you do not pay much attention to the lapse of time. Moving dots, each of which was 1.8 mm in diameter, were presented on the display during the standard duration. The velocity of dots was 16 /s or 2 /s. There were 16 dots, which were randomly assigned in an area 10.7 cm 14.2 cm on the display and moved horizontally from the subject s left to right. Each dot disappeared at the right side of the area and reappeared at the left side of it at the same horizontal level. By combining the two attention levels and the two velocities of dots, four experimental conditions were generated. Each subject participated in all four. Procedure. Each subject was examined individually. At the beginning of the experiment, the subject was asked to remove his or her watch, and not to count during duration estimation. First, seven training trials of duration estimation, with the word association task, were carried out. At the beginning of the training trials, the subject was told that the moving dots had nothing to do either with the duration estimation task or with the word association task, but he or she had to keep looking at the display. After the first training trial without any instruction regarding attention, three training trials were carried out after the subject had been instructed to pay more attention to the lapse of time, and then another three training trials after the subject had been instructed to pay more attention to the word association task. Standard durations in training trials were 3.5 s, 4.5 s, and 5.5 s, in random order. Then, 48 trials were carried out: 36 experimental trials and 12 dummy trials. These trials were equally divided into six blocks. In half the blocks, trials were carried out after the subject had been instructed to pay more attention to the lapse of time, and in the other half after the other instruction. The order of the six blocks was Block X, Block Y, Block Y, Block X, Block X, and Block Y. For half the subjects, Block X was after the subject had been instructed to pay more attention to the lapse of time, and Block Y was after the other instruction, and for the other half of subjects vice versa. At the beginning of each block, the subject received the instruction regarding attentional allocation, and before the fifth trial in each block the instruction was given again. In the six experimental trials in each block, standard durations were 5.5 s, which was much shorter than in Experiment 1 because of the nature of the task set. In these trials the two conditions of velocity of dots were used three times each, in random. In order not to let the subject notice that the standard durations were always the same length, two dummy trials, whose standard durations were 6.5 s and 4.5 s, were included in each block. In these trials the velocity of dots was 4 /s. After the subject had finished all the trials, he or she was asked to answer easy problems of calculation for 1 min, to annul the recency effect in the following recall task. The recall task took 5 min, and verified whether the attentional allocation instructions had been followed. In this task the subject was asked to recall as many S-words and A-words as possible, making no distinction between the two kinds of words. (A recognition task like that in Experiment 1 could not be used, because correct recognition of these words was very easy.) Results and discussion Validity of manipulation. In order to ascertain the effects of the instructions regarding attention, the total numbers of words recalled correctly in the recall task were calculated for the four conditions (Table 3). A two-way ANOVA (attention level velocity of dots) showed neither significant main effects nor a significant interaction, F(1, 11) = 1.15; F(1, 11) = 1.00; Table 3. Mean (SD) total numbers a of words recalled correctly in Experiment 2 Attention level to the lapse of time Velocity of dots High Low High 2.8 (1.9) 4.2 (2.4) Low 4.2 (2.7) 3.9 (2.8) a The maximum possible was 18.

8 Effects of attention and stimuli on duration estimation under a prospective paradigm 151 F(1, 11) = 1.48, respectively, p.10. However, when the dummy trials, where the two conditions of the attention level were the same as in the experimental trials, were added to the experimental trials, the mean (SD) numbers of correctly recalled words under the condition of high attention and the condition of low attention to the lapse of time (combining data under the two conditions of velocity of dots) were 8.6 (4.8) and 11.3 (5.1), respectively, out of a possible maximum of 48, and the difference between them was significant, by F-test, F(1, 11) = 7.78, p.05. The results of these analyses therefore suggest that the instructions regarding attention level did affect the actual attention level to the lapse of time. Because the instructions were basically the same as those in Experiment 1, and there were no significant interactions between the two conditions, we can assume that attention to the lapse of time was changed successfully by the instructions, and that the velocity of dots did not greatly affect attention level. Verbal estimated duration. Mean (SD) verbal estimated durations under the four conditions are shown in Table 4. A two-way ANOVA (attention level velocity of dots) showed significant main effects of both attention level and velocity of dots, F(1, 11) = 6.79; F(1, 11) = 8.28, respectively, p.05, though the interaction between them was not significant, F(1, 11) = 0.67, p.10. That is, the verbal estimated durations were longer after the instruction to pay more attention to the lapse of time than after the instruction of pay more attention to the word association task, as in Experiment 1, and they were longer with faster-moving dots than with slower dots. Table 4. Mean (SD) verbal estimated durations (s) under the four conditions in Experiment 2 Attention level to the lapse of time Velocity of dots High Low High 3.58 (0.98) 3.14 (0.94) Low 3.00 (1.18) 2.68 (0.79) General discussion The purpose of this study was to verify whether there are two independent cognitive factors that affect duration estimation. The present experiments showed that the level of attention to the lapse of time was one factor, and that there was another factor concerning cognition of external stimuli irrelevant to the tasks performed during the estimated interval. The one factor was unable to be embodied in the other. From these findings, it is clear that models are not viable if they are based only on internal clocks or internal tempos (e.g., Fetterman & Killeen, 1990; Hoagland, 1981; Treisman et al., 1992), which are essentially unconcerned with the effects of cognitive variables and contextual factors (Zakay & Block, 1997), or are based on only one cognitive variable, such as the change model (e.g., Block & Reed, 1978; Fraisse, 1967; Poynter, 1983), the storage-size model (Ornstein, 1969), the model of shortterm memory demands (Fortin & Breton, 1995; Fortin & Rousseau, 1987; Fortin et al., 1993), or the attentional model (Hicks et al., 1977; Underwood & Swain, 1973). The present findings strongly suggest that a valid model of duration estimation must include at least two independent cognitive factors. There are a few models in which multiple factors and their interactions are considered, such as the attentional-gate model of Zakay and Block (Block & Zakay, 1996; Zakay & Block, 1997) and the four-multiplicative-factors model of Matsuda (1996b). Both are cognitive models, but also assume internal pulses that are produced autonomously and whose rate is influenced only by arousal level. Zakay and Block (1997) state that the attentional-gate model explains prospective duration estimation, which depends on arousal level and the amount of attention allocated to time. According to this model, increasing the arousal level induces a pacemaker to produce more pulses in a given unit of time, and nonrelevant environmental information can change this arousal level. It is also assumed that the more attention is allocated to the lapse of time, the more widely the attentional gate opens and the more pulses pass

9 152 Y. Kojima and F. Matsuda through it. Therefore, the higher the arousal level and the more attention allocated to the lapse of time, the longer the estimated duration will be. The findings in Experiment 2 can be explained by this model, assuming that looking at dots at high speed raises the arousal level. However, the effect of the presentation rate of sounds in Experiment 1 is not in accord with the expectation from the model, unless one could assume that the higher rate of the external tempo lowered the arousal level. This would be counterintuitive. External tempo may actually affect the time estimation process after the assumed pulse stream passes through the attentional gate in the model. In fact, Zakay and Block (1997) state that it may sometimes be difficult to identify which component of the model (e.g., pacemaker, attentional gate, or working memory) is influenced by a particular manipulation (p. 15). The four-multiplicative-factors model (Matsuda, 1985, 1989a, 1996b) assumes that the length of the estimated duration is basically determined by the following four factors: (a) the actual length of the duration; (b) internal tempo, represented as a frequency (in a particular subject, the tempo is stable and can change only when the level of neurophysiological excitation changes significantly); (c) the level of attention to the lapse of time; and (d) the integrated effect of cognition of nontemporal attributes of events that arise during the duration. Whether the nontemporal attributes lead to cognition of the more (i.e., the larger, the stronger, the longer, the faster, etc.) or the less is critical. That is, the more, the longer the estimated duration. This is a reflection of the assumption that more events need a longer duration (under a constant speed of the event s occurrence), and the over-generalization of that assumption. It has been found that in young children this effect is strong, but in adults very weak (Matsuda, 1989b, 1991; Yamasaki & Miyata, 1984). The model assumes that the four factors multiplicatively affect duration estimation in the following way. The assumed tempo to estimate duration, also represented as a frequency, becomes slower than the internal tempo when less attention is paid to the lapse of time. Conversely, when the level of attention to the lapse of time is higher, more pulses of the internal tempo transfer to the assumed tempo. A representation of the estimated duration is basically constructed on a measure of the actual length of the duration by that assumed tempo, though the representation is distorted slightly by the cognition of nontemporal attributes of events during the target interval in the direction of the more, the longer. Therefore, the longer the actual length of the duration, the higher the internal tempo, the higher the level of attention to the lapse of time, and the stronger the cognition of the more, the longer the estimated duration will be. Since, in the present experiments, the actual length of the standard duration was constant and there seemed to be no changes in the level of neurophysiological excitation, only the latter two factors should relate to duration estimation. The findings in Experiment 2 can be reliably explained by this model as well as by the attentional-gate model, assuming that looking at more (faster) dots resulted in cognition of the more. In addition, the findings in Experiment 1 may also be explained by this model, if one can assume that hearing more sounds at shorter intervals results in cognition of the less in introspective adults. However, this assumption seems to be arbitrary, as there are no measurements to support it. The present experiments clearly show that there are at least two independent cognitive factors that affect duration estimation. However, the existing models with multiple factors, that is, the attentional-gate model (Block & Zakay, 1996; Zakay & Block, 1997) and the four-multiplicative-factors model (Matsuda, 1996b), cannot fully explain the present findings. These two models assume a similar function about the level of attention to the lapse of time in the process of duration estimation. However, the presumed functions of external stimuli irrelevant to tasks during a target interval are different. That is, in the attentionalgate model these external stimuli work at the first step of the duration estimation process, by directly affecting internal tempo, while in

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