THE SYMBOLIC DISTANCE EFFECT FOR ALPHABETIC ORDER JUDGEMENTS : A SUBJECTIVE REPORT AND REACTION TIME ANALYSIS

Quarterly Journal of Experimental Psychology (1978) 30, 33-43. THE SYMBOLIC DISTANCE EFFECT FOR ALPHABETIC ORDER JUDGEMENTS : A SUBJECTIVE REPORT AND REACTION TIME ANALYSIS J. M. E. HAMILTON AND A. J. SANFORD Department of Psychology, The University of Glasgow, Glasgow, Scotland The time needed to decide whether a pair of letters is in the correct alphabetic order varies inversely with alphabetic separation. This is an example of a phenomenon generally found with the comparison of ordered symbols or concepts, called the symbolic distance effect (Moyer and Bayer, 1976). It is argued that when letters are compared, an important determinant of this effect is the degree to which subjects subvocally run-through parts of the alphabet to determine the correctness of the order of a stimulus pair. A trial-by-trial introspection procedure coupled with reaction time measurements is used in the present experiment, in which letter order judgements were made over a range of separations. RTs increased with increasing number of letters in the reported run-through. At small letter separations, run-through occurred more frequently, and this was found to be the basis of the symbolic distance effect. For trials on which no run-through was reported the symbolic distance effect was absent. The data are summarized as a model in which comparisons are made from directly available order information from memory, or with an additional run-through process. The details of the run-through process suggest that groupings learned in childhood are probably involved in the selection of the starting letter for run-through. Introduction An important cognitive function is the ability to compare the memory representations of objects and events along specified dimensions. At the empirical level, in a wide range of situations it has been established that the time needed to compare two things varies inversely with the distance between their referents on the judged dimension. For example, with digit pairs, less time was needed to decide which digit was larger if the difference between the digits was large (e.g. 2 and 8) rather than small (e.g. 7 and 8) (Banks, Fujii and Kayra-Stuart, 1976; Buckley and Gillman, 1974; Moyer and Landauer, 1967). Similar results obtain for decisions about alphabet order (Lovelace and Snodgrass, 1971 ; Parkman, 1971), the normative sizes of objects in memory (Holyoak, 1977; Moyer, 1973) and the comparison of items learned in an arbitrary linear ordering along some dimension 33

34 J. M. E. HAMILTON AND A. J. SANFORD (Potts, 1974). In addition, Holyoak and Walker (1976) found a similar effect with the comparisons of terms for the dimensions of time, quality and temperature. This general finding, termed the symbolic distance effect (Moyer and Bayer, 1976) is one of the primary phenomena which any model of comparison must explain. The present paper is concerned with the origins of the symbolic distance effect in making judgements of the order of letters of the alphabet. Early researchers had pointed out a parallel between alphabetic order judgements and numerical magnitude comparison (Lovelace and Snodgrass, 1971), but while research on the latter has advanced (Buckley and Gillman, 1974; Banks, Fujii and Kayra- Stuart, 1976), our understanding of the former has not progressed very far. This is surprising; while digit comparisons are known components of arithmetical operations (Restle, 1970; Svenson, 1974) the alphabet is of importance as a good example of a string of sequentially ordered information. In addition, making alphabetic judgements produces some pervasive introspective phenomena which could provide a unique cue to the underlying information processing. Consider how alphabetic index systems are used, for example in looking up a word in a dictionary. In trying to look up a word beginning with P a person may find himself at the letter L. How does he know which way to flick the pages? Anecdotal observation suggests that people sometimes discover the order relation by mentally running through the sequence LMNOP. If run-through of this kind is the time-consuming process it seems to be, and if it occurs more often with small letter separations, then it could offer an explanation of the symbolic distance effect. The possible involvement of such run-through was mentioned en passant by Lovelace and Snodgrass ( I~I), but they did not explore the question in any detail. The experiment below requires subjects to decide on the correctness of the order of letter pairs at various alphabetic separations. By asking subjects to give introspections after each and every trial as to processing which may have occurred, the role of run-through as a determinant of the symbolic distance effect was assessed. Method Subjects The 10 subjects were students or staff from the University of Glasgow between 21 and 31 years of age. All were paid volunteers, and received sop for attending one session lasting approximately 40 min. Apparatus and stimuli The stimuli, presented visually, were 2 X z in slides presented in a Scientific Prototype 3 channel automatic tachistoscope type GB. Each subject saw 168 slides, each consisting of a pair of letters separated along the horizontal axis about the centre of the visual field. The letters were black on a white background and in upper-case letraset. Twelve examples of each letter separation from one letter apart (i.e. adjacent letters), to 14 letters apart were randomly selected from the available possibilities. Therefore there were 24 examples of each letter separation, the 12 chosen and their reverse order constituted 12 examples of each letter separation in the correct alphabetic order, i.e. the

ALPHABETIC ORDER JUDGEMENT 35 letter to the left of the pair coming nearer the beginning of the alphabet, and 12 examples of incorrect alphabetic order. The slides were divided into two groups, each group having six examples of each alphabetic separation in the correct order and six in the incorrect order, with the restriction that no one-letter pair was present in both correct and reverse order within the one group. Procedure The 168 slides were presented to each subject in a random order with the constraint that no two consecutive slides contained letters which lay adjacent to each other in the alphabet. Subjects saw a 2-9 blank (warning) field, which was immediately followed by the stimulus field for two seconds. Subjects were instructed to respond as quickly and as accurately as possible by pressing keys to indicate whether or not the pair of letters exposed was in the correct alphabetic order. Reaction times were taken from the onset of the stimulus field to the subjects response. After each and every trial the subject reported to the experimenter verbally, whether he was aware of any conscious process which had led to the response, or whether the answer had come automatically. Subjects were randomly allocated to two slide groups and, randomly within each group, half of the subjects responded to the correct order with their preferred hand and the other half used their non-preferred hand. Results The symbolic distance efect Errors occurred on 7'25% of all trials and were not systematically related to separation; error trials were eliminated from further analyses. Figure I shows that the closer together two letters are in the alphabet, the greater the RT to determine whether they are in the correct alphabetic order. This was confirmed by an analysis of variance on the median RTs (F = 8-96, df = 13, 117, P --I 7 Correct order pairs \ V Incorrect order pars 6-81 L I I I I 1 1 1 1 I I 1 1 1 I 2 3 4 5 6 7 8 9 10 II 121314 Letter separation FIGURE I. Mean RT as a function of letter separation. The dotted lines show the mean RTs produced by a summary model described later.

36 J. M. E. HAMILTON AND A. J. SANFORD <O.OOI). In addition RTs were longer for incorrect pairs (F = 17-98, df = I, 9, P <O*OOI). These two main factors showed a reliable interaction (F = 2-21, df = 13, 117, P <o.ozg). Run-through and separation All subjects reported running-through parts of the alphabet on some of the trials, although the extent of this varied considerably from person to person. No subject showed run-through at all separations, and some subjects had a fairly limited range of run-through length. Length of run-through was defined as the number of letters reported as being thought of prior to emission of the response. Thus if the subject reported running through the sequence PQRS, this would represent a run of 4, regardless of whether one or both of the stimulus letters were included in the run. Figure 2 shows the relation of RT to reported run length for all runs of two or more items. l I I I I I I,, - - L - d 2 3 4 5 6 7 8 More Run- thmugh length FIGURE 2. Mean RT as a function of reported run-through length. The dotted lines indicate the best fitting straight lines by the method of least squares. Although there was not enough data for individual functions to be fitted, pooling data across subjects showed the relationship of RT to reported run-length to be an approximately linear function (slope = 367 ms/item; r = +0.98). This indicates that the process giving rise to run-through is a crucial factor in determining RT. Furthermore, the rate of run-through falls within the range

ALPHABETIC ORDER JUDGEMENT 37 normally obtained for subvocal speech (Landauer, 1962), and is consistent with the introspection that run-through is a species of implicit speech." Figure 3 shows that run-through reports were given most often at small letter separations: Friedman's test (Siegel, 1956) giving x: = 59.9 (P <O*OOI) for correct-order pairs, and x: = 50'47 (P <O.OOI) for incorrect order pairs. In contrast to frequency, mean run-through length was not reliably related to separation, and had a mean value of 4-32 letters. The relationship (or lack of it) could not be tested by an analysis of variance because not enough subjects produced data to provide a value for each separation. However, a Spearman rank-order correlation was calculated between run-length averaged across subjects as a function of separation. The value of rs = 0.193 is both weak and unreliable. There was no difference between correct and incorrect orders. Letter separation FIGUR~ 3. Percentage of run-through trials at each letter separation. Taken together, these data indicate that any contribution run-through might make to the symbolic distance effect originates in its frequency of occurrence at each separation rather than in the length of run-through. In this case, the mean RT at a given separation should show a reliable dependence upon the proportion of run-through trials occurring at that separation. The productmoment correlations between RT and proportion of run-through trials were 0.91 (t = 7.60, N = 12, P <O-OOI) for the correct order data, and 0.93 (t = 8.77, An experiment in which adjacent item naming is required has been carried out and provides support for the idea that the RT/run-length functions are linear for individual subjects (Hamilton and Sanford, in preparation).

38 J. M. E. HAMILTON AND A. J. SANFORD N = 12, P <O.OOI) for the incorrect order data. The magnitude of these correlations provides strong evidence for the involvement of run-through in the symbolic distance effect. Direct trial and separation When run-through does not occur (i.e, on direct only trials) the symbolic distance effect is absent from the data (Fig. 4). An analysis of variance revealed that a main effect of separation remained (F = 2.22, df = 13, 117, P to.025) along with an order effect (F = 23-00, df = I, 9, P <O*OOI) and an interaction of these two factors (F = 1-97, df = 13, 117, P to.05). Independent analysis show that the correct-order separation effect is unreliable (F = 1-65, df = 13, 117) but is reliable for the incorrect order data (F = 2-33, df = 13, 117, P to.025). I" Correct order pairs V Incorrect order pairs 0.81 111111111111111 I 2 3 4 5 6 7 8 9 10 II 1213 14 Letter separation FIGURE 4. Mean RT for direct trials only as a function of letter separation. The analysis indicates no effect of letter separation when a separation of I is excluded from the incorrect order data. This appears to be due entirely to the influence of one data point. This was confirmed by reanalysing the data excluding the first point, in which case there is no longer a reliable separation effect (F = 1-24, df = 12, 115). As a final demonstration of the major role run-through plays in the RT/separation function, a rough model was constructed including the assumptions that RTs for direct trials are not a systematic function of separation and run-through length is not a systematic function of separation. In this way the model generates the symbolic distance effect solely on the basis of proportion of trials on which run-through is reported (the Appendix gives further details). Predictions based on this model are shown in Figure I as dotted lines. The numerical correspondence and fit (as a correlation) is reasonable (Y = 0*93), and provides a convincing demonstration that run-through proportion determines the symbolic distance effect.

ALPHABETIC ORDER JUDGEMENT 39 Discussion The results suggest that alphabetic order comparisons can be made in two ways. Firstly, the order information may be directly available to the subject. In this case, it does not matter how many letters separate the comparison pair, the RT is constant because the information is directly available. In this way direct judgements do not contribute to the symbolic distance effect. Secondly, if the information is not directly available, subjects consistently report running through part of the alphabet prior to responding. Run-through occurs more frequently at small separations, and the corresponding longer RTs produce the symbolic distance effect. It might appear surprising that run-length is not smaller at smaller separations. However, on closer inspection of the introspective reports, there appeared two good reasons why this should not be the case. The first was that running through did not necessarily start with one of the stimulus letters, but may have commenced some way before in the alphabet. Secondly both of the stimulus letters were not invariably included in the recitation. The manner in which run-through operates is therefore fairly complex. The reports can be conveniently divided into three types, however. On 59 2% of runs-through, both of the stimulus letters were included. For example, given the pair MO, one subject reported running through LMNO. (Note that the starting letter was not one of the stimulus pair.) Thus by producing a string in which both target letters were included, decisions about the order of the letters could be reached. On some occasions a second type of run-through occurred in which the direction of the run-through was such as to increase the gap between the two stimuli. For instance, given the pair RP, one report was RSTUV. This outcome occurred on 14 2% of occasions. The remaining type (26.6%) consisted of gap closures not including one of the letters-e.g. given MU, the run-through LMNOP might be produced. These forms of run may lead to a direct comparison of the current letter of run-through with one of the stimulus letters, enabling a decision to be made. As run-through does not always begin with one of the stimulus letters, this suggests that the subject cannot always say which letter follows another in the alphabet. An explanation of this may lie in the way the alphabet is learned by most people as a series of groups bounded by pauses. An example of such a structure might be (ABCDEFG) (HIJK) (LMNOP) (QRST) (UVW) (XYZ), where brackets indicate pause boundaries. It is possible that if a letter (e.g. V) initiates run-through, then the whole of the group of which it is a member is retrieved as a sequential output, starting with the$rst letter of the group, in this case U. Evidence for this assumption comes from Sternberg s (1967) order judgement task and Wilkes and Kennedy (1970) using groups of letters. Both investigations were consonant with the contention that retrieval of order information occurs by run-through from the beginning of the relevant group, although (curiously) none of these authors commented on any introspections of run-through. To test whether such groups did influence starting letters, an attempt was made to check individual subjects groupings against starting letters. This was possible for seven of the subjects. Of all the run-through trials, 49% did not start with

40 J. M. E. HAMILTON AND A. J. SANFORD one of the stimulus letters. Of these 58% of the runs started with a subjective group starting letter. There were of course some runs which started with a stimulus letter which was also a group starting letter. This occurred in fact on 42% of these trials. Closer scrutiny of the still unaccounted-for starting letters revealed that they were almost all due to three subjects consistently using one (different) letter to start their individual runs. (Only 16% of starting letters are then unaccounted for.) A second possible interpretation of the groupings is that direct responses may take place when the stimulus letters fall in different groups, i.e. directly discriminably different portions of the string. This was not the case; 68% of runs occurred when the presented letters were in different blocks. The experiment clearly indicates that the symbolic distance effect results from the use of run-through. For other tasks, quite different accounts fit the data better-for semantic judgements a continuous (analogue) representation of comparison values has been suggested as offering the best fit to the data (Holyoak and Walker, 1976). For digit comparison, Parkman (1971) proposed a counting model. The basis of this was an internal register which was initially set at zero, and progressed by unitary incrementation and comparison cycles until a match was made with the minimum digit. Although superficially it may appear that incrementing and checking is like run-through, the cycle rate inferred by Parkman was very fast (50-100 item+ compared with 3-4 for the present run-through). Parkman also concluded that his model did not fit the alphabet comparison data he obtained. One further difference between digit and letter comparisons is that there is no introspection of run-through in the former (Fenning and Sinclair, unpublished data, University of Glasgow). It is extremely unlikely that a single mechanism, analogue or run-through based, can explain all cases of the symbolic distance effect. So far as the alphabet itself is concerned, the next step is to understand precisely how run-through relates to underlying modes of storage and retrieval. Appendix Reaction times were predicted for each separation by computing RT = A (mx + c) + B.K A+B Where A = number of trials at that separation on which run-through was reported. B = number of direct trials at that separation (A + B = 60). x = mean length of run-through averaged across all separations and both orders. m = slope of RT as a function of run length. c = intercept of RT/run-through function. K = mean RT on direct trials for appropriate order. This formula embodies the principles that run-length and direct trial times are independent of separation. K, c and m have two values each, one for correct- and the other for incorrect-order data. K will entail processes of comparison not entailed in run-through trials, and incorrect order pairs are slower to process than correct order pairs. What is less obvious is that the difference between c (correct order) and c (incorrect order) at 306 ms is considerably greater than the difference between K (correct order) and K

ALPHABETIC ORDER JUDGEMENT 4' (incorrect order) at 173 ms. This implies that whatever the mechanism determining the order difference may be, it is influenced by, or varies with the mode of comparison (direct or run-through) employed. Because of the missing data problem, this cannot be tested statistically, but obviously makes a considerable difference to the fit of the model. The fact that m has two values is trivial; the two values used for nz from the best-fitting linear regressions for individual orders were very close, m (correct order) = 360ms/item; m (incorrect order) = 374 ms/item. References BANKS, W. P., FUJII, M. and KAYARA-STUART, F. (1976). Semantic congruity effects in comparative judgements of magnitude of digits. Journal of Experimental Psychology : Human Perception and Performance, 2, 435-47. BUCKLEY, P. B. and GILLMAN, C. B. (1974). Comparison of digit and dot patterns. Journal of Experimental Psychology, 103, I 131-6. HOLYOAK, K. J. (1977). The form of analog size information in memory. Cognitive Psychology, 9, 31-57. HOLYOAK, K. J. and WALKER, J. H. (1976). Subjective magnitude information in semantic orderings. Journal of Verbal Learning and Verbal Behavior, 15, 287-99. LANDAUER, T. K. (1962). Rate of implicit speech. Perceptual and Motor Skills, 15, 646. LOVELACE, E. A. and SNODGRASS, R. D. (1971). Decision times for alphabetic order of letter pairs. Journal of Experimental Psychology, 88, 258-64. MOYER, R. S. (1973). Comparing objects in memory: Evidence suggesting an internal psychophysics. Perception and Psychophysics, 13, 180-4. MOYER, R. S. and BAYER, R. H. (1976). Mental comparisons and the symbolic distance effect. Cognitive Psychology, 8, 228-46. MOYER, R. S. and LANDAUER, T. K. (1967). Time required for judgements of numerical inequality. Nature, 215, 1519-20. PARKMAN, J. M. (1971). Temporal aspects of digit and letter inequality judgements. Journal of Experimental Psychology, 91, 191-205. POTTS, G. R. (1974). Storing and retrieving information about ordered relationships. Journal of Experimental Psychology, 103, 431-9. RESTLE, F. (1970). Speed of adding and comparing numbers. Journal of Experimental Psychology, 83, 274-8. SIEGEL, S. (1956). Non-Parametric Statistics for the Behavioural Sciences. New York: McGraw-Hill. STERNBERG, S. (I 967). Retrieval of contextual information from memory. Psychonomic Science, 8, 55-6. SVENSON, 0. (1974). Analysis of time required by children for simple additions. Report from the Psychological Laboratories, University of Stockholm. WILKES, A. L. and KENNEDY, R. A. (1970). The relative accessability of list items within different pause-defined groups. Journal of Verbal Learning and Verbal Behaviour, 9, 197-201. Received 7 April 1977