Psychology and Aging 2003, Vol. 18, No. 2, 240 –249

Copyright 2003 by the American Psychological Association, Inc. 0882-7974/03/$12.00 DOI: 10.1037/0882-7974.18.2.240

Subitizing Speed, Subitizing Range, Counting Speed, the Stroop Effect, and Aging: Capacity Differences and Speed Equivalence Chandramallika Basak and Paul Verhaeghen Syracuse University Thirty younger and 29 older adults were tested on reaction times for set size of a display of 1 to 9 digits. On half of the trials, the nominal value of the digits was equal to the size of the set displayed; on the other half, the value differed by 1 from the set size (Stroop interference). We found evidence for age differences in subitizing span (2.83 vs. 2.07). Once individual differences in subitizing range were taken into account, no age differences were found in the rate of either subitizing or counting, and no individual differences were apparent in subitizing speed. There was no age difference in the susceptibility to the Stroop effect. The results suggest that, with advancing age, the size of the focus of attention may shrink, but speed of access to elements in the focus of attention may remain constant.

maximum number that can be subitized by the individual, enumeration can no longer occur in parallel, and the slower, presumably serial and controlled process of counting must be used. Counting typically yields RT ⫻ Set Size slopes of 250 to 350 ms per item. Subitizing is an automatic or near-automatic process; counting is not. The general theoretical agreement in the field of cognitive aging is that more complex tasks yield larger age differences (e.g., Cerella, 1990). Therefore, one might expect that if an age-related dissociation between those two tasks would be observed, counting speed would be more adversely affected by age than subitizing speed. In fact, one could derive an even more precise expectation, namely that age differences in subitizing speed would be zero, or close to zero, because automatic processes are generally exempt from the negative effects of aging (Kausler, 1991). This, however, is not what is found. First, the ratio of old to young subitizing slopes, an indicator of age-related slowing in the subitizing process, is typically larger than one (this ratio is 1.33 in Kotary & Hoyer, 1995 [distractor-absent target-only conditions]; 1.41 in Nebes, Brady, & Reynolds, 1992; 2.01 in Sliwinski, 1997; and 0.82 in Trick, Enns, & Brodeur, 1996; yielding an average age-related slowing factor of 1.39). Furthermore, age-related slowing factors are larger for subitizing than for counting in two out of three studies that compared both (Kotary & Hoyer, 1995: 1.33 vs. 1.19; Trick et al., 1996: 0.82 vs. 1.36; Sliwinski, 1997: 2.01 vs. 1.04; yielding an average old-over-young slope ratio of 1.39 for subitizing and 1.19 for counting).1 Thus, we are left with a puzzling result: Subitizing, a parallel and automatic or near-automatic process, is age sensitive, and, moreover, it seems to be more age sensitive than counting, which is a serial controlled process. At least two explanations can be advanced for this finding. First, it is possible that the dissociation is not due to differential age-related slowing in the two processes but is merely a statistical artifact. In all of the studies that examined age difference in

One of the best established regularities in the field of cognitive aging is the orderly relation between latencies of older adults and those of younger adults, known since the 1980s (Cerella, 1990; Cerella, Poon, & Williams, 1980; Myerson, Hale, Wagstaff, Poon, & Smith, 1990). This regularity can typically be captured quite well by a single highly linear function, relating latencies of older adults to corresponding latencies of younger adults. This result has led to the formulation of a general slowing hypothesis, that is, the hypothesis that a single ratio of old to young processing rates can describe slowing in central cognitive processes (e.g., Cerella, 1990, 1994). The challenge for current theory and research in the field is to find exceptions to this pattern. Recently, distinct age-related slowing ratios have indeed emerged for different types or domains of tasks (Hale & Myerson, 1996; Kliegl, Mayr, & Krampe, 1994; Mayr, Kliegl, & Krampe, 1996; Myerson & Hale, 1993; Sliwinski & Hall, 1998). Such exceptions to the pattern of general slowing have been termed “dissociations” (Perfect & Maylor, 2000). One dissociation that has recently attracted some attention— because the direction of the effect is counterintuitive—is the dissociation between subitizing speed and counting speed. Subitizing is the rapid, probably preattentive (Trick & Pylyshyn, 1994) process that allows people to immediately “see” or “grasp” the number of elements shown in a display, as long as the number of elements is very small (i.e., not more than three or four elements). Subitizing is postulated to be a parallel and near-automatic process, as reflected in very shallow Reaction Time (RT) ⫻ Set Size slopes—that is, 40 to 100 ms per item (Trick & Pylyshyn, 1994). When the number of elements in the display is larger than the

Chandramallika Basak and Paul Verhaeghen, Department of Psychology and Center for Health and Behavior, Syracuse University. This research was supported in part by Grant AG-16201 from the National Institute on Aging. We thank Norbou Buchler, John Cerella, William J. Hoyer, and especially Martin Sliwinski for helpful comments on drafts of this article. Correspondence concerning this article should be addressed to Chandramallika Basak or Paul Verhaeghen, Department of Psychology, 430 Huntington Hall, Syracuse University, Syracuse, New York 13244-2340. E-mail: [email protected] or [email protected]

1

This was the state of the field when we started planning our study. Since then, a new study has been conducted (Watson, Maylor, & Manson, 2002) that conforms better to the predictions derived from aging theory: In this study (distractor-absent conditions only), the old-over-young slope ratio for subitizing was found to be ⫺0.46; the slope ratio for counting was 0.96. 240

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enumeration speed, the young– old comparison is based on the assumption that the maximum number of items that can be subitized (i.e., the subitizing range) is identical across participants and across age groups. Most researchers set the subitizing range at four items (Nebes et al., 1992; Sliwinski, 1997; Trick et al., 1996; but Watson et al., 2002, set it at three items). It is quite possible, however, that the subitizing range varies across individuals, across age groups, or even across trials within individuals. If the subitizing range of a fair number of participants is smaller than the numerosity defined as the cutoff for the subitizing range, then the higher numerosities within the generic subitizing range will be contaminated by the slow serial counting process, and the subitizing slope of the group will be overestimated. Moreover, if the subitizing range of older adults is on average smaller than that of younger adults, one would expect that the subitizing slope is overestimated by a larger amount in older adults than in younger adults. The range hypothesis can be tested directly by estimating each individual’s subitizing range and by calculating subitizing slopes at the individual level. If the observed age differences in subitizing rate are an averaging artifact, then the age difference will disappear when subitizing speed is calculated within each individual’s own subitizing range. An alternative explanation for the subitizing/counting dissociation is that the decline is indeed due to age-related changes in the efficiency of the subitizing process. The most parsimonious explanation for such changes is that although subitizing is an automatic process in younger adults, it becomes an increasingly controlled process in old age. This hypothesis can be tested by including a manipulation that would selectively influence one of these processes and not the other, and by testing for Age ⫻ Manipulation interactions. The manipulation we included was Stroop-like interference (for a review of the Stroop literature, see MacLeod, 1991). In our experiment, the stimuli to be enumerated consisted of digits. On half of the trials the numerosity indicated by the digit differed (by 1) from the number of digits shown; on the other half of the trials, numerosity of the digit and the number of digits shown matched. It can be argued that subitizing, as an automatic process, has very fast mappings from preverbal to verbal representations, which would then be relatively impervious to verbal interference. For counting, however, participants rely more on verbal representations (i.e., they silently enumerate, which is a slow process of verbal number name retrieval; Trick et al., 1996), and therefore the verbal representations shown on the screen might influence this process. The expectation then is that a Stroop manipulation would interfere with the counting process but not with the subitizing process. If the automaticity hypothesis is correct, then the Stroop effect on subitizing would be smaller for younger adults (where subitizing is presumably automatic and driven by fast preverbal-to-verbal mappings) than for older adults (where subitizing might be a more controlled, verbal process).

Method Participants We tested 30 younger adults (mean age ⫽ 19.07, SD ⫽ 1.17; ranging from 18 to 22; 19 females and 11 males) who received course credit for participating, and 29 older adults (mean age ⫽ 74.17, SD ⫽ 5.04; ranging from 66 to 81; 21 females and 8 males) who received $15 in return for their time and effort. Younger adults averaged 13.13 years of education

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(SD ⫽ 1.17), and older adults averaged 15.19 years of education (SD ⫽ 2.17); the difference is significant, t(56) ⫽ 4.51. The older adults scored higher on the Mill-Hill Vocabulary test (Raven, 1982; 22.41, SD ⫽ 4.2) than did the younger adults (15.87, SD ⫽ 4.0), t(56) ⫽ 6.13.

Task and Procedure Each stimulus for the enumeration task consisted of a set of nonzero digits, shown on a computer screen. Set size ranged between 1 and 9. The numerosity indicated by the digit was equal to the set size on half of the trials (congruent trials), and it differed from the set size by 1 on the other half of the trials (incongruent trials). On incongruent trials and for set sizes 2 through 8, the numerosity indicated by the digits was 1 unit smaller than the set size on half of the trials, and it was 1 unit larger on the other half. For set-size 1, the digit shown on incongruent trials was always 2; for set-size 9, numerosity indicated on incongruent trials was always 8. The participants’ task was to enumerate the number of digits shown (i.e., the set size) and press the key on the computer keypad that corresponds to that number. Participants were instructed to be as fast and as accurate as possible. Participants were presented with four blocks of 90 enumeration trials each (45 congruent, 45 incongruent; 5 of each set size within each trial type). Each enumeration block was preceded by a familiarization block of 45 trials of an RT task in which single digits (5 instances of each of the 9 nonzero digits) were presented. The task was to hit the corresponding number on the computer keypad. This task was inserted for two reasons: (a) as a warm-up, and (b) to train the participants in the numerosity-to-keypad mapping. In the enumeration task, congruent and incongruent trials were mixed randomly within blocks. Short 2–5-min tests (a digit span test [Wechsler, 1981], a symbol digit substitution test [Smith, 1982], and the vocabulary test [Raven, 1982]) were interspersed between blocks. Sessions typically lasted between 1 hr and 90 min. All stimuli were projected in white on a black background. Each trial started with the presentation of a fixation cross in the middle of the screen, for a duration of 1,000 ms. Then an array of digits appeared, randomly placed in an invisible 9 ⫻ 9 square grid, about 126 mm wide. Each digit was projected in a sans serif font and was about 14 mm high and 9 mm wide. To ensure irregularity of the display, the digits were displaced by up to 3 mm in the horizontal and/or vertical directions. Participants were allowed to individually optimize their viewing distance from the screen. As a consequence, visual angle of the stimuli differed across participants. Most participants seemed to prefer a distance from the screen of approximately 50 cm. At this distance, the area of the stimulus presentation (for set size greater than 1) varied randomly from 3.44° to 15.94° in height and from 5.72° to 18.18° in width (depending on the dispersion of the stimuli). Although on average the area of stimulus presentation was larger for set sizes greater than 6, there was more variability in smaller set sizes, with their larger areas being the same size as those for larger set sizes. The participant terminated the display of the stimulus array by executing a keypress. This initiated the next trial in the block after a delay that varied randomly between 1,500 and 2,000 ms. For familiarization trials, the target always appeared in the center of the screen. Alpha levels for each test were set at p ⫽ .05.

Results As usual in enumeration research (e.g., Mandler & Shebo, 1982; Trick & Pylyshyn, 1994), the largest set size (i.e., 9) was excluded from the analysis because of possible end effects.

Error Rates Error rates were quite low throughout the experiment. We conducted an Age Group ⫻ Set Size ⫻ Trial Type (i.e., congruent vs. incongruent) repeated measures analysis of variance

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(ANOVA). Older adults were more accurate than younger adults (proportion correct: .98 vs. .96), F(1, 57) ⫽ 10.49, MSE ⫽ 0.0096. Answers to items in the congruent condition were more accurate than answers to items in the incongruent condition (.98 vs. .96), F(1, 57) ⫽ 12.39, MSE ⫽ 0.0043. Accuracy varied across set size, F(7, 399) ⫽ 19.24, MSE ⫽ 0.0022, and both the linear and quadratic components were significant. Accuracy was essentially stable across numerosities 1 to 5 (between .98 and .99) and dropped off slightly for larger set sizes (.96 for set-size 6; .95 for set-size 7, and .94 for set-size 8). These main effects were qualified by a Set Size ⫻ Trial Type interaction, F(7, 399) ⫽ 2.25, MSE ⫽ 0.0022, indicating that the drop-off in accuracy over set size was larger in the incongruent trials than in the congruent trials (averaged over numerosities 1 to 4: congruent, .99; incongruent, .98; averaged over numerosities 5 to 8: congruent, .97; incongruent, .95); and by an Age ⫻ Set Size interaction, F(7, 399) ⫽ 2.81, MSE ⫽ 0.0022, indicating that the drop-off in accuracy over set size was larger in younger than in older adults (averaged over numerosities 1 to 4: younger, .98; older, .99; averaged over numerosities 5 to 8: younger, .94; older, .97). In terms of absolute accuracy, all of these effects are quite small. They do indicate, however, that our sample of older adults was more accurate in enumeration than the younger sample, both in terms of main effects and in being less susceptible to the detrimental effect of larger set sizes.

We conducted a maximum-likelihood hierarchical linear model analysis (HLM; Snidjers & Bosker, 1999), rather than a standard ordinary least squares (OLS) regression analysis, for two reasons. First, HLM allows for the examination of individual differences in parameters, that is, intercepts and slopes. Second, when individual differences in subitizing range are taken into account (see below), the design becomes unbalanced, with each individual contributing an unequal number of observations. HLM is well suited to handle unbalanced data; traditional OLS is not. There are two levels for this dataset. Level 1 contains the within-subject variables (set size, trial type, and block number); Level 2 contains the between-subjects variable (age group). All plausible interactions (both between Level 1 variables and crosslevel interactions) were taken into account. The first model (Model 1: 1-segment random intercepts model) fitted was a random intercepts model (i.e., a model equivalent to traditional repeated measures ANOVA):

Reaction Time in Enumeration Trials

Results from the fitting procedure are reported in Table 1. In the next step, a random coefficients model was fitted to test for individual differences in the set-size-related change in RT. In this model (Model 2: 1-segment random coefficients model), the RT ⫻ Set Size slope was not treated as a fixed effect, as it was in Model 1, but it was allowed to vary from individual to individual. The random coefficients model is represented as

Figure 1 depicts RT (for correct responses only) as a function of set size, separated by age group and trial type. The main analysis of the RT data was done by using multilevel modeling, taking into account the nested sources of variability. Fixed effects were estimated to analyze average RT and the effects of age. Random effects and their variances were used to model individual differences in RT scores and individual differences in enumeration speed. Different covariates were introduced in the models: (a) a dummy variable indicating the participant’s age group (0 ⫽ younger and 1 ⫽ older); (b) block number; (c) set size (ranging from 1 to 8); and (d) trial type (1 ⫽ congruent and 2 ⫽ incongruent).

Yij ⫽ ␥ 00 ⫹ ␥ 10 共set size兲 ⫹ ␥ 01 共age group兲 ⫹ ␥ 20 共trial_type兲 ⫹ ␥ 30 共block兲 ⫹ ␥ 40 共Set Size ⫻ Task_Type兲 ⫹ ␥ 11 共Age Group ⫻ Set Size兲 ⫹ ␥ 21 共Age Group ⫻ Trial_Type兲 ⫹ ␥ 31 共Age Group ⫻ Block兲 ⫹ ␥ 41 共Age Group ⫻ Set Size ⫻ Trial_Type兲 ⫹ U0j ⫹ Rij .

Yij ⫽ ␥ 00 ⫹ ␥ 10 共set size兲 ⫹ ␥ 01 共age group兲 ⫹ ␥ 20 共trial_type兲 ⫹ ␥ 30 共block兲 ⫹ ␥ 40 共Set Size ⫻ Task_Type兲 ⫹ ␥ 11 共Age Group ⫻ Set Size兲 ⫹ ␥ 21 共Age Group ⫻ Trial_Type兲 ⫹ ␥ 31 共Age Group ⫻ Block兲 ⫹ ␥ 41 共Age Group ⫻ Set Size ⫻ Trial_Type兲 ⫹ U0j ⫹ U1j 共Set Size兲 ⫹ Rij .

Figure 1. Reaction time (for correct answers only) as a function of age group and condition.

(1)

(2)

The RT ⫻ Set Size slope (␥10 ⫽ 289.80 ms/item; SE ⫽ 11.59) was identical to the slope obtained from Model 1 (␥10 ⫽ 289.80 ms/item; SE ⫽ 4.32); but the deviance of Model 2 was significantly smaller, ␹2(2, N ⫽ 59) ⫽ 725.00, indicating a better fit. Therefore, all further models fitted were random coefficients model. Because neither the interaction between age group and trial type, F(1, 4120) ⫽ 0.42, MSE ⫽ 1910.66, nor the three-way interaction term between age group, actual set size, and trial type, F(1, 4120) ⫽ 0.62, MSE ⫽ 60.34, was significant, we decided to drop these interaction terms from all subsequent models. Model 3 (1-segment reduced model) is the reduced Model 2, with the nonsignificant interaction terms removed. The intercept of this model reflects the average RT score for younger adults at set-size 0 (627.30, SE ⫽ 36.40). The RT ⫻ Set Size slope was 288.24 ms/item (SE ⫽ 11.42). At set-size 0, older adults were 381.00 ms slower than younger adults (SE ⫽ 47.14); older adults’ RT ⫻ Set Size slopes were 10.17 ms/item larger than those of

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Table 1 Parameter Estimates for Fixed Effects and Variance Components for Random Effects for Models 1–3 1-segment random intercepts model

1-segment random coefficients model

1-segment reduced model

Fixed effect

Coeff.

SE

p

Coeff.

SE

p

Coeff.

SE

p

␥00 ␥10 ␥01 ␥20 ␥40 ␥11 ␥21

619.40 289.80 ⫺365.45 11.02 ⫺18.97 ⫺13.24 ⫺28.27

57.41 4.32 80.51 25.40 6.11 6.06 48.25

⬍.0001 ⬍.0001 ⬍.0001 .7487 .0019 .0291 .5580

619.40 289.80 ⫺365.45 11.02 ⫺18.97 ⫺13.24 ⫺28.27

39.07 11.59 54.78 31.17 5.54 16.26 43.71

⬍.0001 ⬍.0001 ⬍.0001 .7237 .0006 .4155 .5178

627.30 288.24 ⫺381.00 ⫺3.35 ⫺15.85 ⫺10.17

36.40 11.42 47.14 21.86 3.89 15.79

⬍.0001 ⬍.0001 ⬍.0001 .8781 ⬍.0001 .5193

Variance component

SE

Variance component

SE

6,034.56 676.61 1,569.19 2,349.46

25,715.00 3,452.20 ⫺3,870.41 106,873.00

6,034.56 676.61 1,569.19 2,351.85

Random effect

␶0 ␶12 ␶01 ␴2 Deviance 2

Variance component 72,993.00 — — 130,105.00 62,299.80

SE 13,772.00 — — 2,842.86

25,722.00 3,452.43 ⫺3,871.53 106,765.00 61,574.80

Note. For description of Models 1–3, see text. The intercept ␥00 reflects the average reaction time (RT) intercept at set size ⫽ 0. ␥10 is the effect of set size. ␥20 is the effect of trial_type. The slope ␥01 is the average rate of set-size-related RT change for cases with age group ⫽ 0 (i.e., for younger adults). The Level 1 interaction term ␥40 is the interaction between trial_type and set size for cases when trial_type ⫽ 2 (congruent). The slope ␥11 is the coefficient of the interaction term of Age Group ⫻ Set Size, for cases with age group ⫽ 0. The slope ␥21 is the coefficient of the interaction term of Age Group ⫻ Trial_Type, for cases with age group ⫽ 0 and trial_type ⫽ 2 (i.e., the young congruent group). Rij is the residual for each person. U0j is the unique effect on intercept for each person. U1j is the unique effect on slope for each person. Thus, Var (Rij), or ␴2, is the within-person variance; Var (U0j), ␶02, is the between-persons variance in intercepts; Var (U1j), ␶12, is the between-persons variance in slopes. Dashes indicate that this effect was not included in the model. Coeff. ⫽ coefficient.

younger adults (SE ⫽ 15.79). Incongruent trials did not lead to a significantly larger intercept (the effect was 3.35; SE ⫽ 21.86), but they produced a more elevated slope, and this effect was 15.85 ms/item (SE ⫽ 3.89). There was no reliable Age Group ⫻ Set Size interaction (interaction term ⫽ ⫺10.17, SE ⫽ 15.79). Models 1 to 3 assume that there is no distinction in slope between a subitizing and a counting range. Theory, as well as Figure 1, suggests that a model with a node at set-size 3 or 4 would be a better fit to the data. Thus, in Model 4 (2-segment fixed node model), we introduced linear spline functions. This analysis can be considered as piecewise regression, where two lines are estimated simultaneously, one on each side of a designated node. We chose to be conservative in our definition of the subitizing range and placed the node at set-size 3. Numerosities between 1 and 3 are assumed to be in the subitizing range; numerosities between 4 and 8 are considered to be in the counting range. The equation for Model 4 can then be represented as Yij ⫽ ␥ 00 ⫹ ␥ 10 共sub兲 ⫹ ␥ 20 共count兲 ⫹ ␥ 01 共age group兲

Young/congruent: Yij ⫽ 849.45 ⫹ 65.07 ⫻ sub ⫹ 321.86 ⫻ count; Young/incongruent: Yij ⫽ 859.47 ⫹ 40.53 ⫻ sub ⫹ 353.22 ⫻ count; Old/congruent: Yij ⫽ 1,315.56 ⫹ 123.03 ⫻ sub ⫹ 307.28 ⫻ count;

⫹ ␥ 30 共trial_type兲 ⫹ ␥ 40 共block兲 ⫹ ␥ 50 共Sub ⫻ Trial_Type兲 ⫹ ␥ 60 共Count ⫻ Trial_Type兲 ⫹ ␥ 11 共Age Group ⫻ Sub兲

Old/incongruent: Yij ⫽ 1,325.58 ⫹ 98.48 ⫻ sub ⫹ 338.64 ⫻ count .

⫹ ␥ 21 共Age Group ⫻ Count兲 ⫹ U0j ⫹ U1j 共sub兲 ⫹ U2j 共count兲 ⫹ Rij .

The correlation matrix of parameters showed that although the subitizing slope is significantly correlated with the intercept (0.81), counting speed is not correlated with the intercept (0.20). The correlation between the subitizing and counting slopes is nonsignificant (⫺.18). The estimated average RT scores (only the fixed part) for young and old participants for the two trial types were provided by the following set of equations (computed from Table 2).

(3)

The variables sub and count denote set size in the generic subitizing and counting ranges, respectively.

(4)

The tests for the fixed effects show that all of the main effects, with the exception of trial type, are significant, indicating that the intercept is larger than zero, that older adults are slower than younger adults at set-size 0, and that the slopes for both subitizing and counting are larger than zero. Additionally, the interaction

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between age group and rate of subitizing was significant, F(1, 3593) ⫽ 9.51, MSE ⫽ 353.06; the interaction between age group and rate of counting was not significant, F(1, 3593) ⫽ 0.67, MSE ⫽ 317.91. This result indicates that the RT ⫻ Set Size slope of older adults is larger than that of younger adults in the subitizing range, but not in the counting range. There is no difference between the slope of the two trial types in the subitizing range, F(1, 3593) ⫽ 2.68, MSE ⫽ 223.95, but there is a significant interaction between trial type and slope in the counting range, F(1, 3593) ⫽ 30.32, MSE ⫽ 32.43. This indicates that the Stroop effect is operating in the counting range and not in the subitizing range. When the interaction term between age group and trial type was introduced in the model, the interaction was found to be nonsignificant, F(1, 3593) ⫽ 0.02, MSE ⫽ 303.43, indicating that there is no age difference in susceptibility to the Stroop effect in the present sample. Note that qualitatively identical results were obtained when we set the node at set-size 4 rather than at set-size 3. In Model 4, we assumed that the subitizing span is identical across all our participants (an assumption almost invariably taken in the literature). Inspection of the individual graphs, however, suggested that there is a large amount of interindividual variability that is unaccounted for in the forced spline model. Therefore, it seems reasonable to refine our spline analysis by taking the subitizing range for each single individual into account. To determine individual subitizing spans, we fitted the OLS model RTij ⫽ b00 ⫹ b10 共sub兲 ⫹ b20 共count兲 ⫹ b30 共block兲 ⫹ Rij to the data of each of the individual participants, systematically varying the subitizing range between 1 (i.e., no subitizing) and 5. We then computed the difference between the subitizing and the counting slopes at each of these numerosities. The set size that yielded the largest difference was considered to be the maximum subitizing span for that participant. This procedure was repeated including trial type as a covariate, so that maximum subitizing span for both congruent and incongruent numerosities could be assessed. A repeated measures ANOVA showed that there was no difference between the subitizing span derived from the congruent and incongruent conditions, F(1, 54) ⫽ 0.16, MSE ⫽ 0.23; nor was the Trial Type ⫻ Age interaction significant, F(1, 54) ⫽ 0.58, MSE ⫽ 0.26. Therefore, we decided to derive the individual’s subitizing span from RT averaged over trial type. Two older participants and 1 younger participant showed no particular pattern; these participants’ data were removed from further analyses. There were 4 younger adults and 9 older adults whose data were best fitted by a straight-line regression equation, that is, they had a subitizing span of 1. Three younger adults and 6 older adults showed a subitizing span of 2; 17 younger adults and 11 older adults showed a subitizing span of 3; four younger adults showed a subitizing span of 4, and only 1 younger adult showed a subitizing span of 5. The average subitizing span was 2.83 for younger adults (SD ⫽ 0.97) and 2.07 for older adults (SD ⫽ 0.87); the age difference is statistically significant, t(53) ⫽ 3.06, p ⫽ .003.2 Figure 2 represents the data, grouped by subitizing span (between 1 and 4), for each of the age groups separately, as well as the freely estimated regression line for subitizing and counting overlaid on the data, for illustration purposes.3 As a first analysis based on the individual subitizing spans, Equation 4 was fitted to the data of only those participants who had a maximum subitizing span of 3 (Model 5: 2-segment select

participants model). The subitizing span of 3 was selected because it had the maximum number of participants (17 young and 11 old participants). The results, depicted in Table 2, show that the unique effect of subitizing slope (parameter ␶12) is not significant, indicating that there is no significant between-persons variability in the rate of subitizing for this select group of subjects. The fixed effects for the model are Young/congruent: Yij ⫽ 832.47 ⫹ 41.02 ⫻ sub ⫹ 356.29 ⫻ count; Young/incongruent: Yij ⫽ 877.05 ⫹ 30.15 ⫻ sub ⫹ 382.57 ⫻ count; Old/congruent: Yij ⫽ 1,136.79 ⫹ 48.05 ⫻ sub ⫹ 334.98 ⫻ count; Old/incongruent: Yij ⫽ 1,181.37 ⫹ 37.18 ⫻ sub ⫹ 361.26 ⫻ count.

(5)

All main effects, with the exception of trial type, were significant (all p ⬍ .01). The interaction between age group and rate of subitizing was not significant and neither was the interaction between age group and rate of counting. Consequently, in this sample of adults with the same subitizing range, there was no effect of age on the rates of either subitizing or counting. Although there was no significant effect of trial type on the rate of subitizing, its effect was significant on the rate of counting. Qualitatively identical results were obtained when linear spline functions at set-size 3 were fitted for those subjects whose maximum subitizing range was either 3 or 4. In a second analysis (Model 6: 2-segment individually-varying node model), the spline equation (Equation 3) was fitted for all individuals who had a subitizing range between 1 and 4, setting the spline point at each individual’s maximum subitizing span. (The one participant with a subitizing range of 5 was omitted because it is not clear whether this is a believable value.) This left 28 younger and 27 older adults in the analysis. The fixed effects for this fitted model (Model 6, see Table 2) yield the following set of equations:

2 One reviewer commented that individual and age differences, or both, in subitizing span might be due to differences in speed–accuracy trade-off. If this is the case, a small subitizing span would be indicative of high cautiousness. The reviewer suggested correlating subitizing span with error rates; the expectation is that the correlation would be positive: larger spans are associated with more errors. The correlation was indeed positive, but not significant (r ⫽ .23). 3 As stated above, all the analyses were conducted on RTs conditionalized on participants being correct, that is, when they have by definition been successful in applying the subitizing strategy. We redid the analysis including all RTs. The rationale is that subitizing ranges derived from all RTs, correct and incorrect, may reflect the range at which participants attempted the subitizing strategy, though not necessarily successfully. The average attempted maximum subitizing span was indeed larger: 3.36 for younger adults (SD ⫽ 1.28) and 2.30 for older adults (SD ⫽ 0.95); the age difference is statistically significant, t(53) ⫽ 3.47, p ⫽ .001.

SUBITIZING, COUNTING, AND AGING

245

Figure 2. Reaction time (for correct answers only) as a function of age group, separated by subitizing span. Filled circles denote performance of younger adults; open circles, performance of older adults. Lines are freely estimated regression lines over the subitizing and counting ranges.

Young/congruent: Yij ⫽ 782.83 ⫹ 52.82 ⫻ sub

Estimated Enumeration Process Times

⫹ 333.74 ⫻ count; Young/incongruent: Yij ⫽ 809.06 ⫹ 32.94 ⫻ sub ⫹ 352.88 ⫻ count; Old/congruent: Yij ⫽ 1,074.42 ⫹ 39.60 ⫻ sub ⫹ 302.60 ⫻ count; Old/incongruent: Yij ⫽ 1,100.65 ⫹ 19.72 ⫻ sub ⫹ 321.74 ⫻ count.

(6)

The results show a pattern similar to that of Model 5. In the fixed-effects analysis, all the main effects, with the exception of trial type, were found to be significant. The interaction between age group and subitizing, between age group and counting, and between trial type and subitizing slope were nonsignificant. The interaction between trial type and counting, however, was significant. The pattern of results remained robust even when we considered only those participants who displayed a detectable subitizing span, that is, only those select participants who have a maximum subitizing span of 2, 3, or 4.

The results presented in the previous section were obtained using manual RTs. The conclusions as presented, however, are only valid if there are no age differences in patterns of pure RTs (i.e., input/output and response-mapping processes) to specific numerosities and if there are no numerosity effects in pure RTs that could explain the difference in slope between subitizing and counting. To test for these possible contaminating effects, we replicated our main analyses (i.e., Models 4 to 6), using the estimated time for completing the enumeration process rather than the full latency. Time needed for the enumeration process was estimated for each set size by subtracting the average RT for each digit on the familiarization trials from the average RT for the corresponding set size on the enumeration trials for each participant for each of the four blocks. The assumptions (which may or may not hold) are (a) that the total latency on the familiarization trials is a good indicator of response mapping and motor output times, and (b) that the additive-factors logic applies, that is, that the enumeration processes (i.e., the subitizing or counting process proper) are independent of and additive to the input/output and response-mapping processes (or, in case the enumeration and response-mapping/ output processes overlap, that the overlap is identical across age

BASAK AND VERHAEGHEN

246

Table 2 Parameter Estimates for Fixed Effects and Variance Components for Random Effects for Models 4 – 6 2-segment fixed node model

2-segment select participant model

2-segment individually varying node model

Fixed effect

Coeff.

SE

p

Coeff.

SE

p

Coeff.

SE

p

␥00 ␥10 ␥20 ␥01 ␥30 ␥50 ␥60 ␥11 ␥21

1,325.58 98.48 338.64 ⫺466.11 ⫺10.02 24.54 ⫺31.36 ⫺57.95 14.58

39.53 15.35 13.03 53.10 16.96 15.00 5.69 18.79 17.83

⬍.0001 ⬍.0001 ⬍.0001 ⬍.0001 .6479 .1019 ⬍.0001 .0021 .4136

1,181.37 37.18 361.26 ⫺304.32 ⫺44.58 10.87 ⫺26.28 ⫺7.03 21.31

70.58 22.45 22.19 87.57 26.97 23.84 9.05 24.42 27.88

⬍.0001 .1097 ⬍.0001 .0003 .0441 .6484 .0037 .7733 .4448

1,100.65 19.72 321.74 ⫺291.59 ⫺26.23 19.88 ⫺19.14 13.22 31.14

43.73 16.25 13.61 59.19 16.54 16.15 4.82 16.50 18.88

⬍.0001 .2320 ⬍.0001 ⬍.0001 .1128 .2184 ⬍.0001 .4233 .0991

Random effect

Variance component

SE

Variance component

SE

Variance component

SE

␶02 ␶12 ␶22 ␶01 ␶02 ␶12 ␴2

37,329.00 1,889.25 4,209.39 6,815.20 2,555.83 ⫺502.31 71,590.00

7,654.58 961.76 863.15 2,270.96 1,826.70 661.96 1,687.62

46,128.00 0.00 4,617.82 6,177.94 6,648.81 19.34 85,920.00

15,370.00 — 1,424.57 2,839.10 3,599.23 1,162.47 2,916.33

42,261.00 0.00 4,512.80 5,523.70 7,287.42 676.95 70,478.00

9,267.25 — 934.49 1,273.08 2,458.22 837.64 1,706.34

Note. For description of Models 4 – 6, see text. The intercept ␥00 reflects the average reaction time (RT) intercept at sub ⫽ 0. ␥10 is the effect of subitizing (sub). ␥20 is the effect of counting (count). The slope ␥01 is the average rate of set-size-related RT change for cases with age group ⫽ 0 (i.e., for younger adults). ␥30 is the effect of trial_type. The Level 1 interaction term ␥50 is the interaction between trial_task and sub for cases when trial_type ⫽ 2 (congruent), and ␥60 is the interaction between trial_type and count for cases when trial_type is congruent. The slope ␥11 is the coefficient of the interaction term of Age Group ⫻ Sub, for cases with age group ⫽ 0. The slope ␥21 is the coefficient of the interaction term of Age Group ⫻ Count, for cases with age group ⫽ 0. The slope ␥31 is the coefficient of the interaction term of Age Group ⫻ Trial_Type, for cases with age group ⫽ 0 and trial_type ⫽ 2 (i.e., the young congruent group). Rij is the residual for each person. U0j is the unique effect on intercept for each person. U1j is the unique effect on subitizing slope for each person, and U2j is the unique effect on counting slope for each person. Thus, Var (Rij), or ␴2, is the within-person variance; Var (U0j), or ␶02, is the between-persons variance in intercepts; Var (U1j), or ␶12, is the between-persons variance in subitizing slopes; Var (U2j), or ␶22, is the between-persons variance in counting slopes. Dashes indicate that SE cannot be computed because the variance component equals zero. Coeff. ⫽ coefficient.

groups within the subitizing and the counting ranges). Figure 3 presents the data, and Table 3 presents the analyses. The data and the analyses show that a 2-segment random coefficients model with a fixed node at set-size 3 (Model 7: 2-segment

Figure 3. Estimated enumeration process times (i.e., reaction time in the enumeration blocks for correct answers minus reaction time for identifying digits in the familiarization blocks) as a function of age group and condition.

fixed node model) would be a better fit to the data than a 1-segment model akin to Model 3. Analyses of the reduced model (i.e., Model 7) are reported because the three-way interaction between age group, trial type, and set size was found to be nonsignificant in all the models under consideration. The correlation matrix of parameters indicates that the subitizing and counting slopes are not correlated (r ⫽ .01). Similar to what was found in Model 4, the subitizing slopes are correlated with the intercepts (r ⫽ .68), whereas the counting slopes are not significantly correlated with the intercepts (r ⫽ .04). The main effects of intercepts and counting slopes are significant, whereas the main effects of age group, subitizing slope, and trial type are not significant, indicating that the intercept is larger than 0, that the slope of subitizing is not larger than 0, that the slope of counting is larger than 0, that younger and older adults do not reliably differ in the estimated times for completing the enumeration process at setsize 0, and that congruent and incongruent trials yield identical estimated enumeration process times at set-size 0. The tests for fixed effects for the interaction terms yielded results identical to those of Model 4, with the exception of the interaction between age group and rate of subitizing, which was found to be nonsignificant, F(1, 3593) ⫽ 0.00, MSE ⫽ 961.62, indicating that younger and

SUBITIZING, COUNTING, AND AGING

247

Table 3 Parameter Estimates for Fixed Effects and Variance Components for Random Effects for Models 7–9, Using Estimated Time for Completing the Enumeration Process Rather Than Full Latency 2-segment fixed node model

2-segment select participants model

2-segment individually varying node model

Fixed effect

Coeff.

SE

p

Coeff.

SE

p

Coeff.

SE

p

␥00 ␥10 ␥20 ␥01 ␥30 ␥50 ␥60 ␥11 ␥21

205.61 17.31 361.00 ⫺13.33 ⫺10.02 24.54 ⫺31.36 1.10 ⫺1.61

25.36 16.93 13.53 31.01 18.55 16.40 6.23 20.77 18.47

⬍.0001 .3107 ⬍.0001 .6674 .5889 .1347 ⬍.0001 .9578 .9306

152.11 ⫺13.41 380.70 24.94 ⫺25.86 16.64 ⫺27.22 26.82 ⫺14.16

26.63 20.59 28.09 26.63 24.21 21.41 8.13 22.17 35.03

⬍.0001 .5207 ⬍.0001 .3492 .2855 .4371 .0008 .2264 .6862

67.94 ⫺7.88 326.13 60.18 25.96 ⫺15.52 20.23 32.37 12.99

36.43 15.68 16.78 47.61 17.82 15.55 5.39 18.09 23.02

.0686 .6186 ⬍.0001 .2063 .1454 .3186 .0002 .0736 .5725

Random effect

Variance component

SE

Variance component

SE

Variance component

SE

␶0 ␶12 ␶22 ␶01 ␶02 ␶12 ␴2

9,102.62 2,390.81 4,458.69 3,171.31 263.40 28.52 85,627.00

2,613.03 1,174.42 926.34 1,479.19 1,109.61 743.94 2,018.53

509.26 0 7,279.32 ⫺1,438.89 1,103.88 ⫺3,111.00 66,757.00

1,352.48 — 2,658.22 1,190.27 1,560.91 1,039.03 2,307.48

23,133.00 745.13 5,909.46 4,415.88 3,677.35 966.11 71,444.00

5,510.99 1,150.41 1,299.12 2,603.69 1,911.20 755.17 1,856.96

2

Note. For description of Models 7–9, see text. The coefficients (coeff.) and the variance components imply the same as those in Table 2, except that the response variable in these models is the estimated time for completing the enumeration process. Dashes indicate that SE cannot be computed because the variance component equals zero.

older adults subitize at identical rates. The pattern of results did not change when we considered set-size 4 as a node for linear splines. Subitizing span was determined for each individual by using the same procedure as used for full latencies. As was found for Model 5, no significant difference emerged, in either age group, between subitizing spans for congruent and incongruent trails. Therefore, subitizing span was determined for each participant by using the combined congruent/incongruent data. Two younger and 8 older adults had a maximum subitizing span of 1; 2 younger and 2 older adults had a subitizing span of 2; 17 younger and 12 older adults had a subitizing span of 3; 4 younger and 3 older adults had a subitizing span of 4; 2 younger adults and 1 older adult had a subitizing span of 5; 3 younger and 5 older adults displayed no regular pattern. The average maximum subitizing span of younger adults is 3.07 (SD ⫽ 0.92) and that of older adults is 2.46 (SD ⫽ 1.22); the age difference is statistically significant, t(43) ⫽ 2.02, p ⬍ .05. When Model 8 (2-segment select participants model) and Model 9 (2-segment individually varying node model for subitizing spans between 1 and 4) were considered, the pattern of results for fixed effects was found to be identical to those of Models 5 and 6, respectively. Although there is a noticeable change in the unique effect of subitizing slope (␶12) as we move from Model 7 to Model 9, ␶12 is significant in Model 7 (2,390.81; p ⫽ .02) but not in Model 9 (745.13; p ⫽ .26), where the individual maximum subitizing spans were taken into consideration to set the individual nodes for the splines. This indicates that once individual differences in subitizing span have been taken into account, individual differences in subitizing rate disappear. The equations obtained from Model 9 are as follows:

Young/congruent: Yij ⫽ 154.08 ⫹ 8.97 ⫻ sub ⫹ 359.35 ⫻ count; Young/incongruent: Yij ⫽ 128.12 ⫹ 24.49 ⫻ sub ⫹ 339.12 ⫻ count; Old/congruent: Yij ⫽ 93.90 ⫺ 23.40 ⫻ sub ⫹ 346.36 ⫻ count; Old/incongruent: Yij ⫽ 67.94 ⫺ 7.88 ⫻ sub ⫹ 326.13 ⫻ count.

(7)

Discussion Our study was motivated by a peculiar effect in the cognitive aging literature, namely that subitizing shows a larger age-related deficit than counting. We replicated this finding (Model 4: 2-segment fixed node model). When a generic subitizing range was used, fixed at a set size of 3, subitizing slopes of older adults were found to be about 1.5 times larger than those of younger adults, but there were no age differences in the counting slope. The slope relating RT to set size in the generic subitizing range is shallow, varying from 41 (incongruent items) to 65 (congruent items) ms/item in younger adults, and 98 (incongruent items) to 123 (congruent items) ms/item in older adults. For elements beyond the generic subitizing range, rates increase, with an average rate of set-size-related RT change varying from 313 (incongruent items) to 353 (congruent items) ms/item in younger adults, and 307

248

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(incongruent items) to 339 (congruent items) ms/item in older adults. For the young adults, our results—namely of a shallow slope of about 50 ms per item in the subitizing range and a steeper slope of about 350 ms per item—fall within the range of values found in previous enumeration studies (Trick & Pylyshyn, 1994). Qualitatively similar results were obtained when the subitizing range was fixed at 4. The subitizing and counting processes were found to be largely independent, as evidenced by the lack of significant betweenpersons correlation between the subitizing and counting rates in Models 4 and 7. Subitizing rate correlated with the intercept, which can be taken as an index of the relatively low-level perceptual/attentive, response mapping and motor processes in the task that are distinct from the subitizing and counting processes per se. The correlation suggests that subitizing may be an automatic and low-level process, perhaps governed by the same cognitive structures as the other perceptual/attentive processes involved. Additional evidence for the low-level nature of subitizing comes from the finding that there are no reliable individual differences in the rate of this process when individual subitizing spans are considered. Counting was a slow process, compared to subitizing; it did show significant individual differences, even when individual subitizing spans were considered; and its rate did not correlate with either the intercept or the subitizing rate in Models 4 and 7. These three findings suggest that it is a controlled, largely postperceptual process. In the introductory section, we advanced two possible reasons for the counterintuitive result of a larger age deficit in subitizing than in counting. The range hypothesis states that the subitizing range may differ between age groups (more specifically, it may be smaller in older adults). When a generic subitizing range is considered and the range is larger than the actual subitizing range for one of the groups, this will contaminate that group’s subitizing slope estimates with the longer RTs coming from counting processes, thus creating an artifactual difference in subitizing rates between the two age groups. The automaticity hypothesis states that with advancing age, subitizing increasingly becomes a less automatic and more controlled process. Our results are clear. The former hypothesis is correct; the latter is not. With regard to the range hypothesis, we indeed found significant age differences in subitizing span. This was true both for the actual subitizing range (i.e., when we examined RTs conditional on the response being correct) and the attempted subitizing range (i.e., when we examined RTs for both correct and incorrect responses). On average, our younger participants had an actual subitizing range close to 3 (viz., 2.83); our older participants had an actual subitizing range close to 2 (viz., 2.07). This result was further confirmed in an analysis on estimated pure enumeration processing times, where younger participants had a subitizing span of 3.07, and older adults, 2.46. It should be noted that the standard deviation was close to 1 in both age groups, indicating the existence of nonnegligible individual differences in the subitizing span. Future research in the field of subitizing should take such individual differences into account. It is more important to note that we found that the age-related dissociation between subitizing rate and counting rate disappeared when individual differences in subitizing range were taken into account. This was demonstrated in two analyses. In the first, we examined all individuals with an identical subitizing range (viz., 3). No age differences were found in either subitizing or counting

rate in this selected subsample. Second, when the breakpoint between subitizing and counting was set at each individual’s subitizing span, age differences in subitizing and counting rate failed to emerge. (In fact, both subitizing and counting rates were nominally slightly smaller in our older adult sample than in our younger adult sample.) Thus, the observed age-related slowing in subitizing rate is an artifact of not taking individual and group differences in subitizing range into account and of setting generic subitizing ranges at values that are clearly larger than the actual ranges measured here. With regard to the automaticity explanation, we hypothesized that a counting Stroop manipulation would interfere with agerelated differences in the largely verbal process of counting, but not with the automatic and preverbal process of counting. Therefore, if subitizing increasingly becomes a nonautomatic verbal process in older adults, older adults should be more susceptible than younger adults to the Stroop manipulation in the subitizing range. We found the predicted general effect for the Stroop effect: In our final models (Models 5 and 6, and 8 and 9), there was a significant Trial Type (i.e., Stroop manipulation) ⫻ Counting interaction, but no significant Trial Type ⫻ Subitizing interaction. This implies that the interference effects of the Stroop manipulation were limited to the counting range. The age effects, however, did not match the predictions from the automaticity explanation: In none of our final models did the three-way interaction between age group, type of task, and either subitizing or counting become significant. Thus there were no age-related differences in the effects of the Stroop manipulation within either the subitizing rate or the counting rate. Further evidence for the continued automatic nature of subitizing across the adult life span can be found in the existence of a correlation between subitizing rate and the intercept, its independence of counting rate, and the lack of individual differences in subitizing rate. Our results have implications for cognitive aging theory that go beyond excising a curious and erroneous result from the literature. First, a dominant theory in cognitive aging posits that capacity differences between age groups arise from differences in basic processing speed (Salthouse, 1996; Verhaeghen & Salthouse, 1997), although the inverse position has been argued as well (Hasher & Zacks, 1988). In our study, we found clear evidence that group speed differences can arise out of differential capacity limitations. That is, when considered at an absolute level (e.g., in the 1–3- or 1– 4-set-size range), processing speed of older adults was, indeed, on average much lower than that of younger adults (in the 1–3 range, enumeration rates of older adults are twice as slow as those of young adults; see Equation 4), but this speed deficit could be completely explained by individual and group differences in the number of elements that can be processed automatically and immediately. Once this capacity limitation is taken into account, the group of older adults was found to be equally fast as the group of younger adults in both the subitizing and the counting process. Second, some theorists in cognitive psychology (Cowan, 2001; Engle, Kane, & Tuholski, 1999) have claimed that the subitizing range may be an indicator of the size of the individual’s focus of attention (see also Pylyshyn’s, 1989, theory of fingers-ofinstantiation). Age differences in subitizing range may then signal a more narrow focus of attention. This, in turn, may have consequences for the capacity of short-term memory and working memory. Another consequences of the narrowing of the focus of attention may be slowing in higher order processes. That is, even if

SUBITIZING, COUNTING, AND AGING

speed of processing for items in the focus of attention remains constant over age, as our results indicate to be the case, overflow of the focus of attention would occur more often for older adults. This then may be a source for age-related differences in speed of processing in higher order processes: Whereas younger adults are able to process three elements in the focus of attention, older adults can only process two elements before it becomes necessary to transfer elements into the available but not immediately accessible part of working memory. On average, the capacity deficit would then give rise to an age-related slowing factor of 3/2, or 1.5, a value indeed not very different from that typically noted in the literature (Cerella, 1990; Cerella et al., 1980).

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Received March 4, 2002 Revision received August 6, 2002 Accepted October 14, 2002 䡲