A Search-By-Clusters Model of Visual Search: Fits to Data From Younger and Older Adults

Abstract

Objectives.

This study aims to specify the processing operations underlying age-related differences in the speed and accuracy of visual search in a mathematical model.

Method.

Eighteen older and 18 young adults searched for a predesignated target within 24-degree visual arrays containing distractors. Targets were systematically placed in regions that extended 2.5, 5.0, 7.5, and 10 degrees from center. Data were fitted to several versions of a mathematical model in which it was assumed that target search proceeds from the center fixation to peripheral areas in a succession of visual inspections of clusters until the target is located and that clusters can vary in size in response to search difficulty.

Results.

Eccentricity effects on latencies and errors were larger for older adults than for younger adults, especially in the hardest search condition. The best-fitting version of the “search-by-clusters” model accounted for an average of 98.4% and 95.4% of the variance in the young and older adults, respectively. The resulting time, accuracy, and cluster parameters behaved plausibly in each of the 36 data sets.

Conclusions.

A quantitative model that specified how individuals searched for targets in large arrays accurately predicted the search times and accuracies of younger and older adults.

EFFECTIVE performance in many everyday situations depends on visual selective attention, defined as a mixture of visual–cognitive processes that serve to modulate the efficiency of detection, localization, and selection of relevant information in the visual environment. In the laboratory, visual selective attention is often studied using a search task in which individual observers are shown visual arrays containing n elements and are asked to report the presence or absence of a designated target element in each of the arrays. Typically, response latencies and/or errors reliably increase as n increases if targets and distractors are hard to discriminate; the functions are linear with slopes of 20–50 ms per element (for reviews, see Duncan & Humphreys, 1989; Wolfe, 1998). Furthermore, response latencies and/or errors increase only slightly with increases in n if targets and distractors are sufficiently discriminable; the search functions are relatively flat with slopes of 0–10 ms per element. These findings have been interpreted as support for two visual search processes, parallel and serial search (e.g., Egeth, 1966; Palmer, Verghese, & Pavel, 2000; Thornton & Gilden, 2007; Treisman & Gormican, 1988).

In age-comparative studies of visual search performance, search slopes are typically steeper for older adults than for younger adults as n increases when it is relatively hard to discriminate the target from distractors, suggesting an age-related decline in the efficiency or speed of serial search processes or limitations in the use, efficiency, or speed of parallel search processes (e.g., Farkas & Hoyer, 1980; Whiting, Madden, & Babcock, 2007). In these and other studies (e.g., Ho & Scialfa, 2002; Hommel, Li, & Li, 2004; Madden, Pierce, & Allen, 1996; Plude & Doussard-Rossevelt, 1989; Plude & Hoyer, 1986), both younger and older adults produce nearly flat slopes under conditions in which it is easy to discriminate the target from distractors. In addition to the influences of age-related process-general decrements and process-specific deficits in attentional selectivity and feature extraction rates in hard searches, the visual search performance of older adults is affected by an age-related restriction of the useful field of view in the presence of distractors and by an age-related narrowing of the visual-spatial eccentricities (e.g., Burton-Danner, Owsley, & Jackson, 2001; Cerella, 1985; Scialfa, Kline, & Lyman, 1987; Scialfa, Thomas, & Joffe, 1994; R. Sekuler & Ball, 1986; A. B. Sekuler, Bennett, & Mamelak, 2000).

It is important to point out that near-zero search slopes depend on the spatial extent or size of the visual field to be searched as well as on target discriminability (e.g., Carrasco, Giordano, & McElree, 2006; Coeckelbergh, Cornelissen, Brouwer, & Kooijman, 2004; Dosher, Han, & Lu, 2010). If n becomes very large and/or if n elements become distributed over a wide area, an individual's capacity for parallel processing can be exceeded (e.g., Dosher et al., 2010; Humphreys & Muller, 1993; Thornton & Gilden, 2007). For example, Thornton and Gilden showed that visual searches generally involve a parallel limited-capacity process but that sometimes, the demands of the search are such that a serial search process is required.

Age-related differences in visual search performance are known to reflect subject-controlled strategic and top–down factors and stimulus-driven and ocular factors (e.g., Greenwood & Parasuraman, 2004; Madden et al., 2002; Tavassoli, van der Linde, Cormack, & Bovik, 2007). The dynamics of the interplay between subject-controlled and stimulus-driven factors are what we seek to capture in a formal model of visual search. Formal or quantitative descriptions of the effects of aging on recognition memory (e.g., Howard, Bessette-Symons, Zhang, & Hoyer, 2006; Ratcliff, Thapar, & McKoon, 2009), spatial attention (e.g., Greenwood & Parasuraman, 2004; McCalley, Bouwhuis, & Juola, 1995), and decision processes in picture completion, letter discrimination, and brightness discrimination (e.g., Hannon & Hoyer, 1994; Ratcliff, Thapar, & McKoon, 2006) have been advanced in recent years, but it seems to us that available models have little direct bearing on explicating the dynamic interplay between age-sensitive mechanisms and task conditions in the domain of visual search.

In this article, we advance a mathematical description of search performance intended to characterize how younger and older individuals carry out easy and hard searches for a target in large arrays containing distractors. We fit this model to data taken from 18 younger and 18 older adults. The model we propose relies on notions that search operates on contiguous subsets of elements simultaneously (i.e., parallel search) and that search proceeds from one subset or cluster of elements to another in a centrifugal fashion until the target is found. Specifically, the assumptions of the model of “search by clusters” are (a) that several elements are sampled at a time, (b) that search within clusters is parallel, (c) that cluster size depends on the difficulty of distinguishing target from distractors, and (d) that the progression from one cluster to another is serial. In the model, if the cluster size is sufficiently large, parallel search performance will be observed; if the cluster size is small, elevated slopes suggestive of serial search performance will be observed. Note that these outcomes may be only the endpoints of a continuum of search routines spun out by processing variations associated with cluster size. The observed slope may reflect cluster size directly and the target/distractor discrimination only indirectly if the observer can strategically adjust cluster size to accommodate the processing demands of difficult target–distractor discriminations. The available data on aging and visual search do not provide a sufficient basis for advancing specific hypotheses about the extent to which the parameters of cluster size and processing time per cluster in the model will account for age differences in the latencies and accuracy of visual search.

In the present study, we fit the search-by-clusters model to the visual search performance of younger and older adults. Latencies and accuracy were measured in a hard search condition and in an easy search condition under viewing conditions that were otherwise identical. Our focus was on accounting for the effects of target–distractor discriminability on cluster size in the visual search performance of younger and older adults.

METHOD

Participants

The participants were 18 younger adults and 18 older adults. Individuals were excluded from participation if (a) they reported taking any medication that could affect cognitive performance or (b) they had near visual acuity worse than 20/30 with correction. Participants were paid $30. The Human Subjects Committee of the Institutional Review Board at Syracuse University reviewed and approved the procedures used in this research.

Near visual acuity was assessed using a handheld vision screener. Participants completed questions about educational level, health, and current medication usage. The forward and backward digit span subtests from the Wechsler Adult Intelligence Scale-Revised (Wechsler, 1981) and the vocabulary subtest from Shipley (1940) were administered for the purpose of sample description. Table 1 presents a summary of the characteristics of the participants.

Table 1.

Means (and SDs) for Measures of the Characteristics of the Research Participants

Measure	Younger adults	Older adults
Age	19.63 (1.26)	71.05 (6.09)
Education	13.11 (1.33)	13.89 (3.20)
Health	1.47 (0.61)	1.72 (1.02)
Limitations	0.05 (0.23)	1.94 (1.11)
Digit span	17.89 (3.31)	17.16 (3.89)
Vocabulary	29.84 (2.93)	34.61 (3.33)

Notes: Each group contained 18 participants. Education = self-reported number of years of formal education. Health = self-reported using a scale from 1 (excellent) to 5 (poor). Limitations = self-reported number of health-related limitations. Digit Span = combined forward and backward scores on Wechsler Digit Span subtest from the WAIS-R (Wechsler, 1981). Vocabulary = score on the Shipley Vocabulary test (Shipley, 1940).

Stimulus Materials and Procedures

Two visual search tasks that differed in terms of target–distractor similarity were used (see Figure 1). The target stimulus was the same in both search tasks; a white right slash (oriented 45°) with a circle at the bottom. The relation between the target and the distractors was manipulated between tasks by using two different sets of distractors. For the easy search task, the distractors were white left slashes (oriented 315°) with a circle at the bottom. For the hard search task, the distractors were white right slashes (oriented 45°) with a circle at the top. The target was present in the displays on about half of the trials. Displays were shown on a high-resolution 21-inch monitor. The background color was gray, and the luminosity level was 55 cm/m². The display configuration and stimulus materials closely matched those used in Scialfa and Joffe (1998).

Figure 1.

Schematic illustration of hexagonal stimulus array showing eccentricity zones (center panel). Sample search elements: Target is right slash with terminal dot; hard distractors are right slashes with leading dot (left panel) and easy distractors are left slashes (right panel).

For each trial, distractors chosen from the appropriate distractor set were placed at random in an 11 by 11 matrix; the remaining cells were empty. The matrix skeleton was not visible. The size of the overall array was 24° by 24°. Placement of the target within the array was experimentally controlled in the following manner: Four concentric hexagonal-shaped rings of progressively larger size were defined for the inner 10 × 10 cells of the 121-cell matrix. The hexagons had radii of 2.5°, 5.0°, 7.5°, and 10.0°. In each block of trials, the target appeared 12 times at fixation and 6 times within each of the four rings. Each of the search tasks consisted of 3 blocks of 32 target-absent trials and at least 36 target-present trials depending on how many errors were made. That is, target-present trials on which the participant missed the target were represented later in the block, so as to obtain response latency (RT) data for statistical analysis from 12 target-present trials at fixation (0°) and from 6 target-present trials at each of the four eccentricities (2.5°, 5.0°, 7.5°, and 10°). Statistical analyses performed on the percent accuracy did not include represented trials. Completion of the search tasks usually took about 15–20 min. Block 1 of each search task was considered practice, and only the data from Blocks 2 and 3 were recorded and analyzed.

A chin rest was used to stabilize the participant's head and to fix the viewing distance (46 cm). A trial began with the presentation of a fixation point consisting of five dots in the pattern of an X. Participants pressed the space bar to begin the search. Upon pressing the space bar, the fixation stimulus was immediately replaced by a display of about 60 elements. Participants pressed the “/” key when they saw the target and pressed the “z” key if they did not see the target anywhere in the display. The display remained on the screen until a response was made. Immediately after each response, participants were given on-screen feedback (“ok” or “error”). Instructions emphasized both speed and accuracy. Response latencies were timed from the point of onset of the search display.

RESULTS

Averaged data from the two age groups are displayed as a function of target area in Figure 2. The smooth curves in this figure are the fits to the model (discussed later). For each age group, search latencies are shown in the upper panel and search accuracies in the lower panel. In each panel, data are given for the easy search condition (filled symbols) and for the hard search condition (open symbols). Latency and accuracy were analyzed separately in a 2 (age: younger, older) × 2 (search difficulty: easy, hard) by five (target region; eccentricities) repeated measures analysis of variance (ANOVA). In the analyses involving the repeated factor, the Mauchly's test of sphericity was significant and the degrees of freedom were corrected accordingly using the Huynh–Feldt method. Unless otherwise noted, the alpha level for the statistical tests was set to p < .05.

Figure 2.

Means of median response latencies RTs (and SEs) by task and eccentricity for younger and older adults (upper panels). Percent accuracies (and SEs) by task and eccentricity (lower panels). Smooth curves are the estimates derived from the search-by-cluster model fitted to the group data for illustrative purposes only.

Analyses of Response Latencies

The data shown in Figure 2 suggest that the effects of size of the target region on latencies of responses were practically negligible for easy search and were substantially larger for hard search. Output from the ANOVA revealed that the two-way interaction between search difficulty and target region was significant, F(3.2, 108.8) = 132.2, p < .001, η² = .80, confirming that eccentricity effects were amplified in the hard search condition. The ANOVA also revealed a two-way interaction between age and search difficulty, F(1, 34) = 53.15, p < .001, η² = .61, and a three-way interaction between age, search difficulty, and target region, F(3.2, 108.8) = 17.69, p < .001, η² = .34, indicating that the eccentricity effects observed in the hard search condition were larger for the older adults. Main effects of age, F(1, 34) = 127.55, p < .001, η² = .79, search difficulty, F(1, 34) = 289.09, p < .001, η² = .90, and target region, F(3.3, 111.9) = 238.53, p < .001, η² = .875, confirmed the standard effects of these factors on response latencies, but the interpretation of these main effects is qualified by the obtained higher order interactions.

Analyses of Response Accuracies

The ANOVA on the accuracy data revealed the same pattern of effects as that reported for latencies with one exception: The three-way interaction between age, search difficulty, and target region, F(3.5, 120.5) = 1.43, was not reliable. Consistent with the latency data, the accuracy data shown in Figure 2 suggest that the effects of size of the target region were specific to hard search. The two-way interaction between search difficulty and target region was significant, F(3.5, 120.5) = 23.55, p < .001, η² = .41, confirming that the diminished accuracy of target search in the outer regions was amplified in the hard search condition. The ANOVA also revealed a two-way interaction between age and search difficulty, F(1, 34) = 6.81, p < .05, η² = .17, indicating that older adults were more error prone in the hard search condition. The main effects of age, F(1, 34) = 8.44, p < .05, η² = .20, search difficulty, F(1, 34) = 53.37, p < .001, η² = .61, and target region, F(3.1, 104.0) = 35.09, p < .001, η² = .508, were significant. Overall, the similar patterns of the results for latencies and accuracies can be taken to suggest that there was little if any trade-off between speed and accuracy.

Model Derivations and Equations

A series of cluster models were developed and fitted simultaneously to the latency and accuracy data from the easy and hard search conditions, always at the level of individuals. Different cluster diameters were allowed, and the effects on model parameters were determined by statistical testing of nested models in which a given parameter was constrained or not across conditions. The basic processing unit was the sample or cluster, specified by a diameter d in units of cells. The models assumed that a search area of A cells would be segmented into N = A/d² clusters and that search proceeds from the center to the periphery. These assumptions were supported (on average) by the eccentricity effects seen in the data of every participant. Represented in the model was the probability that the target will be detected in the n^th sample, and the cumulative probability that the target will be eventually detected in N samples. We use the term “path length” to refer to the number of samples required for target detection. The mean latency for successfully detected targets was tied to the mean number of samples.

In the development of the final version of the model, we explored two sampling schemes in the derivation of latencies, sampling with replacement (memoryless search) and sampling without replacement (memory-assisted search). Consistent with recent empirical findings suggesting that observers are more likely to visit new locations than to revisit previously inspected locations when searching an array (e.g., Kramer et al., 2006), sampling without replacement revealed better fits than did sampling with replacement scheme.

The best-fitting sampling scheme had six parameters $\left\{s,d_e,d_h,T,B_e,B_h\right\}$: A separate d (cluster diameter) for hard and easy search, a separate B (residual time) for hard and easy search, a single s (cluster accuracy), and a single T (cluster duration). That is, d and B were allowed to take different values in the easy search and the hard search conditions.

This model depicts the observer as adjusting the cluster size to reflect the difficulty of the search while holding constant the processing time per cluster and the detection accuracy per cluster. We present the five equations that fully define this final version of the search-by-clusters model here:

Given that the target is present in the n^th of N possible samples, its probability of detection is

(2)

The cumulative probability of detection in the first M of N possible samples (M ≤ N) is

(3)

Summation over the path length n is here replaced by the equivalent closed integral, which allows n to vary continuously for model-fitting purposes. Solving the integral and setting the cutoff M equal to N (i.e., all N clusters are sampled) gives the formula for target-present accuracy (ACC⁺),

(3′)

The expression for mean path length is similarly converted from an integer summation to a closed integral

(4)

Solving the integral and setting the cutoff M equal to N gives

(4′)

from which target-present latencies (RT⁺) are obtained by a linear mapping

(5)

The equations for $AC{C}^{+}$ and $R{T}^{+}$ are conditional on both the number of elements A and the number of clusters N_A. The latter is an intervening variable tied to the independent variable A (the search area) by the relation

(1)

which depends solely on the cluster diameter d.

For a complete presentation of the model derivation, see the supplementary appendix, which is available online.

Model Fits

Fitted to all 36 of the data sets, the search-by-clusters model achieved a median goodness of fit (the percentage of variance accounted in each data set) of 98.4% (range = 94.1%–99.9%) for the 18 young adults and 95.4% (range = 86.7%–98.4%) for the 18 older adults (see Figure 3, upper left panel). These were excellent fits, especially considering that much of the residual variance was due to noise in the traces for accuracy. (When accuracy is high, fluctuations of one or two errors introduce sizeable zigzags in the trace.) For illustration only, the model was also fitted to the group data. These are the smooth curves in Figure 2.

Figure 3.

Upper left panel gives goodness-of-fit statistics. Remaining panels give values of the fitted parameters by age and condition. The bottom panel gives the “area of elements” searched before a correct “no-target” response is made; this measure is inferred from the model parameters.

Values of the fitted parameters are summarized in Figure 3 (upper five panels). Consider first the results from the 18 young observers. The model expresses the s (cluster-accuracy) parameter only indirectly. Its value of .98 predicts target-detection accuracies that fall from 100% at a path length of 1 to 93% at a path length of 100. In other words, participants were perfectly accurate when targets were found at fixation, path length n = 1, and were progressively less accurate as the number of clusters associated with target detection increased. The values of T, processing time per cluster, are shown in the upper right panel of Figure 3. The value for younger adults, 18 ms per cluster, corresponds to the slope reported for young adults under similar search conditions using similar stimuli (i.e., Scialfa et al., 1994, see Figure 3).

For younger adults, there are two cluster diameters, d_e and d_h. The values for these parameters were 5.4 cells and 0.62 cells, respectively. These are cluster diameters; squaring them gives cluster areas of about 30 cells and ½ element, respectively. Thus, in the easy search condition, only a few samples were needed to detect a target anywhere within the 61-element target region. In the hard search condition, the area of ½ cell suggests that several samples were required to assess each element of those arrays. The effects of search condition on cluster size as well as on the B parameter (residual time) were significant well below the 2.5% level (paired t-tests).

For older adults, the T (processing time per cluster) and s (cluster accuracy) parameters showed deficits as expected (see Figure 3, upper-center and upper right panels). Processing time was slowed by a factor of three in comparison with younger observers (53 ms per element in older observers). The value of B (baseline or residual time) parameter was slowed by a factor of about two (968 ms for the easy search and 1,356 ms for the hard search). The s (cluster accuracy) parameter was reduced from its value of 0.98 in the young to 0.90 in the old. Target-detection accuracies declined from 100% at a path length of 1 to 68.5% at a path length of 100. The age-related differences in each of these parameters were significant well below the 2.5% level (independent t-tests). In the hard search task, older observers returned a cluster diameter of 0.69 cells, which, squared, matched the value of ½ cell seen in the young [the age difference was not significant, t(34) = −0.539]. Cluster size in the easy search condition was significantly larger in older observers (diameter 7.1 cells; squared ≈ 50 cells) than in younger observers, t(34) = −2.246, p ≤ .03.

DISCUSSION

A cluster-search model was established on the basis of fits to data from younger and older adults in two search conditions that differed in difficulty. The best-fitting model took the classical form of target-terminated serial search without replacement, except that the search units were clusters of elements (or parts of elements when the cluster size was less than one) rather than individual elements. The model was distinctive in that it treated the accuracy as well as the latency of target detection. A target present in a cluster might not be detected: The probability of detection declined as a function of path length or number clusters inspected. That is, clusters sampled early in the search sequence were inspected with higher accuracy than those sampled later. The decline was negatively accelerated: As search proceeded, the detection rate roughly stabilized at a value somewhat below its initial level.

What does the model say about visual search and about age-related differences in visual search? First and foremost, the success of the model, built around an assumption of “centrifugal search,” confirms the trends in the data. Given an hexagonal array of 121 cells and a central starting location, observers evidently search outward from the center in a centrifugal pattern, leading to well-defined eccentricity effects in both accuracy and latency in the hard search condition. Our data and model are compatible with Wolfe's “guided-search” model of similar effects (Wolfe, 2007; Wolfe, O’Neill, & Bennett, 1998). In Wolfe's model, the order in which display elements will be accessed by an attention-driven search process depends on the strength of the feature codes in a preattentive map. Eccentricity is one factor that modulates feature strength. If the field is featurally homogeneous, eccentricity will be the only modulating factor, and the inspection of elements will be centrifugal.

Second, the notion of cluster search provides a parsimonious account of visual search in younger and older individuals. Search times and accuracies in young and older individuals were predicted with impressive precision as a function of search area on the basis of just two theoretical variables, cluster size and cluster accuracy, coupled with the two descriptive “mapping” variables T and B. The notion of clustering is similar in many ways to the “group-scanning hypothesis” of Treisman (1982). Treisman suggested that gains in search efficiency would be possible if several elements could be assessed simultaneously and, further, that the search slopes would overestimate the actual dwell time per element in those cases. Field segmentation in Treisman's model of feature integration presumably occurs preattentively on the basis of physical clusters of like elements. In our model, clustering is imposed by the logic of the search process, even in cases where the distribution of features across the field is homogenous.

Strong evidence from nested-model tests supported the specific hypothesis that observers adjusted cluster size to match the difficulty of search while at the same time holding cluster accuracy and cluster duration constant. These findings emerged from comparing latencies and accuracies in the easy and hard search conditions, an easy “parallel” search and a slow “serial” search. This depiction of an optimal or ideal “cluster-casting” observer is intuitively plausible and consistent with an ideal observer model in which observers choose search strategies that maximize information gained about the target's location (e.g., Najemnik & Geisler, 2008). If the observer is missing too many targets, s/he may elect to reduce his cluster size to improve accuracy. If an observer is speeding through large numbers of clusters, s/he may elect to increase cluster size to reduce the serial overhead. Implicit in either adjustment is the notion of a preferred accuracy level and cluster-assessment interval, which serves as a feedback criterion for cluster size settings.

Adjustments in cluster size in response to search difficulty can also be explained in terms of a variable gradient attentional focus rather than in terms of a subject-controlled strategy (Greenwood & Parasuraman, 2004). Greenwood and Parasuraman (1999, 2004) postulated that the size and density of the focus of attention is dynamically scaled in response to search demands. Greenwood and Parasuraman's data from older adults (65–74 years of age) supported their model that suggests that scaling is a malleable movable resource of visuospatial attention that serves to heighten processing within regions of visual field where it is called for by task demands.

Regarding the fitted parameters, the cluster-duration value of 18 ms per cluster in younger observers tells us that the visual samples identified by the model are not to be taken as fixations because they cycle at about 10 times the fixation rate. Clusters must be segregated for processing by a more nimble mechanism of attentional selection that operates within the clumsier sequence of eye movements (see also Dosher et al., 2010). This value falls squarely within the range of search slopes derived from conventional procedures (e.g., Scialfa et al., 1987) and is consistent with eye-tracking data (e.g., Najemnik & Geisler, 2008; Porter et al., 2010).

The cluster size of 30 elements for the younger observers in the easy search condition indicates that only two samples were necessary to inventory the 61-element region that contained the target and only four to inventory the entire 121-element array. Under the tenets of the model, young observers were thus able to compute the predicate “left-tilted disk-plus-stem” (vs. “right-tilted disk-plus-stem”) simultaneously over a sizeable fraction of the easy arrays—an impressive demonstration of parallel search capacity.

The cluster area of ½ element in the hard search condition suggests that the visual predicate “disk at top of stem” (vs.“disk at bottom of stem”) could not be computed from a single sample; it required two samples. This in turn suggests the nature of a relational predicate—perhaps the disk and the blob had to be identified separately before their spatial relationship could be determined. This is in complete accord with conclusions of Treisman and Gormican (1988) regarding two-line predicates like “joining” or “intersecting” or “parallel,” and the dot and enclosed-region predicate “inside-of”(see also Wolfe, Oliva, Horowitz, Butcher, & Bompas, 2002). But what Treisman and Gormican's model sees as serial search, element-by-element, at a low rate, is seen by the cluster model as something less than serial search, half-element-by-half-element, at a standard rate. What the experimenter counts as a single element may be reckoned by the visual system as two elements plus a relation. Thus, the cluster-model's interpretation of these two search conditions seems cogent and parsimonious—a massively parallel computation in one case and a difficult relational predicate in the other. This composite portrait is based on just three theoretical variables: one cluster accuracy and two cluster sizes.

Turning to the older observers, the process-duration parameters, as well as the accuracy parameter, exposed deficits of a sort typically seen in cognitive aging. The cluster size results were more surprising. In the hard search condition, older observers returned a cluster size very close to the value of ½ element seen in the young. The age invariance suggests that the logic of this relational predicate may well require two samples; a requirement that visual system cannot easily accommodate. Hence, there may be little latitude for variation in the cluster size parameter in hard search.

The same is not the case for the easy search condition. Here, logic makes no demands, and one might expect that the easy size parameter will magnify any difference in visual efficiency in which case older observers would show a reduced size. As it happened, the cluster size in the easy search condition was significantly larger in the old (approximately 50 elements). What are we to make of this counter-intuitive result? There is no doubt that older observers suffered from reduced efficiency seen in lower accuracies and longer durations at the cluster level. Under the tenets of the model, the older observer is willing or forced to accept these performance limits but at the same time is free to adjust cluster size to whatever value can sustain the limits. Evidently, the visuospatial attentional system of healthy older adults is able to meet these lowered standards with a larger cluster size than would allow a younger system to achieve its more elevated standards.

This counter-intuitive conclusion regarding optimized “field size” of older observers illustrates an argument of Cerella (1990): The age deficits responsible for a complex set of observations can be misleading or impossible to assess outside of the framework of a quantitative model. The slightly larger eccentricity effects in both the latencies and the accuracies of older observers in the easy search condition seem at first sight to point to an age-related reduction in the visual-field size. In fact, when the complicated trade-off between processing time, processing accuracy, and cluster size is taken into account, the weight of the evidence suggests that older observers cast a larger net than younger observers.

In sum, the model predicted search times and accuracies with impressive precision as a function of search area on the basis of just two theoretical variables, cluster size and cluster accuracy, coupled with the two latency-mapping variables, T and B. Thus, for relatively large homogeneous arrays, it can be inferred that search proceeds outward from center fixation in a centrifugal pattern whose efficiency is determined by target–distractor similarity. The latency and accuracy data used for modeling replicate previously reported findings of exaggerated eccentricity effects in older adults under difficult search conditions (Burton-Danner et al., 2001; Porter et al., 2010; Scialfa et al., 1987, 1994; R. Sekuler & Ball, 1986; A. B. Sekuler et al., 2000).

One obvious limitation of the search-by-clusters model is that it does not deal with the classic search dimension of set size (the number of elements in our displays was fixed at 60 ± 2) nor of the related factor of element density. Systematic variation of the three variables of target eccentricity, set size, and element density would be needed to shape a more comprehensive model. Beyond these factors lies the greater challenge of distractor heterogeneity in which stimulus variations may alter the gradients of attention across the visual field. Other models (e.g., Itti, 2006) or perhaps an extension of the search-by-clusters model is required to characterize visual search in arrays in which performance is influenced by variations in the salience and conspicuity of distractors. The model findings reported here are intended to stimulate further formal descriptions of the effects of aging on the mechanisms of visual search and selective attention.

FUNDING

This research was supported by research grant AG11451 from the National Institute on Aging to W. J. Hoyer.

SUPPLEMENTARY MATERIAL

Supplementary appendix can be found at: http://psychsocgerontology.oxfordjournals.org/

CONFLICT OF INTERESTS

This article was written in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. The authors declared that they had no conflicts of interest with respect to their authorship or the publication of this article.