Psychonomic Bulletin & Review 2010, 17 (5), 756-762 doi:10.3758/PBR.17.5.756
Making strides in modeling individual differences: Reply to Leite, Ratcliff, and White (2007) JOEL MYERSON AND SANDRA HALE Washington University, St. Louis, Missouri AND
JING CHEN Grand Valley State University, Allendale, Michigan Leite, Ratcliff, and White (2007) claimed that the diffusion model (Ratcliff, 1978) could simulate the molar patterns in response times (RTs) from the multiple tasks observed by Chen, Hale, and Myerson (2007). We present our own simulations to clarify the underlying mechanisms and show that, as is predicted by the difference engine model (Myerson, Hale, Zheng, Jenkins, & Widaman, 2003), correlations across tasks are the key to the molar patterns in individual RTs. Although the diffusion model and other sequential-sampling models may be able to accommodate patterns of RTs across tasks like those studied by Chen et al., the difference engine is the only current model that actually predicts them.
The difference engine (Myerson, Hale, Zheng, Jenkins, & Widaman, 2003) is a model of individual differences in processing speed that was proposed to explain the robust, large-scale regularities observed in response time (RT) data from multiple cognitive tasks. According to the difference engine model, individuals’ processing steps on any given task are correlated such that for some individuals, all of the steps tend to be relatively brief, whereas for other individuals, all of the steps tend to take longer than average. The difference engine model posits that these individual differences hold across tasks as well as within tasks; those individuals whose processing steps all tend to be brief on one particular task will tend to have brief processing steps on other tasks as well. These simple postulates about the durations of latent processing steps yield highly constrained predictions about the patterns of RTs from multiple tasks. Let RTij represent the mean RT for the ith participant on the jth task, and let Mj represent the group mean RT on that task. The difference engine predicts that the between-subjects standard deviation (SD) of performance on task j will be SDj m(Mj tr ),
(1)
where m and tr are free parameters whose values are constant across all tasks and individuals.1 This invariance represents a large degree of predicted constraint. The parameter tr may be interpreted as the duration of the nondecisional component of RTs (Luce, 1986; Ratcliff,
1979). Thus, Equation 1 states that SDj is proportional to the average duration of the decisional component (i.e., Mj tr ). Equation 1 provides a group-level prediction of how the dispersion of individual mean RTs varies across tasks. The difference engine also yields highly constrained individual-level predictions of how individuals’ mean RTs vary across tasks. Because the duration of latent processing steps is correlated not just within tasks, but also across tasks, RTij can be predicted on the basis of a measure of individual processing speed, zi (the mean z score for the ith individual across all of the tasks): RTij (1.0 zi m) Mj zi mtr .
Because the values of the parameters m and tr are constant across tasks and individuals, Equations 1 and 2 not only provide constraint at the group and individual levels of performance, they also provide the link between the two levels. Using data from over 100 young adults who varied widely in ability and who performed a diverse set of verbal and nonverbal speeded cognitive tasks, Chen, Hale, and Myerson (2007) conducted a priori tests of these predictions. They found that Equation 1 accurately described the relation between group (between-subjects) SD and mean RT, with no significant difference between the regression lines for verbal and nonverbal tasks (both r 2 .95). They also found that Equation 2, with no free parameters ( just the constants m and tr estimated from fitting Equation 1), accurately predicted the mean RTs of subgroups of the fastest and slowest quartiles of the participants (both r 2s .96). Equation 2 also did a good job predicting individual performances: Not only were individuals’ RTs linear functions of group mean RT (median r 2 .81), but the slopes of the individual lines were strongly correlated with individual mean z score (zi ), as was predicted by Equation 2. Leite, Ratcliff, and White (2007) responded to Chen et al. (2007) by arguing that the diffusion model (Ratcliff, 1978), as well as other sequential-sampling models, can account for similar results. To make this point, they reported simulations showing that the diffusion model can produce a strong correlation between SD and mean RT, as was predicted by Equation 1, and a fanning out of the RTs of hypothetical individuals, as was predicted by Equation 2, as well as the increase in the difference between fast- and slow-group mean RTs as task difficulty increases, which is also predicted by Equation 2. They concluded that such patterns do not provide sufficient support for the difference engine model over alternative models. In fact, they claimed that the positive correlation between SD and mean RT, on the one hand, and the fanning out of the RTs of individual participants, on the other hand, are
J. Myerson,
[email protected]
© 2010 The Psychonomic Society, Inc.
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NOTES AND COMMENT just two ways of describing the same data, not separate predictions of the difference engine model. Thus, according to Leite et al., the relation between SD and mean RT and the fanning out of individual RTs represent a single phenomenon, and data sets that have the first property will also show the other property. We will demonstrate, however, that the linear relation between SD and mean RT and the fanning out of the RTs of individual participants are two separate phenomena. Using diffusion model simulations similar to those of Leite et al. (2007), we will show that one can occur without the other, and we will argue that this dissociation is consistent with Myerson et al.’s (2003) derivations of Equations 1 and 2, each of which describes one of these two separate phenomena. Finally, we will use our simulation results to contrast Leite et al.’s qualitative account of these phenomena with the precise, parameter-free predictions provided by the difference engine model.
Diffusion Model Simulations We replicated the Leite et al. (2007) simulations using the same values for the parameters of the diffusion model and the same estimates of their variability, but because data from three tasks seemed insufficient for assessing linearity or estimating regression parameters, we increased the number of simulated tasks to five by adding additional drift rates while holding all other aspects of the simulation constant. In our first set of simulations, we varied mean drift rate across tasks, holding the z score for each hypothetical participant’s drift rates constant so that individual rankings based on drift rate parameters were the same from task to task. As can be seen in Figure 1, the RTs of our hypothetical participants fanned out with increases in task difficulty (panel A), and the group SD and mean RT were strongly correlated (panel C), just as in the Leite et al. simulations. Moreover, the relation between SD and RT was well described by Equation 1, providing estimates
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Figure 1. Results of the first (panels A and C) and second (panels B and D) sets of diffusion model simulations. Panels A and B show the trajectories of the response times (RTs) of hypothetical individuals across five tasks of increasing difficulty, as indexed by decreases in the group mean drift rates (0.55, 0.45, 0.35, 0.25, and 0.15). Panels C and D show the group SD plotted as a function of the group mean RT.
MYERSON, HALE, AND CHEN
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of the parameters of this equation (i.e., m 0.461 and tr 0.468) that are useful in further analyses of this data. Additional results of this first set of simulations are presented in panels A and B of Figure 2. Panel A of this figure reveals that, as was reported by Leite et al. (2007), the fanning out of individuals’ RTs was associated with an increase in the difference between the mean RTs of fast and slow subgroups formed by a median split. Note that the lines running through the data points from the fast and slow subgroups are not regression lines. Rather, they
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Figure 2. Further results of the first (panels A and C) and second (panels B and D) sets of diffusion model simulations. Panels A and B show the mean response times (RTs) of fast (upright triangles) and slow (inverted triangles) subgroups on the five tasks plotted as a function of the corresponding group mean RTs. The dashed line represents equality ( y x), and in panel A, the solid lines represent the parameter-free predictions of Equation 2. Panels C and D show the slopes of the regressions of the RTs of individual hypothetical participants on the group mean RTs, plotted as a function of their mean z scores. In panel C, the solid line represents the parameterfree prediction line based on Equation 2.
NOTES AND COMMENT regression line. Rather, it is the line that is predicted by Equation 2, according to which the slope of the regression of an individual’s RTs on the corresponding group mean RTs is equal to 1.0 mzi . Again, this prediction uses no free parameters, just an individual’s mean z score and the slope (m 0.421) of the regression of group SD on group mean RT (Equation 1). For our second set of diffusion model simulations—the results of which are presented in panels B and D of both Figure 1 and Figure 2—we changed the procedure so as to disrupt the extremely high correlations between the RTs on the different tasks (median r .96) that resulted from the Leite et al. (2007) procedure: Instead of holding the boundary separation and the z score for each hypothetical individual’s drift rates constant from task to task, we allowed them to vary randomly while keeping the mean drift rate and variability for each task the same as that in the previous simulations. As can be seen in panel B of Figure 1, the RTs of the hypothetical participants again became more disperse with increases in task difficulty, but their RTs did not fan out as they did in the previous simulations. Instead, the trajectories of hypothetical participants’ RTs crisscross randomly. Nevertheless, as can be seen in panel D of Figure 1, the slope and intercept parameters of the linear relation between group SD and mean RT were virtually unchanged from the first set of simulations. Panels A and B of Figure 2 show that the differences between the RTs of the fast and slow subgroups were greatly decreased from the first to the second set of simulations, despite the fact that the relation between SD and mean RT was identical in the two sets of simulations. Comparing the bottom panels of this figure shows that, on the one hand, a hypothetical participant’s mean z score strongly predicted the slope of the regression of that participant’s RTs on the group mean RT in the first set of simulations. On the other hand, the hypothetical participants’ mean z scores were only weakly related to their regression slopes in the second set of simulations. Taken together, the two sets of simulations demonstrate that, contrary to Leite et al. (2007), the increase in the group SD as task difficulty increases and the fanning out of the RTs of individual participants are not just different ways of measuring the same phenomenon. Rather, they are two clearly dissociable phenomena; hence, their cooccurrence in the Chen et al. (2007) data and elsewhere (e.g., Hale & Jansen, 1994; Zheng, Myerson, & Hale, 2000) is of some theoretical interest. These simulations also demonstrate that even though the diffusion model, with typical values for the model’s parameters, can generate results similar to those reported by Chen et al. (2007), the same parameter values can also generate results quite dissimilar to those reported by Chen et al. Which of these outcomes will occur depends on what assumptions are made about the relation between performances on different tasks. In the present simulations, only assumptions that were similar to those of the difference engine (i.e., stable rankings of individual parameter values, leading to performance that was highly consistent from task to task) led to results similar to those reported by Chen et al., whereas other assumptions (i.e., individual
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parameter values that varied from task to task, leading to inconsistent performance) did not. Of course, the diffusion model is a model of performance on individual tasks, and as such, it makes no assumptions or predictions about what it is that changes (or does not) from task to task, although empirical research on this issue has now begun (for a brief review, see Ratcliff, Thapar, & McKoon, 2006). Thus, the diffusion model is relatively unconstrained at this more molar level and, in principle, is as compatible with results like those of the second set of simulations as it is with results like those of the first set. In contrast, the difference engine model makes precise, highly constrained predictions about how individual performance varies across tasks that differ in difficulty. The present results show that when the assumptions of the difference engine model are met, and (importantly) only when they are met, the model’s predictions can be extremely accurate. What is remarkable is that the accuracy of these predictions, seen here in simulations in which the sources of variability are known, is not greatly diminished in studies with real participants (e.g., Chen et al., 2007; Hale & Jansen, 1994; Zheng et al., 2000), in which the sources of variability are unknown and are, in fact, the object of study. Dissociating Fan Out and Dispersion The difference engine model provides an account of the different mechanisms underlying the increase in group SD as task difficulty increases and the fanning out of individual participants’ RTs. This account explains when these phenomena will co-occur, when they will not, and why. Of immediate relevance, the difference engine account explains why changing the rankings of the drift rates of hypothetical participants from task to task in the second set of simulations had virtually no effect on the relation between group SD and mean RT, whereas it greatly decreased the fan out of the RTs of hypothetical participants and the differences between the RTs of the fast and slow subgroups from those observed in the first set of simulations, in which the rankings were held constant. More specifically, the explanation may be found in the derivations of Equations 1 and 2 provided by Myerson et al. (2003). With respect to the fact that changing the rankings of the drift rates of hypothetical participants from task to task did not affect the relation between SD and RT described by Equation 1, note that this manipulation decreases the correlation of processing step durations across tasks, but it does not affect the within-task correlations. Correlations of processing step durations within a task are inherent in the diffusion model, if (following Chen et al., 2007) one defines the duration of a step as the time spent accomplishing another X% of the computation for that task. By this definition, individuals with faster average diffusion rates for a given task have briefer processing steps throughout the task than do individuals with slower diffusion rates. Changing which individuals have faster and slower diffusion rates from task to task, as in the second set of simulations, has no effect on these within-task correlations.
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According to Myerson et al.’s (2003) derivation of the relation between group SD and mean RT, which is based on the formula for the variance of the sum of multiple correlated measures, the within-task correlations are what determines the linearity of this relation; the between-tasks correlations are irrelevant. For example, the same formula, when applied to the variance of total test scores, reveals that the variance for a given test depends on the number of items and their variances and intercorrelations and not on correlations with scores on other tests, regardless of whether such intertest correlations exist or not. In this example, the number of items is analogous to the number of processing steps in the difference engine model, and the scores on individual items are analogous to the durations of individual processing steps. Thus, manipulations that do not affect the within-task correlations or variances of processing steps should not affect the parameters of Equation 1, and the fact that the slope and intercept of Equation 1 were virtually unchanged from the first set of simulations to the second set is consistent with the difference engine model. The RTs and slopes predicted by Equation 2, in contrast, do depend on the assumption that processing step durations are correlated across tasks (Chen et al., 2007; Myerson et al., 2003). Accordingly, this equation accurately predicted the data from the first set of simulations but not the data from the second set. Taken together, the results of the two sets of simulations show the dissociation of dispersion, as measured by the group SD, and fan out, as exemplified by the trajectories of individuals’ RTs across tasks, that is expected on the basis of the theoretical mechanisms postulated by the difference engine model. Two Sports Analogies To illustrate how the observed relation between group SD and mean RT follows from having individuals who are either consistently fast or consistently slow, Leite et al. (2007) offered a sports analogy based on a 10-K race. Although we differ in our take on this analogy, it is useful for understanding the nature of individual differences in speeded cognition and the theoretical implications of parameter constraints. Leite et al. noted that the linear increase in group SD emerges naturally from the same mathematical process that describes the increase in the difference between the elapsed running times of fast and slow runners as they run the first kilometer of the race, then the first 5 K, and finally the whole 10-K course. We agree completely. What we would point out, however, is that this analogy focuses on the increasing size of individual differences within a single race or task. Applied to multiple races (e.g., separate 5-K and 10-K races) or multiple tasks, this analogy only captures the process described by the difference engine model; it does not capture the process underlying individual differences in Leite et al.’s diffusion model simulations. Moreover, in Leite et al.’s (2007) simulations, and therefore in our systematic replications of their simulations as well, what was varied across tasks was not the distances to the boundaries corresponding to the decision criteria,
although these distances would appear to correspond to the race distances in their sports analogy. Instead, what was varied across tasks was the average drift rate.2 Perhaps more importantly, what Leite et al. overlooked is that the elapsed times for successive portions of the race in the example bear little obvious relation to the kind of data fit by Chen et al. (2007). Such data are better exemplified by total elapsed times for different events, from cross-country races to track events like sprints and long-distance races. There is general agreement that although sprint performance depends primarily on pure speed, long-distance races are more dependent on stamina; moreover, crosscountry running requires a different stride length, leg action, and foot plant from track events of equivalent distance. Similarly, there is general agreement that the speeded tasks studied by Chen et al. (2007), including both verbal tasks (category judgment, lexical decision, and rhyme judgment) and visuospatial tasks (shape comparison, visual search, and abstract matching), involve quite different cognitive processes. Just as physiological differences (e.g., proportions of fast- and slow-twitch muscle fibers) are associated with better performance on different racing events, different abilities (e.g., verbal vs. visuospatial) and skills (e.g., semantic vs. phonological encoding) are associated with better performance on the different types of tasks studied by Chen et al. From this perspective, one would not have expected the same individuals to be consistently the fastest or the slowest across these cognitive tasks, just as the best sprinters are rarely very good distance runners. Thus, it takes a considerable theoretical leap to treat the differences among diverse tasks as if they were like the successive portions of a long race. Myerson et al. (2003) specifically made that leap in formulating the difference engine model. The model assumes that different tasks and conditions differ only in the numbers of processing steps and that individuals differ only in the durations of these steps. It is obvious, of course, that these differences are not the only ones that exist. The success of our modeling enterprise so far, however, suggests that they may be the key determinants of the size of individual differences in speeded cognition (Chen et al., 2007). It remains unclear whether Ratcliff and his colleagues (Leite et al., 2007; Ratcliff & McKoon, 2008) are ready to make the same theoretical leap or whether they remain agnostic on the issue of the nature of individual differences, despite recent reports of cross-task correlations in the estimated values of individuals’ diffusion model parameters (Ratcliff et al., 2006). Theoretical Models and Model Selection In comparing the Chen et al. (2007) and Leite et al. (2007) efforts, we would note that whereas the predictions of the difference engine model take the form of equations describing precise mathematical relations, the results of the diffusion model simulations are described by Leite et al. in relatively vague terms. For example, contrast Equation 2, from the difference engine model (Myerson et al., 2003), with what Leite et al. termed fan out. Cer-
NOTES AND COMMENT tainly, Equation 2 predicts the fanning out of individual participants’ RTs, but its predictions are not only quantitative rather than qualitative, they are also parameter free. Moreover, we would stress the fact that simulating results is not the same as being able to predict them. Whereas the difference engine model was able to predict Chen et al.’s findings, Leite et al. attempted to show that, under certain circumstances already specified by Chen et al., the diffusion model could accommodate the same results. Such a demonstration, even when somewhat successful (e.g., our first set of simulations), is relatively empty, because other simulations of the diffusion model lead to different results (e.g., our second set of simulations). For a theoretical model, the ability to predict multiple outcomes can be a source of strength, at least when the model specifies the circumstances under which different outcomes will occur and why. In the absence of such theoretically motivated constraints, a model may be hard to assess, precisely because it can accommodate a wide range of outcomes, as appears to be the case with the diffusion model here. In contrast, the difference engine model represents a theory that stands or falls on the accuracy of its highly constrained quantitative predictions. Importantly, these predictions focus on aspects of individual differences that have been largely ignored by other approaches, such as the absolute size of individual differences in speeded cognition and the systematic increase in these differences with increases in task difficulty. As was reported above, the difference engine model predicted the results of diffusion model simulations extremely accurately only when the assumptions of the difference engine regarding correlations among processing step durations were met. We believe that the fact that the difference engine model predicts poorly when its assumptions are not met is an extremely positive finding. A model that can simulate too broad a range of outcomes raises questions as to whether it predicts these outcomes or merely accommodates them. This philosophy underlies recent developments in model selection (Zucchini, 2000), including minimum description length (Grünwald, 2000) and Bayesian approaches (Myung & Pitt, 1997; Wasserman, 2000). Although the simulation results presented here demonstrate that the difference engine is clearly restricted in terms of what it can predict, the range of predictable outcomes fortunately appears to correspond reasonably closely to what has actually been observed to date. That is, although the difference engine model predicts individual performance only when there are reasonably strong intertask correlations, such correlations tend to be the rule (Carroll, 1993; Faust, Balota, Spieler, & Ferraro, 1999), and in such cases, the predictions of the difference engine model appear to be quite accurate (Chen et al., 2007; Myerson et al., 2003). Note that neither the results reported by Chen et al. nor those presented here provide evidence against the diffusion model or other sequential-sampling models of individual RTs. At the molar level, however, it is only the difference engine model that provides a precise, quantitative account of how a general speed factor determines the large-scale regularities observed in individual differences in RTs from multiple tasks.
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Conclusion Our diffusion model simulations reveal that the dispersion of individual mean RTs on individual tasks (as indexed by group SDs) and the diversity among individuals with respect to their general processing speed (as indexed by the slopes of the trajectories of individual mean RTs across tasks) are separate, dissociable phenomena, contrary to what was claimed by Leite et al. (2007). Nevertheless, according to the difference engine model, these separate phenomena have a common cause. The difference engine model posits that because individuals tend to be generally either fast or slow in performing many different kinds of cognitive operations, the durations of their processing steps will be correlated within tasks even though those tasks are rarely process pure (Jacoby, 1991), and they will be correlated across tasks as well, even when such tasks are designed to require use of different processes. These correlations, within each task and across different tasks, give rise to the orderly effects of task difficulty on the dispersion and diversity of speeded performances. The fact that separate phenomena have a common cause, however, does not mean that they are two measures of the same thing. To continue the use of sports analogies, running can be hard on the knees but good for the heart, and separate mechanisms clearly underlie these different effects. Similarly, the present results show that separate mechanisms give rise to the systematic increases in both the dispersion of RTs and the size of stable individual differences as task difficulty increases. According to the difference engine model, these separate mechanisms are captured by a set of interrelated equations that, as Chen et al. (2007) showed, are able to predict individual, subgroup, and group behavior with a degree of quantitative precision that is rare in psychology. AUTHOR NOTE We thank Jeff Rouder for providing the code for the diffusion model simulations and also for his helpful comments and suggestions. Correspondence concerning this article should be addressed to J. Myerson, Department of Psychology, Washington University, St. Louis, MO 63130 (e-mail:
[email protected]). REFERENCES Carroll, J. B. (1993). Human cognitive abilities: A survey of factoranalytic studies. New York: Cambridge University Press. Chen, J., Hale, S., & Myerson, J. (2007). Predicting the size of individual and group differences on speeded cognitive tasks. Psychonomic Bulletin & Review, 14, 534-541. Faust, M. E., Balota, D. A., Spieler, D. H., & Ferraro, F. R. (1999). Individual differences in information-processing rate and amount: Implications for group differences in response latency. Psychological Bulletin, 125, 777-799. Grünwald, P. (2000). Model selection based on minimum description length. Journal of Mathematical Psychology, 44, 133-152. Hale, S., & Jansen, J. (1994). Global processing-time coefficients characterize individual and group differences in cognitive speed. Psychological Science, 5, 384-389. Jacoby, L. L. (1991). A process dissociation framework: Separating automatic from intentional uses of memory. Journal of Memory & Language, 30, 513-541. Leite, F. P., Ratcliff, R., & White, C. N. (2007). Individual differences on speeded cognitive tasks: Comment on Chen, Hale, and Myerson (2007). Psychonomic Bulletin & Review, 14, 1007-1009.
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Luce, R. D. (1986). Response times: Their role in inferring elementary mental organization. New York: Oxford University Press. Myerson, J., Hale, S., Zheng, Y., Jenkins, L., & Widaman, K. W. (2003). The difference engine: A model of diversity in speeded cognition. Psychonomic Bulletin & Review, 10, 262-288. Myung, I. J., & Pitt, M. A. (1997). Applying Occam’s razor in modeling cognition: A Bayesian approach. Psychonomic Bulletin & Review, 4, 79-95. Ratcliff, R. (1978). A theory of memory retrieval. Psychological Review, 85, 59-108. Ratcliff, R. (1979). Group reaction time distributions and an analysis of distribution statistics. Psychological Bulletin, 86, 446-461. Ratcliff, R., & McKoon, G. (2008). The diffusion decision model: Theory and data for two-choice decision tasks. Neural Computation, 20, 873-922. Ratcliff, R., Thapar, A., & McKoon, G. (2006). Aging and individual differences in rapid two-choice decisions. Psychonomic Bulletin & Review, 13, 626-635. Wasserman, L. (2000). Bayesian model selection and model averaging. Journal of Mathematical Psychology, 44, 92-107. Zheng, Y., Myerson, J., & Hale, S. (2000). Age and individual differences in visuospatial processing speed: Testing the magnification hypothesis. Psychonomic Bulletin & Review, 7, 113-120. Zucchini, W. (2000). An introduction to model selection. Journal of Mathematical Psychology, 44, 41-61. NOTES
(2007). The equation in Leite, Ratcliff, and White (2007) contained a different typographical error and was also corrected in an erratum. 2. We have also conducted simulations in which, as was implied by Leite et al.’s (2007) sports analogy, tasks are defined by boundary separation (i.e., more difficult tasks involve greater distance between boundaries). The results of these simulations were very similar to those of the simulations presented here. If a hypothetical individual’s processing speed (i.e., drift rate) is held constant across tasks, results like those of our first set of simulations are obtained: Group SD increases systematically with group mean RT, individual RTs fan out as task difficulty increases, and the difference between fast and slow subgroups’ RTs also increases systematically. Moreover, individual regression slopes (Equation 2) can be predicted with no free parameters, just the individual’s mean z score, averaged across tasks, and the value of m estimated by the slope of the regression of SD on group mean RT. In contrast, when a hypothetical individual’s processing speed varies randomly from task to task, results like those of our second set of simulations are obtained: Group SD still increases systematically with mean RT, but individual RTs do not fan out, and the difference between fast and slow subgroups’ RTs shows little change as task difficulty increases. Moreover, hypothetical individuals’ mean z scores are uncorrelated with the slopes of the regression of their RTs on group mean RT. These results, taken together with those of our simulations in which we used procedures like those of Leite et al. (2007), testify to the generality of our conclusions regarding the critical role of correlations among processing step durations in determining the relation between task difficulty and the size of individual differences in RTs.
1. The equation for the relation between SD and mean RT that originally appeared in Myerson et al. (2003) contained a typographical error that was later corrected in an erratum and in Chen, Hale, and Myerson
(Manuscript received December 19, 2009; revision accepted for publication July 23, 2010.)
