Delay of Gratification as Reputation Management

Abstract

Although delay-of-gratification tasks have long been used as measures of self-control, recent evidence suggests that performance on these tasks is also driven by rational decision processes. The present research examined whether the effects of rational decision processes extend beyond costs and benefits embedded in the task itself to include anticipated consequences for the child’s reputation. Across two studies, 3- and 4-year-olds from China (N = 273) were assigned to a standard delay-of-gratification condition or to a reputation condition in which they were told that their teacher or a peer would find out how long they had waited. Children waited longer in the reputation conditions and longer in the teacher condition than in the peer condition. This is the first evidence that children’s performance on a delay-of-gratification task is sensitive to reputational concerns and to the identity of potential evaluators of their behavior.

Keywords

delay of gratification social cognition open data preregistered

Delay-of-gratification tasks, in which young children are presented with a choice between obtaining a smaller reward immediately or waiting for a larger reward (Shoda, Mischel, & Peake, 1990), have drawn the attention of the research community and the public at large. One reason for the widespread interest is that longer waiting times predict a diverse set of long-term outcomes, including better stress tolerance and higher SAT scores (Shoda et al., 1990). Although delay-of-gratification tasks have traditionally been interpreted as measures of the ability to exercise self-control, recent evidence suggests that this account is an oversimplification (Watts, Duncan, & Quan, 2018) and that it neglects the role played by rational decision-making processes (Kidd, Palmeri, & Aslin, 2013; McGuire & Kable, 2013). In the present research, we sought to determine whether these rational decision-making processes extend to a different form of cost-benefit analysis: potential benefits to one’s reputation. Specifically, we postulated that children’s waiting times are sensitive to a desire to be perceived favorably by other people, which we refer to as the reputation hypothesis. The reputation hypothesis predicts that children will associate longer waiting times with positive social evaluations and will wait longer if they are told that how long they wait will be conveyed to another person.

Recent models of rational decision-making on delay-of-gratification tasks point to the possibility that it may not always be adaptive to wait for a desired outcome and that the extent to which children are likely to wait depends on a range of task-related factors, such as their expectations about the magnitude of the promised reward and when it will be delivered (McGuire & Kable, 2013) as well as the likelihood that it will never be delivered at all (Kidd et al., 2013; Michaelson & Munakata, 2016).

There is evidence that children’s cost-benefit calculations are sensitive to information that relates to their social identity. Using a minimal-groups paradigm, Doebel and Munakata (2018) found that 3- to 5-year-old children waited longer if they were told that members of an in-group had waited a long time but members of an out-group had not. This finding is consistent with evidence that following group norms can have important implications for group acceptance (Watson-Jones, Whitehouse, & Legare, 2016; Wen, Herrmann, & Legare, 2016) as well as evidence of selective imitation of in-group members, which has been documented among children as young as 14 months (Buttelmann, Zmyj, Daum, & Carpenter, 2013). However, it is far less clear whether young children can make nonobvious inferences about how different courses of action might affect their relationships with familiar others and then use these inferences to guide their behavior (see Sperber & Baumard, 2012). Assessing this possibility speaks to the developmental roots of reputation management, a social skill that plays a fundamental role in allowing people to coordinate and cooperate with each other (Engelmann, Herrmann, & Tomasello, 2012, 2018; Engelmann, Over, Herrmann, & Tomasello, 2013; Leimgruber, Shaw, Santos, & Olson, 2012; Piazza, Bering, & Ingram, 2011; Shaw et al., 2014).

We tested the reputation hypothesis in a delay-of-gratification context by telling 3- and 4-year-olds that their teacher or one of their peers would find out how long they waited. Obtaining empirical support for the reputation hypothesis would expand the scope of rational considerations that can account for children’s behavior on delay-of-gratification tasks and would contribute to our understanding of children’s persistence more broadly (see McGuire & Kable, 2013). In the present research, we also sought to gain valuable information about how social motivation functions within real-world relationships by assessing children’s behavior with reference to real people they care about and interact with regularly (i.e., teachers and peers). This approach maintained ecological validity in a controlled experimental context.

Study 1

In a test of the reputation hypothesis, 3- and 4-year-old preschool children in China participated in a sticker version of the delay-of-gratification task (see Imuta, Hayne, & Scarf, 2014; Ma, Chen, Xu, Lee, & Heyman, 2018). Children were given a choice of obtaining one sticker immediately or waiting until the experimenter returned and obtaining two stickers. In one of three conditions, the standard condition, participants completed a standard version of the delay-of-gratification task. There were also two reputation conditions in which the experimenter provided additional information: Children in the peer condition were told that one of their friends would find out how long they had waited for the sticker, and children in the teacher condition were told that their teacher would find out how long they had waited. As is typical in delay-of-gratification tasks, participants were not told whether they should wait or whether anyone else might prefer for them to wait.

We predicted that children would wait the shortest amount of time in the standard condition, an intermediate amount of time in the peer condition, and the longest amount of time in the teacher condition. We made the latter prediction because we expected children to care more about their teacher’s evaluation and would assume that a teacher would value waiting more.

Method

Participants

The final sample consisted of 193 children between the ages of 3 and 4 years, each of whom attended one of two preschools in Hangzhou, China, a city with a population of about 9 million. There were 61 children in the standard condition (age: M = 4.32 years, SD = 0.51, range = 3.61–4.99 years; 31 boys), 70 children in the peer condition (age: M = 4.19 years, SD = 0.49, range = 3.05–4.99 years; 36 boys), and 62 children in the teacher condition (age: M = 4.18 years, SD = 0.52, range = 3.10–4.99 years; 31 boys). Children attended schools that serve middle-class families. For the experimental manipulation that we tested, there was no precedent in the literature to help us determine an appropriate sample size, so we selected a sample size that would allow us to detect relatively small effects. A total of 10 children were excluded: four who decided not to continue after the initial executive-function assessment phase and six who said that they did not want to wait at all in the delay-of-gratification task. The study was approved by the institutional review board of Zhejiang Sci-Tech University, and only children who gave their oral assent to participate and had consent from parents or legal guardians were included.

Procedure

The procedure was conducted by a female experimenter in individual sessions in a quiet room at the child’s school, and each session was recorded by a hidden camera. The entire procedure was conducted in Chinese, and the quotations that appear in this section are translations. Each participant was randomly assigned to one of the three conditions. The standard condition followed the procedure used by Ma et al. (2018), in which children have an opportunity to earn an extra sticker by waiting. The peer and teacher conditions were identical to the standard condition, except that participants were also informed that either a peer or their teacher would be told how long they had waited.

Each session began with a hammer-knocking executive-function task (a revised version of the peg-tapping task; see Diamond & Taylor, 1996), which measures components of executive function, such as inhibitory control, that are thought to play an important role in the ability to delay gratification (Blair, Granger, & Razza, 2005). This executive-function task also served as a pretext to reward children with a sticker of their choice, which helped to set up the delay-of-gratification task. Children were shown a plastic hammer that matched one held by the experimenter and were told, “Whenever I knock the hammer once, I want you to knock it twice; whenever I knock it twice, I want you to knock it once.” The experimenter then presented two practice trials. Eleven children failed the practice trials and were then taught the rule a second time, at which point all of them were able to follow the rule. After the practice trials, 16 scored trials that consisted of either 1 or 2 knocks each were presented in the following order: 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2 (see Ma et al., 2018).

In all conditions, after concluding the executive-function task, the experimenter said, “You can choose one sticker right now, or if you wait until I come back, you can have another sticker, too. Two stickers!” The experimenter asked, “Do you want to wait for another sticker?” A total of six children declined to wait. These children did not continue in the study, and they were excluded from the final sample. All of the other children were told, “Okay, let’s continue. If you want to stop waiting, then ring the bell and I will come back right away, but you will get only this one sticker.” The experimenter asked the following three questions to make sure that children understood the rule: (a) “How many stickers will you get if you can wait here until I come back?” (b) “What will you do if you want to stop waiting for me?” and (c) “How many stickers will you get if you ring the bell before I come back?” Most of the children answered all three questions correctly on their first attempt. The 15 children who did not were then corrected and asked again, and at that point, all of these children answered the three questions correctly. Most of the incorrect answers concerned the second question, for which several children just named the bell or pointed to it without offering any further information.

Next, the experimenter said that she needed to leave. She asked the child not to leave his or her seat and not to play with the stickers. At this point, there was an additional step for participants who were assigned to the peer condition or the teacher condition. Children in the peer condition were told, “Your friend [a child who the participant had mentioned as being a friend during the warm-up period] will find out how long you wait.” Children in the teacher condition were told, “Your teacher [the name of the child’s current teacher] will find out how long you wait.”

After confirming that the participant understood that he or she would need to wait to be able to obtain a second sticker, the experimenter then left the room for 15 min or until the child rang the bell. The primary dependent measure was the number of seconds the child waited before the task ended because of either (a) the child ringing the bell early and earning one sticker or (b) the child waiting the full 15 min and earning two stickers (see Ma et al., 2018).

Results

We first examined the percentage of children who waited long enough to obtain the second sticker, which was 68% of the children in the teacher condition, 40% of children in the peer condition, and 16% of children in the standard condition. Kruskal-Wallis tests indicated that the percentage of children who waited for the full 15 min differed across the three conditions, χ²(2, N = 193) = 33.33, p < .001. Pairwise comparisons indicated that the percentage of children who waited was greater in the teacher condition than in the peer condition, χ²(1, N = 132) = 10.08, p < .01; greater in the teacher condition than in the standard condition, χ²(1, N = 123) = 32.95, p < .001; and greater in the peer condition than in the standard condition, χ²(1, N = 131) = 8.75, p < .01.

We next examined whether waiting times varied by condition. Waiting times across the delay period were not normally distributed¹ because the data for children who waited the full 15 min were right censored. Therefore, we used the Mantel-Cox log-rank test to test for differences in waiting times across the three conditions.² There were significant differences across the three conditions, Mantel-Cox χ²(2, N = 193) = 43.81, p < .001. Pairwise comparisons indicated that differences in waiting times were significantly different between each pair of conditions—between the teacher condition and the peer condition: χ²(1, N = 132) = 13.33, p < .001; between the teacher condition and the standard condition: χ²(1, N = 123) = 47.01, p < .001; between the peer condition and the standard condition: χ²(1, N = 131) = 9.36, p < .01. As predicted, children waited the longest in the teacher condition (mean waiting time = 790.23 s, nonparametric bootstrapped 95% confidence interval, or CI = [739.31, 841.14]), followed by the peer condition (mean waiting time = 569.44 s, nonparametric bootstrapped 95% CI = [492.77, 646.12]), with the shortest times in the standard condition (mean waiting time = 409.10 s, nonparametric bootstrapped 95% CI = [334.39, 483.81]; for the methods of data analysis, see Michaelson & Munakata, 2016). Figure 1 shows the percentage of children in each condition who waited as a function of time.

Fig. 1.

Survival functions for the 900-s waiting period in Study 1, separately for each condition.

In a final set of analyses, we examined the relation between executive function and delay of gratification (see Table 1). We first examined whether executive function varied as a function of condition using a one-way analysis of variance (ANOVA) with condition as the independent variable and executive-function score as the dependent variable. The results showed that there was no significant effect of condition.

Table 1.

Mean Waiting Time and Executive-Function Score for Each Condition in Study 1

Condition	Waiting time	Executive-function score
Teacher	790.23 (206.23)	10.73 (4.02)
Peer	569.44 (329.67)	10.29 (4.45)
Standard	409.10 (300.18)	10.44 (4.26)

Note: Waiting times are given in seconds. Standard deviations are given in parentheses.

We then assessed the roles of executive function and condition on waiting times in the delay-of-gratification task (see Table 2) using a hierarchical Cox regression model (see Duckworth, Tsukayama, & Kirby, 2013). Age (in months) and gender were entered in the first step, followed by executive function in the second step, and condition (1 = teacher condition, 2 = peer condition, 3 = standard condition, with the standard condition as the reference condition) in the third step. Waiting time was the predicted variable. The first block showed a significant effect of age, χ²(2, N = 193) = 13.41, p = .001, with older children waiting longer. The second block was significant, χ²(3, N = 193) = 21.90, p < .001, which suggests that higher executive-function scores were associated with longer waiting times. The third block was also significant, χ²(5, N = 193) = 78.97, p < .001, which suggests that even after analyses controlled for age and executive function, condition affected children’s waiting times. Specifically, compared with children in the standard condition, children in the peer condition were only 0.401 times as likely to stop waiting, and children in the teacher condition were only 0.145 times as likely to stop waiting.

Table 2.

Results of the Cox Regression Model for Study 1

Block and predictor	b	SE	Wald	Exp(b)	95% CI
1
Age	−0.65	0.19	11.55***	0.521	[0.357, 0.759]
Gender	−0.26	0.19	1.83	0.774	[0.534, 1.122]
2
Age	−0.42	0.21	3.81	0.660	[0.434, 1.002]
Gender	−0.22	0.19	1.33	0.804	[0.554, 1.166]
Executive function	−0.06	0.02	7.15**	0.939	[0.897, 0.983]
3
Age	−0.71	0.22	10.58***	0.491	[0.320, 0.754]
Gender	−0.17	0.19	0.77	0.846	[0.582, 1.229]
Executive function	−0.07	0.02	9.01**	0.932	[0.891, 0.976]
Condition			51.44***
Condition (1)	−1.93	0.28	49.09***	0.145	[0.084, 0.249]
Condition (2)	−0.91	0.22	17.90***	0.401	[0.262, 0.612]

Note: CI = confidence interval.

p < .01. ***p < .001.

These findings support the reputation hypothesis. Specifically, they indicate that the reputation-management manipulation had a large effect on children’s behavior and that it was stronger when the target audience was a teacher rather than a peer.

Study 2

Study 2 was designed to investigate whether the results of Study 1 would extend to a version of the task that uses a biologically relevant reward in the form of a cookie. These types of rewards have typically been used in classic delay-of-gratification studies (see Mischel, Shoda, & Rodriguez, 1989), and biological relevance is important to consider in light of theoretical accounts that differentiate between affect-based and cognition-based responses (Metcalfe & Mischel, 1999).

A second question that emerged from the results of Study 1 was whether children might have waited longer in the reputation conditions simply because the concept of waiting was made salient in those conditions but not in the standard condition. This possibility was addressed in Study 2 by telling children in all conditions that the task was a waiting game.

Method

Participants

Following the preregistration for this study (https://aspredicted.org/ht93y.pdf), we recruited a final sample of 80 children. These children did not participate in Study 1, but each attended one of the same preschools. Half of the children were assigned to a standard condition (age: M = 4.53 years, SD = 0.49, range = 3.24–5.13 years; 24 boys), and the other half were assigned to a teacher condition (age: M = 4.57 years, SD = 0.45, range = 3.20–5.12 years; 19 boys). In contrast to Study 1, there was no peer condition. The sample size was determined on the basis of the results of Study 1. A total of 27 other children were initially recruited for the study but then excluded: five who decided not to continue after the initial executive-function assessment phase and 22 who said that they did not want to wait at all in the delay-of-gratification task.³ Children who said that they did not want to try to wait offered a wide range of explanations, such as not liking the kind of cookie they were offered or the belief that it would be inappropriate for a child to eat more than one treat. The study was approved by the institutional review board of Zhejiang Sci-Tech University, and only children who gave their oral assent to participate and had consent from parents or legal guardians were included.

Procedure

The procedure was the same as in Study 1, with the following exceptions. First, the reward took the form of cookies rather than stickers, with the initial cookie being sealed inside a transparent snack box and placed in front of the child. Second, as noted above, each participant was randomly assigned to one of two conditions, standard and teacher, rather than the three conditions of Study 1. The peer condition was omitted because staff at the school expressed a desire to minimize the number of children who would be offered cookies in light of concerns about possible food allergies and because some parents were apprehensive about their children being exposed to what they considered unhealthy food. Third, in both conditions, the experimenter described the task as a waiting game. Specifically, she told children, “Let’s play a waiting game. You can get this snack right now, or if you wait until I come back, you can get another snack. Two snacks!” An additional manipulation check was added to make sure that children remembered this instruction: “What game are we playing now?” Most of the children answered all four questions correctly on their first attempt, but 13 children got at least one question wrong. For these children, all of the instructions and questions were repeated, and all 13 were then able to answer each of the questions correctly.

Results

We first examined the percentage of children who waited long enough to obtain the second snack, which was 58% of the children in the teacher condition and 23% of the children in the standard condition. A Mann-Whitney test indicated that the percentage was higher in the teacher condition than in the standard condition, Mann-Whitney U = 520.00, p < .001.

We next examined whether waiting times varied by condition. Waiting times across the delay period were not normally distributed⁴ because the data for children who waited the full 15 min were right censored. Therefore, a Mantel-Cox log-rank test was used to test the differences in waiting times between the two conditions.⁵ There were significant differences between the conditions, Mantel-Cox χ²(1, N = 80) = 12.35, p < .001. As predicted, children waited longer in the teacher condition (mean waiting time = 634.90 s, nonparametric bootstrapped 95% CI = [528.79, 741.01]) than in the standard condition (mean waiting time = 341.80 s, nonparametric bootstrapped 95% CI = [234.93, 446.67]). Figure 2 shows the percentage of children in each condition who waited as a function of time.

Fig. 2.

Survival functions for the 900-s waiting period in Study 2, separately for each condition.

In a final set of analyses, we examined the relation between executive function and delay of gratification (see Table 3). We first examined whether executive function varied by condition using a one-way ANOVA with condition as the independent variable and executive-function score as the dependent variable. The results showed that there was no significant effect of condition.

Table 3.

Mean Waiting Time and Executive-Function Score for Each Condition in Study 2

Condition	Waiting time	Executive-function score
Teacher	634.90 (346.76)	11.38 (3.42)
Standard	341.80 (342.72)	10.88 (4.68)

Note: Waiting times are given in seconds. Standard deviations are given in parentheses.

We then assessed the roles of executive function and condition on delay of gratification using a hierarchical Cox regression model. Age (in months) and gender were entered in the first step, followed by executive function in the second step, and condition (1 = teacher condition, 2 = standard condition, with the standard condition as the reference condition) in the third step. Waiting times were the predicted variable. The first block showed a significant effect of age, χ²(2, N = 80) = 6.74, p < .05, with older children waiting longer. The second block was significant, χ²(3, N = 80) = 24.59, p < .001, which suggests that the increased executive-function ability was associated with longer waiting times. The third block was also significant, χ²(4, N = 80) = 38.82, p < .001, which suggests that, as in Study 1, even after analyses controlled for age and executive function, the reputation manipulation influenced children’s waiting times. Specifically, compared with children in the standard condition, children in the teacher condition were only 0.346 times as likely to stop waiting (see Table 4).

Table 4.

Results of the Cox Regression Model for Study 2

Block and predictor	b	SE	Wald	Exp(b)	95% CI
1
Age	−0.63	0.29	4.79*	0.534	[0.305, 0.936]
Gender	0.41	0.30	1.92	1.506	[0.843, 2.689]
2
Age	−0.03	0.32	0.01	0.974	[0.522, 1.818]
Gender	0.44	0.30	2.21	1.559	[0.868, 2.800]
Executive function	−0.16	0.04	16.99***	0.849	[0.785, 0.918]
3
Age	−0.23	0.32	0.54	0.793	[0.427, 1.473]
Gender	0.33	0.30	1.20	1.391	[0.771, 2.507]
Executive function	−0.15	0.04	16.10***	0.863	[0.803, 0.927]
Condition	−1.06	0.32	11.26***	0.346	[0.186, 0.643]

Note: CI = confidence interval.

p < .05. ***p < .001.

The results of Study 2 provide further evidence for the reputation hypothesis and show that it extends to a food-reward context. The results also indicate that the effects of Study 1 were not simply due to children in the standard condition being unaware that the task had anything to do with waiting.

General Discussion

We evaluated the reputation hypothesis, which proposes that a desire to be perceived favorably by other people can influence children’s waiting times on delay-of-gratification tasks. In Study 1, preschool-age children were told that they could either have one sticker right away or wait and obtain a second one as well. Each child was assigned either to a standard delay-of-gratification condition or to one of two reputation conditions, in which the child was told that either the teacher or a peer would find out how long he or she had waited. In Study 2, a new group of children was told that they could have one cookie right away or wait to obtain a second one, and each child was assigned to a standard delay-of-gratification condition or a teacher condition. Both studies showed strong reputation effects. For this pattern of results to be produced, children in the two reputation conditions presumably needed to infer that the teacher or peer would (a) consider the information to be relevant (a reasonable assumption according to the maxim of relevance; Grice, 1975) and (b) use the information to make an evaluation in which longer waiting times are judged more favorably.

The present results add to evidence that rational decision processes play a role in children’s ability to wait for things they want by showing that children’s waiting times are sensitive to the expected judgments of other people. These findings shed light on children’s motivation to engage in behaviors that promote affiliation with their social group (Wen et al., 2016) by suggesting that one early-emerging goal is to promote their reputation within the group. This is particularly notable given that only a small number of experimental studies to date have found that children younger than 5 years can systematically modify their behavior depending on whether or not an audience is present (Botto & Rochat, 2018; Rapp, Engelmann, Herrmann, & Tomasello, 2017; Zhao, Heyman, Chen, & Lee, 2018; see Engelmann & Rapp, 2018). Our findings show that young children can be responsive to reputational concerns that relate to persistence, even if the expected evaluator of their behavior is not physically present (see Rapp, Engelmann, Herrmann, & Tomasello, 2019).

In Study 1, children waited longer when the individual who was to be told how long they waited was identified as a teacher rather than as a peer, which is noteworthy because the capacity to adjust one’s behavior depending on the audience has been identified as a hallmark of a fully strategic reputation-management ability (Engelmann et al., 2013; see also Watling & Banerjee, 2007). It is also notable that systematic reputational effects were observed in relation to peers. Although children might have inferred that teachers would approve of longer waiting times on the basis of seeing teachers encourage patience in other contexts, this explanation is less plausible for peers. It seems likely that children either applied their understanding of adults’ values to the peer or inferred from the task context that the peer would value waiting (e.g., by noting that waiting is being rewarded on the task and that rewarded behaviors tend to be socially valued).

Executive-function scores predicted waiting times in both studies, which suggests that young children need to have certain basic cognitive capacities in place to be able to wait for an extended period of time. However, the results from the reputation conditions indicate that they were capable of waiting much longer than has typically been seen in the delay-of-gratification literature. Perhaps children would wait even longer still if they were directly told that waiting is socially valued (see Leonard, Berkowitz, & Shusterman, 2014) or if their waiting behavior were directly observed.

It is not clear why fewer children were willing to begin waiting for a second cookie in Study 2 than for a second sticker in Study 1. Although this might simply reflect differences in the relative desirability of the rewards, some participants offered normative explanations, such as indicating that children should not eat more than one cookie, which points to the potential role of cultural learning. A related question for future research is how the present results will generalize across cultures. Because teachers in China tend to be highly respected (see Li, 2005), it would not be surprising if Chinese children were more motivated to impress their teachers than are children in some other countries.

One limitation of this research is that our paradigm did not allow us to assess whether reputational concerns affect children’s performance on standard delay-of-gratification tasks. It is possible that children are motivated to wait longer when they care how they are perceived by the experimenter and think that the experimenter will be pleased if they wait longer. This possibility could be evaluated by developing a new paradigm to manipulate children’s beliefs about whether the experimenter will find out how long they have waited. If reputational effects do indeed play a role, it will be important to examine whether individual differences in children’s reputational concerns contribute to the predictive power of these tasks.

The present research raises important questions about how to interpret studies that use the delay-of-gratification paradigm, and it may have implications for understanding other measures of self-control as well (see Loewenstein, 2018). Moreover, the present findings contribute to our understanding of the rational decision-making processes that affect waiting times on delay-of-gratification tasks by showing that children’s cost-benefit analyses can go beyond factors associated with the explicit costs and benefits that serve to structure the task. Finally, the results shed light on the ontology of reputation management by providing evidence that children as young as 3 years can make sophisticated inferences about how they will be judged by important people in their lives and act on these inferences in a way that shows sensitivity to the identity of their potential audience.

Supplemental Material

Heyman_OpenPracticesDisclosure_rev – Supplemental material for Delay of Gratification as Reputation Management

Supplemental material, Heyman_OpenPracticesDisclosure_rev for Delay of Gratification as Reputation Management by Fengling Ma, Dan Zeng, Fen Xu, Brian J. Compton and Gail D. Heyman in Psychological Science

Footnotes

Acknowledgements

The authors thank Joseph Kable, Cristine Legare, and Piotr Winkielman for helpful comments on an earlier version of the manuscript.

Transparency

Action Editor: Bill von Hippel

Editor: D. Stephen Lindsay

Author Contributions

F. Ma worked out the details of the study design and supervised the data collection and analysis. D. Zeng collected the data. F. Xu worked with F. Ma on the design. G. D. Heyman conceived the study hypothesis and the general strategy for testing it. G. D. Heyman and B. J. Compton drafted the manuscript. All the authors approved the final manuscript for submission.

ORCID iD

Gail D. Heyman

Notes

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