Manon Laroche
email: manon.laroche@univ-amu.fr

Peggy Roussel

Nicolas Mascret

Francois Cury

Manon <au>

Laroche

François Cury

Health Regulatory Focus, Selection Optimization and Compensation Strategy and Sports Practice: A Mediational Analysis

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<ABS> Abstract

This study aimed to investigate associations of health promotion and prevention regulatory foci with sports practice, and examined the Selection, Optimization and Compensation (SOC) process behind the positive relationship between health promotion focus and sports practice. A cross-sectional study was conducted with 513 French volunteer sports participants aged from 18 to 82. Participants completed an online self-report survey measuring health regulatory foci, SOC strategy, Amount of Sports Practice (ASP), health condition, and educational level. Path analysis main results (χ 2 = 16.64; df = 5; p < .01; RMSEA = 0.067; CFI = 0.98; R 2 = .24) demonstrated that ASP was positively related with health promotion focus (β = .13, p < .01), SOC strategy (β = .28, p < .001) and negatively related with health prevention focus (β = -.20, p < .001). SOC strategy was positively related with health promotion focus (β = .39, p < .001) and bootstrapping analyses revealed that this strategy partially mediated the positive relationship between health promotion focus and ASP, 95% CI [.13, .29]. Finally, additional analyses showed that it was specifically the elective selection, optimization and compensation sub-components of SOC strategy which played mediating roles in this link, [.13, .29] < 95% CI < [.13, .29]. For the first time in the literature, these results evidenced direct links between health regulatory foci and sports practice, and a mechanistic pathway between health promotion focus and sports practice. The theoretical and applied implications of these results for sports promotion are discussed.

<HIS> Received 22 March 2018; Revised XXXX; Accepted XXXX.

<KWD> Key words: health regulatory focus, SOC strategy, sports practice.

Undertaking regular physical activity (PA) is one of the recommendations of public health information campaigns. Among the available forms of PA (e.g., those related to work, household or leisure), sport activity in their free time is one means that many individuals use to respond to these health messages. But each person responds in his or her own way, engaging in their preferred sport with greater or lesser amounts of practice. One of the most prominent factors determining individuals' participation in their sport is their motivation. Many motivational theories and models have brought real contributions to explain the antecedents and processes that give rise to sports practice [START_REF] <refj> Ryan | The Oxford handbook of human motivation[END_REF]. In France, the motivations most often mentioned by adults who engage in sport are pleasure and health [START_REF] <refj> Gleizes | Pratiques physiques ou sportives des femmes et des hommes: Des rapprochements mais aussi des différences qui persistent [Physical and sporting practices of women and men: Similarities but also differences that persist[END_REF]. The present study focuses on these health-related motivations, and the questions addressed are the following: What is the influence of health motivation on the amount of sports practice? What process can explain this influence?

Regulatory focus theory [START_REF] Higgins | Beyond pleasure and pain[END_REF], a well-established motivational theory and a useful framework in the health research area [START_REF] <refj> Fuglestad | Getting there and hanging on: The effect of regulatory focus on performance in smoking and weight loss interventions [Special issue[END_REF][START_REF] <refj> Leder | Self-regulation and protective health behavior: How regulatory focus and anticipated regret are related to vaccination decisions[END_REF][START_REF] <refj> Uskul | Regulatory fit and health behavior[END_REF], argues for the existence of two regulatory systems (promotion and prevention) which guide people in their self-care behaviors.

Self-regulation with a promotion focus is associated with growth and accomplishment needs and is reflected in a concern for the presence or absence of positive outcomes (gains and nongains). Self-regulation with a prevention focus is associated with security and safety needs and is reflected in a concern for the presence or absence of negative outcomes (losses and nonlosses) [START_REF] Higgins | Beyond pleasure and pain[END_REF]. In the literature, regulatory focus has been studied as a situationally induced orientation (e.g., experimentally manipulated by framing health communications, [START_REF] <refj> Cesario | Regulatory fit and persuasion: Transfer from "feeling right[END_REF] and as a chronic individual-difference variable assessed with dispositional measures (e.g., [START_REF] Higgins | Achievement orientations from subjective histories of success: Promotion pride versus prevention pride[END_REF]. To assess chronic regulatory focus, several studies in the health research area have used general measures of regulatory focus (e.g., [START_REF] <refj> Fuglestad | Getting there and hanging on: The effect of regulatory focus on performance in smoking and weight loss interventions [Special issue[END_REF][START_REF] <refj> Joireman | Promotion orientation explains why future-oriented people exercise and eat healthy: Evidence from the two-factor consideration of future consequences-14 scale[END_REF]. Because the health domain involves specific situations that are not necessarily captured by general measures of regulatory focus, [START_REF] <refj> Gomez | Avoiding poor health or approaching good health: Does it matter? Conceptualization, measurement and consequences of health regulatory focus[END_REF][START_REF] <refj> Gomez | Avoiding poor health or approaching good health: Does it matter? Conceptualization, measurement and consequences of health regulatory focus[END_REF][START_REF] <refj> Ferrer | Developing a scale to assess health regulatory focus[END_REF] have recently highlighted the relevance of capturing chronic regulatory foci in the specific health context to study health behaviors. Chronic health regulatory foci measures have thus been developed. A chronic health promotion focus measure is supposed to capture concerns for improving the health state or attaining health-related gains. On the other hand, a chronic health prevention focus measure is supposed to capture concerns for protecting health state or avoiding healthrelated losses [START_REF] <refj> Ferrer | Developing a scale to assess health regulatory focus[END_REF][START_REF] <refj> Gomez | Avoiding poor health or approaching good health: Does it matter? Conceptualization, measurement and consequences of health regulatory focus[END_REF]. [START_REF] <refj> Gomez | Avoiding poor health or approaching good health: Does it matter? Conceptualization, measurement and consequences of health regulatory focus[END_REF] examined the influence of these two chronic health regulatory foci on the adoption of several health behaviors such as health information seeking, consumption of organic and functional foods, number of physician and pharmacist visits or drug consumption. More recently, Ferrer et al. (2017) examined the influences of these two chronic health regulatory foci on intentions to perform several health behaviors including regular exercise.

The first aim of the present study was to extend Ferrer et al.'s (2017) study on regular exercise intention by examining for the first time in the literature the influences of chronic health promotion and prevention regulatory foci on the amount of sports practice. According to [START_REF] <refj> Rothman | How persons and situations regulate message framing effects: The study of health behavior[END_REF], sports practice, balanced diet or sunscreen are among the health behaviors that promote health. These authors suggest that when people think about engaging in sport, they are more likely to attend to the presence or absence of favorable outcomes than to the presence or absence of unfavorable outcomes. Sports practice, being an activity that sustains a positive-outcomes approach could consequently fit better with a health promotion focus. One of the positive effects of fit is that people engage more strongly in what they are doing [START_REF] Higgins | Value from regulatory fit[END_REF]. Thus, sports practice could be easier to undertake for people who experience a fit [START_REF] <refj> Aaker | Understanding regulatory fit[END_REF]. Moreover, while sports practice can lead both to immediate (e.g., pleasure) or distal positive consequences (e.g., general health), [START_REF] Hall | Temporal self-regulation theory: A model for individual health behavior[END_REF] have nevertheless evidenced that this health behavior has the particularity of involving an intertemporal trade-off by accruing present costs to achieve delayed rewards. In particular, those authors examined the perceived temporal proximity of benefits associated with physical activity and show that benefits become salient the equivalent of several hundred hours later (2007( , p. 11). Fuglestad et al. (2008) ) also suggested that engaging in a regular exercise program will probably entail a longer initiation phase to fully master the new, effortful behaviors and to achieve positive outcomes. Thus, the importance that individuals attach to the future consequences of their behavior influences their willingness to engage in this health behavior (e.g., Gellert, Ziegelmann, Lippke, & Schwarzer, 2012).

Promotion-focused individuals tend to adopt a distal temporal perspective whereas preventionfocused individuals tend to adopt a proximal temporal perspective [START_REF] <refj> Pennington | Regulatory focus and temporal distance[END_REF]. [START_REF] <refj> Joireman | Promotion orientation explains why future-oriented people exercise and eat healthy: Evidence from the two-factor consideration of future consequences-14 scale[END_REF] also showed that a chronic general promotion focus was positively related to consideration of future consequences whereas a chronic general prevention focus tends to be positively related to consideration of immediate consequences. Thus, sports practice seems more in line with the distal temporal perspective of promotion-focused individuals. In addition, Ferrer et al.'s (2017) results evidenced that health promotion focus was positively related to the intention to exercise regularly whereas health prevention focus was negatively related to this variable. Based on these theoretical supports and Ferrer et al.'s (2017) empirical results, we thus expected that health promotion focus should be positively related with the amount of sports practice whereas health prevention focus should be negatively related with this variable. Furthermore, while Ferrer et al. 's results (2017) support the idea that health promotion focus seems to be more favorable to physical activity, they do not inform us about the processes explaining this association. Thus, the second aim of this study was to explore for the first time in the literature a potential process behind the positive relationship expected between health promotion focus and sports practice. This process issue is important, both from a theoretical standpoint of acquiring, a clear and detailed understanding of how health promotion focus acts on sports practice, and from an applied standpoint of knowing how to support and facilitate sports practice. We proposed that the general set of resource allocation strategies SOC (Selection, Optimization and Compensation) could play a role in this process. Indeed, sports practice is a part of lifestyle behaviors which implies in individuals a self-management to overcome the perceived obstacles to practice (e. g., lack of time, family constraints, work -Salmon, Owen, Crawford, [START_REF] <refj> Salmon | Physical activity and sedentary behavior: A population-based study of barriers, enjoyment, and preference[END_REF][START_REF] <refj> Gómez-López | Perceived barriers by university students in the practice of physical activities[END_REF]. Now, the SOC model [START_REF] <refj> Baltes | Psychological perspectives on successful aging: The model of selective optimization with compensation[END_REF]) evidences a set of four interrelated action regulation strategies that help individuals to self-manage in an optimal way:

Elective selection (i.e., developing and committing to a hierarchy of personal goals), loss-based selection (i.e., changes in the goal or the goal system in response to loss), optimization (i.e., engaging in goal-directed actions and means), and compensation (i.e., acquiring alternative means or support in response to a loss). For Freund and [START_REF] <refj> Freund | Life-management strategies of selection, optimization and compensation: Measurement by self-report and construct validity[END_REF], SOC is "viewed as a coordinated ensemble of contextualized processes. Therefore, S, O, and C are considered conjointly and in their interaction for fully understanding their role in positive development" (p. 643). In the health domain, several studies have evidenced that use of SOC strategy has a positive influence on sports practice in a rehabilitation context [START_REF] <refj> Ziegelmann | Planning and strategy use in health behavior change: A life span view[END_REF], on maintenance of sports practice among older adults [START_REF] <refj> Gellert | An age-tailored intervention sustains physical activity changes in older adults: A randomized controlled trial[END_REF] and on long-term adherence to a sport program among older women [START_REF] Evers | Long-term adherence to a physical activity intervention: The role of telephone-assisted vs. selfadministered coping plans and strategy use[END_REF]. The adoption of sports practice involves an initial motivation process that results in intention formation and subsequent self-regulation processes that address the pursuit of these goals [START_REF] <refj> Schwarzer | Self-efficacy in the adoption and maintenance of health behaviors: Theoretical approaches and a new model[END_REF]. SOC strategy has been found to act as a mediator variable between planning (i.e., plans on 'when', 'where', and 'how' to perform a behavior) and exercise behavior [START_REF] <refj> Ziegelmann | Planning and strategy use in health behavior change: A life span view[END_REF]. It is thus proposed that SOC strategy constitutes an important step in the process leading to sports practice. Furthermore, SOC strategy calls for flexibility for development of new goals in response to difficulties, refinement of goal-relevant means, and acquisition of new skills/resources [START_REF] <refj> Freund | Life-management strategies of selection, optimization and compensation: Measurement by self-report and construct validity[END_REF]. Individuals with a promotion focus tend to approach desired end-states by engaging in flexible, explorative cognitive processing [START_REF] <refj> Crowe | Regulatory focus and strategic inclinations: Promotion and prevention in decision-making[END_REF]. In addition, SOC is a strategy which serves a goal of achieving success [START_REF] Baltes | Future time perspective, regulatory focus, and selection, optimization, and compensation: Testing a longitudinal model[END_REF], which is a goal congruent with promotion focus. Indeed, individuals with a promotion focus are attuned to emotions relating to the successful pursuit of positive outcomes [START_REF] Higgins | Ideal versus ought predilections for approach and avoidance distinct self-regulatory systems[END_REF]. Thus, promotion focus and SOC strategy are matching because they are goal-congruent. Finally, Baltes et al.'s (2014) results

showed that a chronic general promotion focus was positively related to the adoption of SOC strategy at work. Based on these theoretical supports and Baltes et al.'s (2014) empirical results, we hypothesized thus that chronic health promotion focus should also be positively related to the adoption of SOC strategy in sport. Moreover, given the link expected on the one hand between health promotion focus and SOC strategy, and the link evidenced in the literature between SOC strategy and sports practice on the other hand, SOC strategy was hypothesized to mediate the positive relationship between health promotion focus and amount of sports practice.

In sum, the aims of the present study were twofold: (a) To explore the direct links between promotion and prevention health regulatory foci and amount of sports practice; and (b) to examine whether SOC strategy has a mediating role in the relationship between health promotion focus and amount of sports practice. These links will be examined among a large sample of sport practitioners. In addition, as age, gender, health condition and education level may affect physical activity practice (e.g., [START_REF] Shaw | Examining the association between education level and physical activity changes during early old age[END_REF], the effects of these variables were controlled in the present study.

In accordance with the above-mentioned frameworks, we formulated the following hypotheses:

Direct effects: Health promotion focus was expected to be positively related with amount of sports practice, whereas health prevention focus was expected to be negatively related with this variable. Indirect effects: SOC strategy was expected to mediate the positive link between health promotion focus and amount of sports practice.

<H1>

Method

<H2> Participants

To be eligible for the study, participants should practice a sport. A total of 588 French volunteer participants took part in the study. Of the initial 588 participants, 75 were excluded because of missing data. The responses of 513 participants (51.7% men, 56.5% healthy) aged from 18 to 82 years (mean age = 36.96, standard deviation = 16.33, median score = 33, skewness value = .74) were consequently considered in this study. Most of the participants (97%) had completed secondary education. Demographic characteristics and health condition of the participants are presented in Table 1. All participants were recruited from various organizations, clubs and leisure centers and represented different sports, namely gym-based exercise (23%), ball sports (21%), racquet sports (17%), running (14%), walking (10%), nature sports (6%), swimming (3%), cycling (2%), fighting sports (2%) and yoga (2%). Furthermore, a high proportion of our sample had a relatively high level of sports practice in comparison with the amount of sports practice of the French population (European Commission, 2018): 50.7% of the participants practiced with high intensity, 91.6% practiced regularly, and 79.3% performed sports sessions for 60 minutes and more. The practice characteristics of the sample are presented in Table 2.

<Insert Table 1 Here> <Insert Table 2 Here> <H2> Procedure Data were collected anonymously via a cross-sectional, online self-report survey. The protocol met France's ethical requirements for human research. Permission to conduct the study was first obtained from the administration. All participants provided informed consent prior to data collection.

<H2> Measures

Health regulatory focus. Gomez et al.'s (2013) French health regulatory focus questionnaire was used to assess participants' health regulatory focus. The questionnaire is composed of five items assessing health promotion focus (e.g., "I do not hesitate to embrace new experiences if I think they can improve my health"), and three items assessing health prevention focus (e.g., "I frequently think about the health problems I may have in the future"), presented in a random order. Participants responded on a scale from 1 = "not at all true of me" to 6 = "very true for me". A Confirmatory Factor Analysis (CFA) was performed on the covariance matrix and the solution was generated by using maximum likelihood estimation.

The two-factor model delineated by [START_REF] <refj> Gomez | Avoiding poor health or approaching good health: Does it matter? Conceptualization, measurement and consequences of health regulatory focus[END_REF] did not provide an acceptable fit to the data (RMSEA = .101; GFI = .947; CFI = .935). The health prevention focus item (i.e., "When I implement a health behavior, it's because I want to protect myself from getting sick.") exhibited high modification indices as well as moderate loadings on both factors. These results were similar to those obtained by [START_REF] <refj> Schmalbach | Psychometric properties of the German version of the Health Regulatory Focus scale[END_REF]. In line with these authors' procedure, a model excluding this item was tested and showed an improvement over the model previously examined (RMSEA = .089; GFI = .965; CFI = .959). Internal consistency was satisfactory for both the health promotion focus (α = .78) and the two-item health prevention focus (α = .77) subscales. Average scores were thus computed for each health regulatory focus subscale.

SOC strategy. Ziegelmann and Lippke's (2007) questionnaire, a version of Freund and

Baltes ' (2002) original questionnaire adapted to the physical activity context, was used to assess participants' SOC strategy. The items were translated into French and slightly adjusted.

The scale was composed of the stem "In general in my sport activity..." which was followed by three items assessing optimization strategy (e.g., "... I do everything possible to make my objectives come true"), and three items assessing elective selection strategy (e.g., "… I define my goals exactly and stick to them"). The second stem, "When it is getting more difficult in my sport activity...," was followed by three items assessing compensation strategy (e.g., "... then I increase my efforts even more to continue to practice my sport activity"), and three items assessing loss-based selection strategy (e.g., "... then I temporarily reduce the practice of my sport activity"). Participants responded on a scale from 1 = "not at all true of me" to 6 = "very true for me". A CFA was performed on the covariance matrix and the solution was generated by using maximum likelihood estimation. In line with Freund and [START_REF] <refj> Freund | Life-management strategies of selection, optimization and compensation: Measurement by self-report and construct validity[END_REF], who proposed the SOC model as an integrated system, a hierarchical model of SOC strategy was constructed comprising a higher-order factor (SOC strategy) and four first-order factors (loss-based Selection, elective Selection, Optimization and Compensation). The model provided an acceptable fit to the data (RMSEA = .088; GFI = .927; CFI = .957) and internal consistency was satisfactory (α = .71). An average score was then computed for SOC strategy.

Amount of sports practice. Three items adapted from the sport and leisure activities subscale of the French Dijon Physical Activity Score [START_REF] <refj> Robert | Le score d'activité physique de Dijon: Reproductibilité et corrélations avec l'aptitude physique de sujets sains âgés [The Dijon Physical Activity Score: Reproducibility and correlation with exercise testing in healthy elderly subjects[END_REF] were used to assess amount of sports practice. This questionnaire is appropriate to assess usual sports practice in samples including elderly subjects aged up to 93 (Emile, Chalabaev, Stephan, Corrion, & d'Arripe-Longueville, 2014). One item assessed sport activity intensity (i.e., "The sport activity that you practice is of: (3) High intensity: significant muscular fatigue, to (1) slight intensity: Without muscular fatigue"). One item assessed sport activity frequency (i.e., "You are used to practicing your sport activity: (4) Every day, to (1) irregularly"). One item assessed sport activity duration (i.e., "The average duration of your sessions for sport activity is: (4) 60 minutes and more, to (1) less than 15 minutes"). In line with Robert et al.'s (2004) procedure, the scores for the three items were added. The possible total score ranges vary between 3 (minimum) and 11 (maximum). The total score of sport and leisure activities subscale of this questionnaire has been shown to be reproductible in test-retest and to be correlated with peak VO2 and maximal power attained during effort test on a cycle ergometer and in walking tests [START_REF] <refj> Gremeaux | Le score d'activité physique de Dijon: Reproductibilité et corrélations avec l'aptitude physique de sujets coronariens [The Dijon Physical Activity Score: Reproducibility and correlations with physical fitness in patients with coronary artery disease[END_REF][START_REF] <refj> Robert | Le score d'activité physique de Dijon: Reproductibilité et corrélations avec l'aptitude physique de sujets sains âgés [The Dijon Physical Activity Score: Reproducibility and correlation with exercise testing in healthy elderly subjects[END_REF]. The distribution of the sample on the scores of this measure is presented in Table 2.

Health condition. Consistent with prior research (e.g., Kempen, Jelicic, & Ormel, 1997), a checklist of 18 diseases was used. Participants were asked to report whether they currently suffered from diseases diagnosed in the last 12 months. The total number of diseases diagnosed was used as a health condition index. The index ranged from 0 (none of the diseases) to 18 (all of the diseases). Educational level. Participants indicated their highest educational qualification attained from 1 (no qualification) to 10 (PhD).

<H2> Statistical analysis

A model for evaluating the combined contribution (direct and indirect effects) of each variable-the promotion and prevention regulatory foci, SOC strategy, age, gender (coded as 0 for women and 1 for men), educational level and health condition-on amount of sports practice was tested. This model tested was built from our hypotheses. A covariance matrix, using Statistica-12, was generated from raw data (observed scores in each variable). A path analysis using Lisrel 9.1 (Scientific Software International) was used to examine the fit of this first model. The .05 level of significance was used for all statistical hypothesis testing. Beta represents the standardized regression coefficient. We used the model fit tests using conservative cutoffs for root-mean-square error of approximation (RMSEA) and comparative fit index (CFI). Finally, a bootstrapping method resample [START_REF] <refj> Preacher | Asymptotic and resampling strategies for assessing and comparing indirect effects in multiple mediator models[END_REF] was employed to test the significance of the mediation effect of SOC strategy on the relationship between health promotion focus and amount of sports practice. Bootstrapping was set at 5000 samples with bias-corrected 95% confidence intervals.

<H1> Results

<H2> Descriptive statistics and correlations

Means, standard deviations and Pearson's correlation coefficients for the variables are presented in Table 3. Furthermore, Khi-2 tests were conducted to examine gender differences.

Only the amount of sports practice was impacted by gender. Men reported significantly higher amount of sports practice than women, χ 2 (8) = 45.67, p < .001.

<Insert Table 3 Here>

<H2> Hypothesized model path analysis

The results from the hypothesized model presented an acceptable fit (χ 2 = 36.15; df = 5; Direct effect. Amount of sports practice was positively related with health promotion focus (β = .13, p < .01), SOC strategy (β = .28, p < .001), and negatively related with health prevention focus (β = -.20, p < .001). 2 Moreover, amount of sports practice was positively 1 In this adjusted model, a hypothetical path between age and SOC strategy was added, and the direct link between age and amount of sports practice was deleted. In accordance with Freund and [START_REF] <refj> Freund | Life-management strategies of selection, optimization and compensation: Measurement by self-report and construct validity[END_REF], we expected a negative association between age and SOC strategy, suggesting that age-related decline in resources and plasticity might limit the expression of this strategy. 2 Analyses examining direct effect of health regulatory foci on amount of sports practice successively for three age groups of nearly equal subsample sizes (younger adults: 18-24 years, N = 174; middleaged adults: 25-43 years, N = 175; older adults: 44-82, N = 164) confirmed globally the results obtained in the whole sample. The results confirmed the positive link between health promotion focus and amount of sports practice for younger and older age groups and indicated a nearly significant positive link for the middle age group. On the other hand, the results indicated a significant negative related with gender (β = .28, p < .001), weakly negatively related with health condition variable (β = -.08, p = .052), and not significantly related with educational level variable.

Indirect effect. SOC strategy was positively related with health promotion focus (β = .39, p < .001) and negatively related with age (β = -.24, p < .001).

Bootstrapping analyses indicated that the indirect contribution of health promotion focus to amount of sports practice through SOC strategy was significant, with a point estimate of .040 and a bias-corrected bootstrapped 95% CI from .13 to .29. Therefore, since zero was not contained in the bootstrapped CI (see [START_REF] <refj> Preacher | Asymptotic and resampling strategies for assessing and comparing indirect effects in multiple mediator models[END_REF], SOC strategy may be considered a significant partial mediator of the relationship between health promotion focus and amount of sports practice, while controlling for educational level, gender and health condition. 3 <Insert Figure 1 Here> <Insert Table 4 Here>

<H2> Additional sub-components of SOC strategy path analyses

As Freund and [START_REF] <refj> Freund | Life-management strategies of selection, optimization and compensation: Measurement by self-report and construct validity[END_REF] consider the SOC model an integrated system, the preceding analyses examined the composite score of SOC strategy as mediator of the link between health promotion focus and amount of sports practice. However, an existing empirical study suggests that specific strategies are more beneficial for exercise than others [START_REF] Evers | Long-term adherence to a physical activity intervention: The role of telephone-assisted vs. selfadministered coping plans and strategy use[END_REF]. Thus, it is possible that sub-components of SOC strategy are more or less involved in this process. This possibility was not the central interest in the present work, but an ancillary analysis was conducted to examine it. Four path models examining the mediating role of each of the four sub-components of SOC strategy in the positive link between health promotion focus and amount of sports practice were thus successively run. Age, 4 gender, health condition and educational level effects were also controlled in these models. Results of these analyses link between health prevention focus and amount of sports practice for the older group and nearly significant negative links for the younger and middle age groups (Table 4). 3 Analyses examining indirect effect successively for the three age groups confirmed that health promotion focus was indirectly associated with amount of sports practice through SOC strategy in the younger and middle age groups. However, they showed that for the older group, the indirect effect was not significant (Table 4). 4 Concerning the age variable, the results, in line with Freund and [START_REF] <refj> Freund | Life-management strategies of selection, optimization and compensation: Measurement by self-report and construct validity[END_REF], indicated that age was negatively related with compensation. In addition, they showed that age was negatively related with optimization and elective selection and positively related with loss-based selection, whereas Freund and Baltes' (2002) results showed that age was not related with optimization, was positively related with elective selection, and tended to be positively related with loss-based selection. Thus, our results partially support those obtained by the authors, even if a comparison between the two studies must be undertaken with precaution, since the two studies did not use the same measures of SOC strategy. confirmed the mediating roles of elective selection, optimization and compensation in the positive link between health promotion focus and amount of sports practice. Specifically, elective selection and compensation partially mediated the link between health promotion focus and amount of sports practice, and optimization totally mediated this link. On the other hand, loss-based selection was not associated with health promotion focus and was negatively associated with amount of sports practice. Fit indices, beta coefficients and bootstrapped CI for each model are shown in Table 5.

<Insert Table 5 Here>

<H1> Discussion

This study investigated the links between health promotion and prevention regulatory foci and amount of sports practice and examined a potential process that may account for the positive relationship between health promotion focus and amount of sports practice. Our findings provided strong support for our predictions. Firstly, in line with Ferrer et al.'s (2017) results based on intentions, our results indicated that health promotion focus was positively related to amount of sports practice whereas health prevention focus was negatively related to this variable. Analyses conducted in sub-groups of participants indicated that these links were globally consistent across age. [START_REF] <refj> Ferrer | Developing a scale to assess health regulatory focus[END_REF] have hypothesized that prevention focus could be associated "with less willingness to engage in any behavior, which would explain why it predicted both lower healthy intentions here, and lower risky intentions in previous work" (p. 54). Our results evidence that as regards self-reported amount of sports practice (and not intention), health prevention focus is also negatively related to this variable. Our results thus provide additional evidence that sports practice is a health behavior positively related to health promotion focus and negatively with health prevention focus. These results could be attributed to the fact that, in contrast to other health behaviors whose primary function is to protect individuals against potential health threatse.g., screening procedures [START_REF] <refj> Ferrer | Developing a scale to assess health regulatory focus[END_REF]), vaccination (Leder et al., 2015), use of drugs [START_REF] <refj> Gomez | Avoiding poor health or approaching good health: Does it matter? Conceptualization, measurement and consequences of health regulatory focus[END_REF])physical activity is part of health behaviors whose primary function is the attainment of positive health outcomes (Rothman et al., 2008). This behavior consequently fits better with a health promotion focus than a health prevention focus.

The second aim of this study was to examine a process through which health promotion focus is positively linked with sports practice, an issue that has received no attention in the literature. As hypothesized, our main findings indicated that health promotion focus was positively related to SOC strategy considered as a coordinated ensemble of processes. These first findings were consistent with Baltes et al.'s (2014) result evidenced in a work-related context. It supports the idea that promotion focus generates in individuals a flexible strategic tendency [START_REF] <refj> Crowe | Regulatory focus and strategic inclinations: Promotion and prevention in decision-making[END_REF] and a concern for achieving success [START_REF] Higgins | Ideal versus ought predilections for approach and avoidance distinct self-regulatory systems[END_REF] emphasized by SOC strategy. In addition, these results indicated that SOC strategy partially mediated the positive relation between health promotion focus and amount of sports practice This mediational finding provides for the first time in the literature an explanation of how the direct positive link between health promotion focus and amount of sports practice occurs.

However, additional sub-components of SOC strategy path analyses showed that when the four SOC strategies are examined separately, only three of them (i.e., elective selection, optimization and compensation) played mediating roles. Concerning loss-based selection, the results showed first that this strategy was not related with health promotion focus. Secondly, in line with Evers et al.'s (2012) research, the results showed that this strategy was negatively associated with amount of sports practice. It seems that this strategy, which consists in changing the goal or the goal system in response to loss [START_REF] <refj> Baltes | Psychological perspectives on successful aging: The model of selective optimization with compensation[END_REF], is compatible neither with the concern of health promotion focus centered on achieving success nor with the usual amount of sports practice. Further research is needed to explore these issues.

Although the findings of the present study provide greater insights into (a) the direct links between health regulatory foci and sports practice, and (b) a mechanistic pathway between health promotion focus and sports practice, there are some limitations. First, our study focused on the determinants favorable to sports practice and was therefore quite naturally oriented toward analysis of the positive dynamic linked to health promotion focus. The negative link between health prevention focus and amount of sports practice also requires further study. Additional research is needed to investigate the potential processes through which health prevention focus is negatively related to sports practice. For example, one could start from the fact that the prevention focus tends to be positively associated with immediate consequences of behavior which is negatively associated with exercise attitude and intention [START_REF] <refj> Joireman | Promotion orientation explains why future-oriented people exercise and eat healthy: Evidence from the two-factor consideration of future consequences-14 scale[END_REF]. Thus, immediate future consequences of behaviors [START_REF] <refj> Strathman | The consideration of future consequences: Weighing immediate and distant outcomes of behavior[END_REF] could be a plausible candidate contributing to mediate the negative relationship between health prevention focus and sports practice. Secondly, our study was interested in the relationship between health regulatory foci and amount of sports practice considered as a health behavior integrated into our participants' lifestyles. [START_REF] <refj> Fuglestad | Getting there and hanging on: The effect of regulatory focus on performance in smoking and weight loss interventions [Special issue[END_REF] have shown that the relationship between regulatory foci and other health behaviors (i.e., smoking cessation, weight loss) were moderated by the different phases of behavioral change. They showed that promotion-focused individuals were more likely to initiate smoking cessation and weight loss, and that prevention-focused individuals were more likely to maintain these behaviors longer. Based on these works, it would be interesting to examine in a future study the moderating effect of length of time of practice (i.e., beginners vs long-term practitioners) on the links between health regulatory foci and amount of sports practice.

Thirdly, the cross-sectional design of the present study does not allow for causal claims among the variables. Experimental manipulation of the health regulatory focus to determine the effects on sports practice and SOC strategy would be of great value. Also, the specificity of our sample (i.e., highly educated participants, high levels of sports practice) limits the generalization of our results. Further study could extend these results to other populations. Finally, while our results show the predictive value of health regulatory foci on the amount of sport habitually practiced by the participants, other motivations of practice (e.g., fun, fitness, competition, making friends, etc.see [START_REF] Ryan | Intrinsic motivation and exercise adherence[END_REF] could broaden the scope of our observations. A further study examining to what extent health regulatory foci are related to amount of sports practice by controlling the weight of other sports motives could be very interesting and would complement our present results.

Beyond these limitations, our findings contribute to the literature in health psychology by evidencing for the first time links between health regulatory foci and sports practice, and a specific process underlying the link between health promotion focus and this behavior. In addition to these theoretical implications, our study also suggests practical steps. To make communications recommending sports practice more effective and to promote this health behavior, it is preferable to stress the improvement of health (promotion argument) rather than the avoidance of disease (prevention argument). The positive dynamic associated with promotion inspires practitioners' capacities to organize and manage themselves to cope with the specific constraints of sports practice. So, whether in large-scale public health campaigns (mass communication), in the discussions that doctors and sports educators may have with people or in support for practitioners through health apps and sports coaching, messages and arguments on the sport-health theme should highlight the health gains and benefits that can be achieved. It remains to be verified, however, how effective such communication is on the amount of sports practice depending on the length of time a person has engaged in sports activity.

p

  < .001; RMSEA = 0.110; CFI = 0.94). Considering the exploratory nature of this hypothesized model, a second model (adjusted model) was then tested including modification indices 1 proposed by Lisrel. The second model tested, adjusted to the specific data, presented an excellent fit (χ 2 = 16.64; df = 5; p < .01; RMSEA = 0.067; CFI = 0.98). This model explained 24% of the variance of amount of sports practice. The results from the adjusted model, detailed below, are presented in Figure 1.

Figure 1 .

 1 Figure 1. Path Diagram of Adjusted Model

Table 1 .

 1 Demographic Characteristics and Health Condition of the Sample Note. Unhealthy participants could suffer from one or several diseases

	N	%

Table 3 .

 3 Descriptive Statistics and Matrix of Pearson r Correlation Coefficients among the Variables (N = 513).

	M	SD	1	2	3	4	5	6	7

Note. Total score of amount of sports practice is the sum of scores of intensity (from 3 = high intensity to 1= slight intensity), frequency (from 4 = every day to 1 = irregularly) and duration (from 4 = 60 minutes and more to 1= less than 15 minutes).


Manon Laroche

Peggy Roussel

Francois Cury

Identifying a motivational process surrounding adherence to

come

INTRODUCTION

Diabetes is one of the most common metabolic disorders in the world [START_REF] Guariguata | Global estimates of diabetes prevalence for 2013 and projections for 2035[END_REF]. In France, there are an estimated 3 million patients with diabetes. The most common is type 2 diabetes, accounting for > 90% of all diabetes cases [START_REF] Mandereau-Bruno | Prévalence du diabète traité pharmacologiquement (tous types) en France en 2015. Disparités territoriales et socio-économiques. [Prevalence of pharmacologically treated diabetes (all types) in France in 2015. Territorial and socioeconomic disparities[END_REF]. Exercise and diet are key aspects of its treatment [START_REF] Colberg | Physical activity/exercise and diabetes: a position statement of the American Diabetes Association[END_REF][START_REF] Evert | Nutrition therapy recommendations for the management of adults with diabetes[END_REF]. Indeed, exercise and weight loss have both been shown to decrease insulin resistance and to improve glycemic control [START_REF] Snowling | Effects of different modes of exercise training on glucose control and risk factors for complications in type 2 diabetic patients: a meta-analysis[END_REF][START_REF] Espeland | Reduction in weight and cardiovascular disease risk factors in individuals with type 2 diabetes: one-year results of the look AHEAD trial[END_REF], which in turn is associated with lower risk of short-term complications (e.g., hyperglycemia), long-term comorbid conditions (e.g., cardiovascular diseases), and mortality [START_REF] Andersson | Long-term glycemic control relates to mortality in type II diabetes[END_REF][START_REF] Klein | relation of glycemic control to diabetic microvascular complications in diabetes mellitus[END_REF]. However, exercise and diet are perceived as costly in terms of time, organization, and personal investment [START_REF] Booth | Diet and physical activity in the self-management of type 2 diabetes: barriers and facilitators identified by patients and health professionals[END_REF]. Nadeau [START_REF] Nadeau | Management of type 2 diabetes mellitus in self-motivated patients: optimized diet, exercise, and medication for weight loss and cardiometabolic fitness[END_REF] underlines that "in clinical practice only a small percentage of the population with T2DM is sufficiently selfmotivated to undertake a rigorous exercise and diet program" (p.50). While physicians are aware of the importance of the patient's self-management in his/her treatment, the author stresses the lack of precise information about the motivators that promote exercise and diet in the patient.

The overall objective of this paper is thus to provide physicians with knowledge about the motivational processes surrounding exercise and diet for patients with type 2 diabetes and offer methods of support to favor their patients' self-management (e.g., What message should they privilege? What should be emphasized in patient support?)

To take exercise and follow a diet effectively, type 2 diabetes patients must be capable of making decisions and setting themselves targets appropriate to the constraints they face. This capacity to adapt to constraints and to reduced personal resources refers to a psychological process formalized by the theoretical model of Selective Optimization and Compensation (SOC, [START_REF] Baltes | Successful aging: Perspectives from the behavioral sciences[END_REF]). This model proposes that the combined use of four strategies helps individuals to optimally allocate their limited resources [START_REF] Baltes | Successful aging: Perspectives from the behavioral sciences[END_REF][START_REF] Freund | Life-management strategies of selection, optimization and compensation: Measurement by self-report and construct validity[END_REF]: elective selection (i.e., developing and committing to a hierarchy of personal goals), optimization (i.e., engaging in goal-directed actions and means), loss-based selection (i.e., changing the goal or the goal system) and compensation (i.e., acquiring alternative means in response to a loss). Several studies have shown that the use of SOC strategy has a positive influence on health behaviors such as the exercise participation of individuals in orthopedic rehabilitation [START_REF] Ziegelmann | Planning and strategy use in health behavior change: A life span view[END_REF] or older women's longterm adherence to a program of exercise [START_REF] Evers | Long-term adherence to a physical activity intervention: the role of telephone-assisted vs. self-administered coping plans and strategy use[END_REF]. In the light of these works, it could be supposed that the use of this SOC strategy would be favorable to exercise and diet among patients with type 2 diabetes. So far as we know, no study has so far examined the links between SOC strategy and exercise and diet among patients with type 2 diabetes. The first aim of this study is therefore to examine these links in this population.

On the other hand, our second aim is to identify for the first time in the literature the motivators which favor the use of this adaptative strategy. Regulatory Focus Theory [START_REF] Higgins | Beyond pleasure and pain[END_REF], a well-established motivational model to study health behaviors both among healthy [START_REF] Ferrer | Developing a scale to assess health regulatory focus[END_REF][START_REF] Joireman | Promotion orientation explains why futureoriented people exercise and eat healthy: Evidence from the two-factor consideration of future consequences-14 scale[END_REF][START_REF] Leder | Self-regulation and protective health behaviour: how regulatory focus and anticipated regret are related to vaccination decisions[END_REF] and patients with type 2 diabetes [START_REF] Avraham | regulatory focus and adherence to self-care behaviors among adults with type 2 diabetes[END_REF], identifies two kinds of motivational orientations that guide people in their self-care behaviors. The first called "promotion focus" is associated with growth and accomplishment needs [START_REF] Higgins | Beyond pleasure and pain[END_REF]. In a health-related context, it is reflected by concerns for improving the health state or attaining health-related gains [START_REF] Ferrer | Developing a scale to assess health regulatory focus[END_REF][START_REF] Gomez | avoiding poor health or approaching good health: Does it matter? Conceptualization, measurement and consequences of health regulatory focus[END_REF]. The second called "prevention focus" is associated with security and safety needs [START_REF] Higgins | Beyond pleasure and pain[END_REF]. In a health-related context, it is reflected by concerns for protecting health state or avoiding health-related losses [START_REF] Ferrer | Developing a scale to assess health regulatory focus[END_REF][START_REF] Gomez | avoiding poor health or approaching good health: Does it matter? Conceptualization, measurement and consequences of health regulatory focus[END_REF].

In this paper, we consider that promotion focus could be a motivator favorable to the use of the SOC strategy. First, promotion-focused individuals tend to engage a flexible cognitive process [START_REF] Crowe | Regulatory focus and strategic inclinations: Promotion and prevention in decision-making[END_REF] which is compatible with the development of new goals in response to difficulties, refinement of goal-relevant means, and acquisition of new skills/resources emphasized by the SOC strategy [START_REF] Freund | Life-management strategies of selection, optimization and compensation: Measurement by self-report and construct validity[END_REF]. Secondly, SOC strategy provides guidance toward success [START_REF] Baltes | Future time perspective, regulatory focus, and selection, optimization, and compensation: Testing a longitudinal model[END_REF], which is a goal congruent with promotion focus [START_REF] Higgins | Ideal versus ought predilections for approach and avoidance distinct self-regulatory systems[END_REF]. Thirdly, Baltes et al. [START_REF] Baltes | Future time perspective, regulatory focus, and selection, optimization, and compensation: Testing a longitudinal model[END_REF] have shown in a work context that promotion focus is a positive predictor of SOC strategy.

On the other hand, we propose that prevention focus would be either not or negatively associated with the adoption of SOC strategy. Prevention-focused individuals tend to engage a rigid cognitive process and not to persist for long in a difficult task [START_REF] Crowe | Regulatory focus and strategic inclinations: Promotion and prevention in decision-making[END_REF]. These two inclinations are likely not to be compatible with the flexibility and the tenacity required by the SOC strategy [START_REF] Freund | Life-management strategies of selection, optimization and compensation: Measurement by self-report and construct validity[END_REF]. Finally, given that promotion focus has been found to be a motivational orientation favorable to exercise and diet among both healthy patients [START_REF] Ferrer | Developing a scale to assess health regulatory focus[END_REF][START_REF] Joireman | Promotion orientation explains why futureoriented people exercise and eat healthy: Evidence from the two-factor consideration of future consequences-14 scale[END_REF] and patients with type 2 diabetes [START_REF] Avraham | regulatory focus and adherence to self-care behaviors among adults with type 2 diabetes[END_REF], the following specific mediational hypothesis was tested. We predicted that the positive relation between promotion focus, exercise and diet would be mediated by the use of SOC strategy.

In sum, in this paper our three main hypotheses are: H1: SOC strategy should be positively associated with exercise and diet behaviors.

Participants

To be eligible for the study participants had to: (i) be older than 18, (ii) have a history of at least three months of diagnosed type 2 diabetes mellitus, and (iii) have French-reading abilities. A total of 491 participants (311 men) aged from 26 to 86 (M = 61.7, SD = 9.6) were selected in the study. Most of the participants (93.3%) had completed secondary education, 56% were retired, and 67.6% lived with a partner. About 72% of the participants were treated with diabetic oral medications, 22% used insulin, and 30% were not medicated for their diabetes. The average number of comorbidities per participant was 1.48 (SD = 1.96); the most common was arterial hypertension (44.8%). The average body mass index of the sample was 29.8 (SD = 5.9). Demographic characteristics and health condition of the sample are presented in Table 1.

Measures

SOC strategy. Reuter et al.'s [START_REF] Reuter | Planning bridges the intention-behaviour gap: Age makes a difference and strategy use explains why[END_REF] questionnaire, a version of Freund and Baltes' [START_REF] Freund | Life-management strategies of selection, optimization and compensation: Measurement by self-report and construct validity[END_REF] original questionnaire adapted for leading a healthy lifestyle, was used to assess participants' SOC strategy. Using a standard "forward-backward" translation procedure, the English-language version of the questionnaire was translated into French. The questionnaire is composed of a total of four items assessing SOC strategy (e.g., "I have defined my goals exactly and stick to them"). Participants responded on a scale from 1 = "completely disagree" to 4 = "completely agree".

First, the reliability of each measure was examined. For SOC strategy and Regulatory focus measures, two confirmatory factorial analyses (CFA) were performed on the covariance matrix of the items to examine whether the factorial structures delineated by Gomez et al. [START_REF] Gomez | avoiding poor health or approaching good health: Does it matter? Conceptualization, measurement and consequences of health regulatory focus[END_REF] and Reuter et al. [START_REF] Reuter | Planning bridges the intention-behaviour gap: Age makes a difference and strategy use explains why[END_REF] generated adequate fit with the observed data. A model was considered adequate if the comparative fit index (CFI) and the goodness of fit index (GFI) were greater than or equal to .90, and if the root mean square residual (RMSEA) was lower than or equal to .08 [START_REF] Meyers | Applied multivariate research[END_REF]. In addition, for all measures (i.e., SOC strategy, Regulatory focus, and Exercise and diet diabetes self-care behaviors) the internal consistency of the items was examined. The internal consistency could be considered satisfactory when Cronbach alphas < .65 [START_REF] Nunally | Psychometric Theory[END_REF].

Once the reliability of the measurements was verified the descriptive statistics (mean, standard deviation, distribution) and correlations of the key variables were examined. Then, a path model for evaluating the combined contribution (direct and indirect effects) of each variable -SOC strategy, promotion focus, prevention focuson exercise, general diet, fruit and vegetable consumption, high-fat food consumption, and spacing of carbohydrates was run. In this model, age, gender, number of comorbidities and educational level were included as control variables. This path analysis was conducted by using Lisrel 9.1. The .05 level of significance was used for all statistical hypothesis testing. Beta represents the standardized regression coefficient. As for previous analyzes, the recommendations of Meyers et al. [START_REF] Meyers | Applied multivariate research[END_REF] were applied to assess the adequacy of the model (CFI and GFI ≥ .90; RMSEA ≤ .08).

Finally, using SPSS software 18.0, a bootstrapping method [START_REF] Preacher | SPSS and SAS procedures for estimating indirect effects in simple mediation models[END_REF] resample set at 5000 samples with bias-corrected 95% confidence intervals was employed to test the significance of the indirect effects. Point estimates of indirect effects are considered significant when zero is not contained in 95% confidence intervals [START_REF] Preacher | SPSS and SAS procedures for estimating indirect effects in simple mediation models[END_REF].

RESULTS

Reliability of the measures

SOC strategy. The results of the CFA showed that the one-factor model delineated by Reuter et al. [START_REF] Reuter | Planning bridges the intention-behaviour gap: Age makes a difference and strategy use explains why[END_REF] provided an excellent fit with the data: ²/df = 1.96; RMSEA =.04; GFI = 1; CFI = 1. Internal consistency was satisfactory (α =.88). An average score was thus computed for SOC strategy.

Regulatory focus. The results of the CFA showed that the two-factor model delineated by Gomez et al. [START_REF] Gomez | avoiding poor health or approaching good health: Does it matter? Conceptualization, measurement and consequences of health regulatory focus[END_REF] provided a good fit with the data: ²/df = 4.13; RMSEA =.08; GFI =.96; CFI =.98. Internal consistency was satisfactory for both the promotion focus (α =.86) and prevention focus (α =.85) subscales. Average scores were thus computed for each regulatory focus.

Exercise and diet diabetes self-care behaviors. Internal consistency was satisfactory for both exercise (α =.68) and general diet (α =.96) subscales. The mean number of days was thus computed for each of these subscales. However, the internal consistency value for the three items of the specific diet subscale was low (α =.32). This low consistency did not allow us to average the scores of these three items. This result is similar to that obtained by Toobert et al. [START_REF] Toobert | the summary of diabetes self-care activities measure: Results from 7 studies and a revised scale[END_REF]. In accordance with these authors' procedure, the items were examined separately in the subsequent analyses.

Descriptive analyses

Means, standard deviations, and Pearson's correlation coefficients for the variables are presented in Table 2. Pearson's correlations showed that (i) SOC strategy was positively correlated with exercise, general diet, fruit and vegetable consumption, and spacing of carbohydrates, and was not correlated with high-fat food consumption; (ii) promotion focus was positively correlated with SOC strategy, exercise, general diet, fruit and vegetable consumption, and spacing of carbohydrates, and was not correlated with high-fat food consumption; and (iii) prevention focus was not correlated with SOC strategy, slightly positively correlated with exercise, general diet, and spacing of carbohydrates, and was not correlated either with fruit and vegetable consumption or high-fat food consumption. The results of the distribution analyses (see Table 3) show that skewness and kurtosis coefficients are respectively < ±2 and < ±7 for all the variables. In accordance with the recommendations of Tabachnick and Fidell [START_REF] Tabachnick | Using Multivariate Statistics (5th edn)[END_REF], the normality of the distribution of our sample was thus verified for all the variables. For each variable, the proportion of the sample having checked the minimum of the scale was relatively low. Thus, our participants were relatively concerned by SOC strategy, Regulatory focus, and Exercise and diet diabetes self-care behaviors.

Hypothetical model path analysis

The hypothetical path model provided a good fit to the data (χ 2 = 107.1; df = 25; ²/df = 4.28; RMSEA =.08; GFI =.96; CFI =.93). The results indicated that when controlling age, gender, number of comorbidities, and educational level, SOC strategy was positively related with exercise (β =.20, p <.05, R 2 =.15), general diet (β =.49, p <.001, R 2 =.24), fruit and vegetable consumption (β =.27, p <.001, R 2 =.10), and spacing of carbohydrates (β =.40, p <.001, R 2 =.18), and not related with high-fat food consumption (β = -.03, ns. , R 2 =.02).

On the other hand, promotion focus was positively related with SOC strategy (β =.69, p <.001, R 2 =.49) whereas prevention focus was not associated with this variable (β = -.01, ns.).

Finally, bootstrapping analyses indicated that promotion focus was indirectly related positively with exercise (point estimate of .07, bias-corrected bootstrapped 95% CI of .11 to .40), general diet (point estimate of .08, bias-corrected bootstrapped 95% CI of .52 to .81), fruit and vegetable consumption (point estimate of .08, bias-corrected bootstrapped 95% CI of .22 to .54), and spacing of carbohydrates (point estimate of .09, bias-corrected bootstrapped 95% CI of .37 to .70) through SOC strategy. SOC strategy partially mediated the direct link between promotion focus and exercise (β =.15, p <.05), and totally mediated the direct links between promotion focus and general diet (β = -.03, ns.), fruit and vegetable consumption (β =.03, ns.), and spacing of carbohydrates (β = -.02, ns.) 1 . Significant indirect relations are illustrated in Figure 1.

DISCUSSION

The overall objective of this study was to provide physicians with knowledge about motivational processes favoring self-management for exercise and diet among patients with type 2 diabetes. First, as hypothesized, the results show that SOC strategy is positively related with exercise and most diet behaviors (including general diet, consumption of fruit and vegetables and spacing of carbohydrates). For the first time in the literature, these results show thus that SOC strategy is beneficial for exercise and diet in patients with type 2 diabetes and extend the work done on other populations (e.g., patients in orthopedic rehabilitation, [START_REF] Ziegelmann | Planning and strategy use in health behavior change: A life span view[END_REF]; older women, [START_REF] Evers | Long-term adherence to a physical activity intervention: the role of telephone-assisted vs. self-administered coping plans and strategy use[END_REF]). Furthermore, for physicians, they suggest that the SOC selfmanagement strategy (i.e., selecting and planning one's objectives by order of priority, assigning the necessary time and effort for the fulfilment of those objectives, adapting one's level of demand if difficulties arise) should be privileged in order to promote exercise and diet in this population.

Secondly, as hypothesized, the results of this study show that (i) promotion focus is positively related with SOC strategy whereas prevention focus is not and (ii) promotion focus is positively related with exercise and most diet behaviors through the SOC strategy. For the first time in the literature, these results thus identify a motivational determinant of the SOC strategy used among patients with type 2 diabetes for exercise and diet. In addition, they extend the study of Avraham et al. [START_REF] Avraham | regulatory focus and adherence to self-care behaviors among adults with type 2 diabetes[END_REF] by documenting a process in the positive link between promotion focus, exercise and diet among patients with type 2 diabetes. In so doing, they advance knowledge about the consequences of promotion focus, and provides perhaps the most detailed and specific explanation to date of the promotion focus-health behaviors relation. Furthermore, the identification of this motivator favorable to the use of the SOC strategy to promote exercise and diet makes it possible to advise physicians on what should be prioritized in patient support to favor their patients' self-management. Specifically, physicians can encourage their patients' self-management by guiding them towards promotion rather than a prevention motivation.

Concretely, in terms of methods of support (e.g., arguments to be highlighted in a therapeutic education program, interactions with patients), it first implies that physicians should adopt a positive motivational discourse centered on the health benefits of exercise and diet (e.g., "Take exercise and eat healthily to improve your quality of life", "Go ahead with exercise and healthy eating") rather than an alarmist discourse focused on the risks of complications (e.g., "Take exercise and keep to a diet to avoid worsening your state of health", "Keep active and eat healthily to avoid complications linked to diabetes"). Such alarmist discourse may be effective among patients with type 2 diabetes in promoting other self-care behaviors such as medication [START_REF] Gomez | avoiding poor health or approaching good health: Does it matter? Conceptualization, measurement and consequences of health regulatory focus[END_REF][START_REF] Leder | Self-regulation and protective health behaviour: how regulatory focus and anticipated regret are related to vaccination decisions[END_REF], but it does not favor exercise and adopting a healthy diet. To guide patients towards a motivational orientation promotion, physicians could also encourage them to focus on their progress and success in their exercise and diet (e.g., stressing the progress made, setting new targets for moving forward). By privileging this type of motivation, physicians should enable their patients to be more capable of self-managing in their treatment behaviors and so facilitate their adherence to constraining health behaviors such as exercise and diet.

Limitations

Despite the scope of these results, both theoretically and practically, some limitations of the study should be mentioned. First, the cross-sectional design of the study does not allow for causal claims among the variables. Future experimental study should thus confirm the effects of promotion focus on exercise and diet through SOC strategy. In addition, future longitudinal studies on patients with type 2 diabetes could strengthen our study by examining, over a year for example and with different measuring times, the long-term impact of promotion focus on exercise and diet, since these are two health behaviors that are beneficial when practiced over the long term. The question of maintenance over time is therefore essential.

Thirdly, the results of this study show that high-fat food consumption was not related either with promotion focus or with SOC strategy. This result is not consistent with those obtained with other items assessing diet (i.e., general diet, consumption of fruits and vegetables, and spacing of carbohydrates). Previous research also did not find the results expected regarding high-fat food consumption [START_REF] Gonzalez | Symptoms of depression prospectively predict poorer self-care in patients with Type 2 diabetes[END_REF][START_REF] Sarkar | is self-efficacy associated with diabetes selfmanagement across race/ethnicity and health literacy?[END_REF]. Gonzales et al. [START_REF] Gonzalez | Symptoms of depression prospectively predict poorer self-care in patients with Type 2 diabetes[END_REF] state that the item of the SDSCA capturing this behavior "is problematic because it gives 'red meat' and 'full-fat dairy

CONCLUSION

Beyond these limitations, these findings contribute to the literature in health psychology by evidencing for the first time among patients with type 2 diabetes (i) positive links between SOC strategy and adherence to exercise and diet, (ii) motivators of SOC strategy, and (iii) a specific process underlying the link between promotion focus and adherence to exercise and diet. In addition to these theoretical implications, this study also suggests practical steps for physicians about motivators that promote self-management for exercise and diet among patients with type 2 diabetes. Physicians should privilege an approach to patients oriented towards promotional motivation so as to favor self-management regarding exercise and diet.

H2:

  promotion focus should be positively associated with SOC strategy whereas prevention focus should be either not or negatively associated with this variable. H3: promotion focus should be indirectly positively associated with exercise and diet through SOC strategy Data were collected via a cross-sectional online self-report survey. Questionnaires were mailed by a polling institute (Dynata, https://www.dynata.com, ISO 20252:2019) to a sample of French adults with type 2 diabetes constituted on a voluntary basis. Participants were treated in accordance with the ethical requirements of the Declaration of Helsinki and the French Psychological Society with respect to consent, confidentiality, and anonymity of the answers. Prior to data collection, all participants signed an informed consent form. They were informed of the goal of the study and of their right to stop their participation at any time. The responses were anonymous, as the individuals were only identified by the day and time of completion of the questionnaire. Prior to data collection, the study was approved by the CNIL (no. 1545711).

  products' as the only two examples of high-fat foods. This item may not capture other sources of high-fat (e.g. fried foods, fast food)" (p.1105). Future research should analyze high-fat food consumption behavior in relation to promotion focus more closely by capturing other high-fat foods. Finally, glycemic control was not measured in this study. A further study examining to what extent regulatory foci and SOC strategy are related to exercise and diet behaviors by controlling the potential effect of glycemic control could complement our present results.

The effects of diabetes duration and diabetes treatment (no treatment vs. insulin vs. oral medication) were also checked in complementary analyses. The results of structural analyses showed that diabetes duration was not associated with exercise, nor with general diet, fruit and vegetable consumption, high-fat food consumption, or spacing of carbohydrates. On the other hand, the results of ANOVAs showed that diabetes treatment had no effect on exercise, general diet, fruit and vegetable consumption, or high-fat food consumption. However, patients with insulin as diabetes treatment reported more spacing of carbohydrates than patients with oral medication. Given this result, we therefore compared the contribution of promotion focus, prevention focus, and SOC strategy on spacing of carbohydrates in these two sub-groups. The results of these analyses indicated that whatever the group, SOC strategy was positively associated with spacing of carbohydrates, promotion focus was positively associated with spacing of carbohydrates through SOC strategy, and prevention focus was not associated with SOC strategy.

Regulatory focus. Gomez et al.'s [START_REF] Gomez | avoiding poor health or approaching good health: Does it matter? Conceptualization, measurement and consequences of health regulatory focus[END_REF] French regulatory focus questionnaire was used to assess participants' regulatory focus. This questionnaire is composed of a total of eight items which assess regulatory focus in the health-specific context: five items assessing promotion focus (e.g., "I do not hesitate to embrace new experiences if I think they can improve my health"), and three items assessing prevention focus (e.g., "I frequently think about the health problems I may have in the future"), presented in a random order. Participants responded on a scale from 1 = "completely disagree" to 7 = "completely agree". Because a previous study [START_REF] Schmalbach | Psychometric Properties of the German Version of the Health Regulatory Focus Scale[END_REF] has shown that a prevention focus item (i.e., "When I implement a health behavior, it's because I want to protect myself from getting sick") exhibited unsatisfactory psychometric properties, this item was slightly adjusted.

Exercise and diet diabetes self-care behaviors. The exercise and diet subscales of the Summary of Diabetes Self-Care Activities questionnaire (SDSCA, [START_REF] Toobert | the summary of diabetes self-care activities measure: Results from 7 studies and a revised scale[END_REF]) were used to assess participants' adherence to exercise and diet. Using a standard "forward-backward" translation procedure, the English-language version of the items was translated into French. Respondents are requested to indicate on how many days of the week (0-7) they performed each activity when they were not sick. Two items assessed adherence to exercise behavior (e.g., "Did you participate in at least 30 minutes of physical activity?"), two items assessed adherence to general diet (e.g., "Have you followed a healthful eating plan?"), and three items assessed adherence to three specific diet behaviors (i.e., "Did you eat five or more servings of fruits and vegetables?", "Did you eat high-fat foods such as red meat or full-fat dairy products?", "Did you space carbohydrates evenly through the day?"). As recommended by Toobert et al. [START_REF] Toobert | the summary of diabetes self-care activities measure: Results from 7 studies and a revised scale[END_REF], responses to the item related to high-fat food consumption were reversed (0=7, 1=6, 2=5, 3=4, 4=3, 5=2, 6=1,7=0).

Statistical analysis


Fabien Jourdan
email: fjourdan@lirmm.fr

Guy Melanc
email: guy.melancon@lirmm.fr

Multiscale hybrid MDS

We present a multiscale MDS method extending Chalmers . Our multiscale strategy is itself based on a O(N log N ) hybrid MDS approach. Our algorithm clearly improves over its predecessors with respect to time, while producing layouts of a comparable quality.

Introduction

Multidimensional scaling (MDS) is concerned with the representation of multi-variate data sets as sets of 2D or 3D euclidean points.That is, high dimensional distances or dissimilarities between pairs of data elements are used to compute a 2D or 3D view. Although most of the MDS literature focuses on the analysis of the representation and of its objective quality [START_REF] Kruskal | Multidimensional Scaling[END_REF][START_REF] Borg | Modern multidimensional scaling[END_REF], efforts have been devoted to the design of methods and algorithms used to actually embed the data in a euclidean space.

Among different approaches, force based placement algorithms have recently gained popularity, partly because of their underlying intuitive model and their ease of implementation. The quality of the drawings they produce can also be accounted for their popularity (see [START_REF] Basalaj | Proximity Visualization of Abstract Data[END_REF]Section 3.5.4]). Although originally designed to produce a drawing of a graph in 2D or 3D space [START_REF] Eades | A heuristic for graph drawing[END_REF][START_REF] Kamada | An algorithm for drawing general undirected graphs[END_REF][START_REF] Frick | A fast adaptive layout algorithm for undirected graphs[END_REF], force based placements generalize to abstract datasets, where they can be seen as variations of simulated annealing algorithms. When dealing with a graph, attractive forces correspond to actual links between nodes. Nodes usually are seen as charged masses and repulse each other. A placement algorithm simply runs a simulation of the corresponding physical system, until the overall system reaches a stable state which is usually admitted to provide a good view of the data. Algorithms defining variations of the physical model and of the simulation have been published in the past two decades (see [START_REF] Battista | Graph Drawing: Algorithms for the Visualization of Graphs[END_REF][START_REF] Kaufmann | Drawing Graphs[END_REF]).

These approaches can be used when dealing with abstract data where forces are defined according to dissimilarities between data elements, where dissimilarities can be computed from contextual attributes or from any other source. Authors frequently compute dissimilarities as highdimensional distance between vectors of attributes associated with data elements. As a particular case, nodes of a graph can be embedded in euclidean using this approach, by making the dissimilarity between two nodes coincide with their distance in the graph.

More recent efforts on force-based placements have addressed scalability. Indeed, a straightforward implementation of the simulation requires to iterate a visit of all N 2 pairs of data elements leading to a O(N 3 ) time complexity for the whole simulation (since most algorithms run the simulation over O(N ) iterations). These costly solutions are obviously not relevant as soon as data sets comprise thousands of elements.

Fruchterman and Reingold [START_REF] Fruchterman | Graph Drawing by Force-directed Placement[END_REF] were among the first to observe that the full visit of the N 2 pairs of elements at each iteration could be avoided, by limiting the computation of the forces to the close neighborhood of nodes, arguing that distant nodes had but a small effect on the overall displacement induced from the repulsive force. Chalmers, Morrison and Ross have fully exploited this avenue in a series of papers on MDS [START_REF] Chalmers | A linear iteration time layout algorithm for visualising high-dimensional data[END_REF][START_REF] Morrison | A hybrid layout algorithm for sub-quadratic multidimensional scaling[END_REF][START_REF] Morrison | Fast multidimensional scaling through sampling, springs and interpolation[END_REF]. Their contribution adds to the ideas introduced by Fruchterman and Reingold by restricting the simulation to a subset of randomly chosen elements in a first phase before opening the process to the full data set. Chalmers The algorithms presented here are in continuity with the improvements brought by Chalmers, Morrison and Ross. First, we describe a hybrid MDS approach inspired from their work running in O(N log(N )) and producing output of comparable quality. However, the major contribution of our work is a multiscale hybrid MDS algorithm. Indeed, the sampling approach introduced by Chalmers et al. naturally leads to a multiscale schema: instead of applying Chalmers' 1996 algorithm on the subset of size O( √ N ), we recursively apply our O(N log N ) algorithm. As we will show, this approach reveals to be significantly faster and provides output of a quality that compares well with previous approaches.

Data sampling, dissimilarities and parentfinding strategies

The 1996 paper by Chalmers [START_REF] Chalmers | A linear iteration time layout algorithm for visualising high-dimensional data[END_REF] had already decreased the complexity of traditional MDS force-based algorithms by designing a linear time iteration approach. This first improvement compared to Fruchterman and Reingold [START_REF] Fruchterman | Graph Drawing by Force-directed Placement[END_REF], in that at each iteration the placement of a data element depended on a constant number of close neighbors, thus making the overall algorithm O(N 2 ). The 2002 and 2003 papers by Chalmers, Morrison and Ross [START_REF] Morrison | A hybrid layout algorithm for sub-quadratic multidimensional scaling[END_REF][START_REF] Morrison | Fast multidimensional scaling through sampling, springs and interpolation[END_REF] introduced a new idea. In a first phase, the computation of the placement is limited to a subset S containing O( √ N ) randomly selected data elements. Applying Chalmers' 1996 quadratic MDS algorithm to the sampled elements thus induced a cost of O(N ) for this first phase. The remaining elements were then added iteratively, their position being refined through a series of neighborhood restricted force-based simulations.

Algorithm 1 Chalmers et al. 2002 and 2003 algorithms main steps : Data :

A set of data elements

V (of size N ) A dissimilarity measure δ(u, v) for all pairs u, v ∈ V Result : Euclidean coordinates (x u , y u ) for each element u ∈ V Randomly select a subset S ⊂ V of size O( √ N ) Embed S in euclidean space using Chalmers 1996 linear time per iteration MDS algorithm. For each u ∈ V \ S do : Find v ∈ S such that δ(u, v) is minimum. Place u according to v's location.
The quality of the placement is partly insured by adequately choosing where to initially position elements u not in S (u ∈ V \ S). The 2002 strategy of Chalmers et al. was to search for the candidate element in S closest to u (with respect to the dissimilarity measures), introducing a O(N (N )) phase dominating the overall complexity. Their 2003 paper modified this parent-finding strategy. Instead of going through the whole set S they first organized it into smaller components to improve the search. The set S was itself sampled to extract a constant size subset of pivots, all elements of S being assigned to buckets associated with pivots. Each pivot p had thus a list of associated buckets, b

(1) p , b (2) p , . . . , b (k)
p , naturally ordered according to the dissimilarity measures δ(-, p) or high-dimensional distances from p.

The search for the elements in S being closest to u ∈ V \ S could then be performed by selecting for each pivot p, a candidate element u p that competed against all others to act as the closest element to u. A search through all emerging candidates u p , astutely using the bucket structure, allowed to elect the element declared as closest to u.

Remark. Others have suggested improvements of forcedirected placements based on similar approaches. Kobourov et al. [START_REF] Gajer | A Fast Multi-Dimensional Algorithm for Drawing Large Graphs[END_REF], for instance, divide the original data set into layers (subsets) that are iteratively taken into account when embedding elements in euclidean space. In another paper [START_REF] Gajer | GRIP : Graph dRawing with Intelligent Placement[END_REF], Kobourov et al. also promote intelligent placement of nodes when they are initially introduced in the simulation, as opposed to a simple random initial placement, to improve the overall behaviour of the algorithm.

Further improving the parent-finding strategy

It is the parent-finding strategy we suggest to change in order to reach a O(N log N ) complexity. Instead of organizing the set S into a subset of pivots with associated buckets, we suggest to associate with each pivot p the list L p of all sampled elements ordered according to the dissimilarity δ(-, p) or high-dimensional distance from p. The cost of sorting the list associated with a pivot element p is O( √ N log(N )) on average and O(N ) in the worst case. The number of pivots being constant, the worst case complexity for this preliminary step is in O(N ). Note that, as Chalmers et al. admit, their strategy 2003 has a worst case complexity of O(N (N )) thus falling back onto their original 2002 algorithm, which happens in the rare case where all data elements merge into a single bucket.

Using ordered lists as we suggest, the search for the closest candidate to u can be performed by going through each ordered list L p to find a best candidate u p , which is done in O(log N ) time. At this point, our strategy differs from that of Chalmers et al. Indeed, the candidate u p is selected by comparing its dissimilarity to p, δ(u p , p) to the dissimilarity δ(u, p). That is, we select the element u p which is as dissimilar to p as u is. The two strategies actually bear some similarity, as we now explain. Assume for a moment that the dissimilarity map δ satisfies the triangular inequality. Assume also that d(u p , p) compares well with δ(u p , p). Then having δ(u p , p) close to δ(u, p) is equivalent to δ(u, u p ) being small, as illustrated in Figure 1. The assumptions we make actually are reasonable as soon as there are reasons to expect a good placement of the data set in the euclidean space. Experimental results presented in the next section confirm that these two strategies produce output of comparable quality.

Comparison

As expected, our O(N log N ) strategy improves on Chalmers et al. Clearly, an experimental study is mandatory in order to assess of these theoretical predictions. Indeed, a comparison of the actual running time of the algorithm showed that Chalmers et al. 2003 hybrid MDS and our O(N log N ) behave similarly. In other words, the real benefits of a O(N log N ) parent-finding strategy only appear for extremely large N . The actual running time of the three algorithms are reported in Figure 9 (section 4). (They correspond to the three intertwined curves at the top of the figure. The curve drawn in the lower part reports the running time of the multiscale variant described in the next section.) This can be explained. The two algorithms we consider here [START_REF] Morrison | Fast multidimensional scaling through sampling, springs and interpolation[END_REF] pivot-based MDS and our's) mainly differ on the parent-finding phase. All the other steps of the algorithms behave similarly. Hence, what we ought to compare is the time spent by each algorithm on the parentfinding phase. Incidentally, Chalmers et al. made a similar remark and observed that roughly 40% of the time is devoted to the parent-finding process. Now, Chalmers et al. parent-finding strategy is in O(N1/4 ) while ours is in O(log N ). Consequently, looking at the curves N 1/4 and log N is more relevant when comparing the two algorithms, since these curves predict the time the algorithms respectively spend on the parent-finding process. As Figure 3 shows, the log N strategy ultimately provides better performance. However, the benefits of the log N search strategy reveal themselves only for rather large datasets (∼ 7 × 10 5 ).

This prediction is confirmed by our experiments. Figure 4 reports the actual time spent on the parent-finding process of the O(N 5/4 ) and O(N log N ) algorithms. As one can see, the curves grow similarly but start to separate when the datasets reach the predicted size. As we will see in a later section, the improvement in time becomes clear and acts as well for smaller dataset when applied through a multiscale schema.

Quality of the output

The improvement offered by the O(N log N ) algorithm requires that we can assess of the quality of its output. Indeed, improving the running time of force-based MDS only makes sense if we remain able to produce nice views of the data. The MDS theory relies on an objective measure to assess of the quality of a placement. This measure is traditionally called the Stress of a configuration (or placement) and is defined as:

σ(X) = u,v (δ u,v -d u,v ) 2 u,v d 2 u,v (1)
where X : V → R 2 denotes the map defining the embedding of the dataset in euclidean space and where the sums run over all pairs of distinct data elements u, v. Intuitively, the lower the stress of a placement X is, the better the algorithm outputting X was able to satisfy the constraints given by the dissimilarities δ u,v . 5. We used as a benchmark a set of randomly generated small world networks of sizes ranging from 500 to 80000 nodes. The algorithm used to generate those graphs randomly selects points in a 2D euclidean space according to a multiple gaussian distribution. More precisely, the distribution used is such that it selects groups of closely situated points 1 . What should be observed is the organization of the points in loosely connected clusters.

Thus, each point in the original data comes equipped with its own 2D coordinates, which moreover determines how it links with its close neighbours and optionally with more distant points. Thus, a MDS algorithm can be fed with dissimilarities exactly equal to the 2D euclidean distance between the points. This type of data clearly acts as a benchmark for any MDS algorithm which should be able to recover the initial positions of the points, up to obvious symmetries (local combinations of rotations and reflections). Figure 7 illustrates the result of our O(N log N ) algorithm on the 500 elements network of Figure 6. Each of the clusters in the original map can be easily identified by visual inspection 2 .

Multiscale MDS

Chalmers et al. algorithms and the O(N log N ) algorithm described above are two-steps algorithms. They concentrate on a subset of randomly sampled elements, embed it in euclidean space, before developing different strategies to agglomerate the remaining data elements around this kernel configuration. As the kernel configuration is computed using Chalmers' 1996 algorithm, and because this algorithm is in O(N 2 ), the size of the sampled subset must necessarily be of size O( √ N ) (thus insuring a O(N ) cost for this first phase).

In our view, these choices follow a basic assumption : a simulation taking into account a larger range of dissimilarity values produces output of a better quality. Hence, that Chalmers' 1996 algorithm performs better on that aspect than the 2002 and 2003 variants, or the one presented 2 The experiments were run using the GVF Java API (see http://gvf.sourceforge.net). Using the framework we were able to test the whole algorithm and compute the Stress of configurations for dataset of size up to 10000. The framework allowed us to run the parent-finding phase of the algorithms with datasets having a size up to 80000. in the previous section, for instance. This assumption is obviously hard to verify and, in our opinion, is incorrect. Depending on the original dataset, it might well be better to first limit the number of dissimilarities defining the forces acting in the simulation, and incrementally expand the simulation to a larger set of dissimilarities. Intuitively, iteratively displacing elements among a crowd may be counterproductive whether incrementally assigning and updating positions may produce good quality output, as our work and that of Chalmers et al. assess. Indeed, Chalmers et al.

had already observed that their O(N 3/2 ) strategy gave output showing a lower stress than that of the 1996 algorithm. Figure 7 corroborates our claim and shows the stress values reached by all four algorithms. Clearly, the Stress of our multiscale MDS is lower than that of all three other algorithms (darker curve at the bottom). Hence, there are evidence that a multiscale (hybrid) force-based MDS can produce output of a quality at least equal if not better than any other force-based MDS. The second benefit we observed from the multiscale schema is a significant improvement in time. Recursively calling the multiscale MDS strategy to compute the kernel configuration clearly improves the performance of the algorithm. As illustrated Figure 9, the improvement brought by the multiscale schema increases with the size of the dataset. Our experiments indicate a ratio stabilizing around 0.4 in favor of the multiscale schema (thus being 60% faster than the twophase approaches).

Conclusion and future work

The O(N log N ) hybrid MDS algorithm we presented turns out to be faster than Chalmers et al. mainly concerns the parent-finding process involved in the second phase of those hybrid MDS approaches, while all other subroutines show similar performance. However, when embedding data through a multiscale schema, the improvement in time of the overall algorithm becomes clearer. Furthermore, our experiments enabled us to verify that the output produced both from the simple two-phase O(N log N ) and the multiscale approach are of a quality similar to [START_REF] Morrison | A hybrid layout algorithm for sub-quadratic multidimensional scaling[END_REF][START_REF] Morrison | Fast multidimensional scaling through sampling, springs and interpolation[END_REF] hybrid MDS algorithms.

The multiscale approach should be further examined. Indeed, at the top level, the multiscale schema is recursively applied to embed the O( (N )) sampled elements into a kernel configuration. The placement of the remain-ing elements is then accomplished just as in the simple O(N log N ) algorithm. Parameters such as the size of the kernel configuration should also be studied in order to "measure" their impact on the overall quality of the layout. Finally, a comparative study of Stress, normalized Stress and Energy [1, Section 3.1] could eventually lead to finer conclusions about the relative behavior and objective quality of all those hybrid, simple and multiscale MDS approaches.

  had already proposed a forcebased MDS algorithm running in time O(N 2 ). The 2002 and 2003 MDS algorithms by Chalmers, Morrison and Ross both used a same idea: run Chalmers' 1996 algorithm on a subset of size O( √ N ) (thus taking O(N ) time) and incrementally add all other elements. By astutely selecting the initial positions and active neighborhoods of the added elements, Chalmers et al. were able to bring the complexity of their MDS force-based approach as low as O(N 5/4 ) [14]. (More detailed comments on the work by Chalmers and his collaborators are provided in the forthcoming sections.)

Figure 1 .

 1 Figure 1. Selecting u p such that δ(u, p) approaches δ(u p , p) is similar to selecting u p such that δ(u, u p ) ∼ ǫ.

Figure 2 Figure 2 .

 22 Figure 2. Comparison of the N 5/4 and N log N curves on the scale 10 3 -10 6 .

4 Figure 3 .

 43 Figure 3. Comparison of the N 1/4 and log N curves on the scale 10 3 -10 6 .

Figure 4 .

 4 Figure 4. Comparing the actual time spent on the parent-finding phase.

Figure 5 .

 5 Figure 5. Comparison of Stress for the O(N 3/2 ), O(N 5/4 ) (grayed) and the O(N log N ) (darker) MDS strategies.

Figure 6 .

 6 Figure 6. Small world network induced from randomly selected points in 2D (500 node elements).

Figure 7 .

 7 Figure 7. MDS output obtained from the network in Figure 6.

Figure 8 .Figure 9 .

 89 Figure 8. Comparison of Stress for all four MDS algorithms considered here. The darker curve at the bottom reports Stress values reached by the multiscale MDS.

  Chalmers et al. use a different crite-rion and select the candidate u p having a smallest dissimilarity δ(u, u p ). Thus, our algorithm can be summarized as follows. Euclidean coordinates (x u , y u ) for each element u ∈ V Randomly select a subset S ⊂ V of size O( √ N ) Embed S in euclidean space using Chalmers 1996 algorithm Select k pivot elements p in S (put them in P ). For each element p ∈ P do : Sort all s ∈ S in a list L p according to δ(s, p) For each u ∈ V \ S do : Find u p in L p such that |δ(u, p)δ(u, u p )| is minimum. Place u according to u p 's location.

	Algorithm 2 Main steps of our O(N log N ) algo-
	rithm:
	Data :	A set of data elements V (of size N )
		A dissimilarity measure δ(u, v)
	Result :	for all pairs u, v ∈ V

See Figures6 and 7. The size and resolution of the images are clearly inadequate. Moreover, giving an example of a 10000 nodes network is useless here. Other images can be found at the URL http://www.lirmm.fr/∼fjourdan).

Acknowledgments

We wish to thank Nicolas Colliot for providing the small world networks used for benchmarking the algorithms.