Nicolas Gillet
email: nicolas.gillet@univ-tours.fr

Alexandre J S Morin

Tiphaine Huyghebaert

Lucie Burger

Axel Maillot

Aurélie Poulin

Elodie Tricard

Running head: NEED SATISFACTION TRAJECTORIES University Students' Need Satisfaction Trajectories: A Growth Mixture Analysis

Keywords: Need satisfaction trajectory profiles, Self-determination theory, Psychological needs, Undergraduate students, Achievement

This study examines trajectory profiles of University students over the course of a University semester defined based on global levels of psychological need satisfaction, as proposed by self-determination theory (SDT). This study also documents the implications of these trajectories for a variety of educational outcomes. A sample of 461 first-year undergraduates completed all measures three times over the course of a University semester. Longitudinal growth mixture analyses (GMA) revealed three distinct need satisfaction trajectories (Low-Decreasing, Moderate-Decreasing, and Moderate-Increasing). The Moderate-Increasing profile was associated with the highest levels of positive affect and effort. In contrast, the Low-Decreasing profile was associated with lower levels of positive affect, effort, and achievement, and higher levels of negative affect than the Moderate-Increasing profile.

Self-determination theory (SDT; [START_REF] Deci | The "what" and "why" of goal pursuits: Human needs and the selfdetermination of behavior[END_REF] proposes that the satisfaction of the three psychological needs for autonomy (the need to experience a sense of volition and psychological freedom), competence (the need to feel effective in interacting with one's environment), and relatedness (the need to feel connected with others) plays a crucial role in the emergence of selfdetermined goal-directed behaviors. Variable-centered research has supported the existence of welldifferentiated effects of the satisfaction of these psychological needs as well as autonomous (engaging in an activity out of pleasure and/or volition and choice) and controlled (engaging in an activity for internal or external pressures) motivations on multiple educational outcomes (Cordeiro, Paixão, Lens, Lacante, & Luyckx, 2016;[START_REF] Wang | Will materialism lead to happiness? A longitudinal analysis of the mediating role of psychological needs satisfaction[END_REF]. In particular, students' psychological need satisfaction appears to be able to explain important educational outcomes such as engagement and achievement (e.g., [START_REF] Jang | Can self-determination theory explain what underlies the productive, satisfying learning experiences of collectivistically oriented Korean students[END_REF], in a variety of cultural contexts (e.g., Belgium, Canada, Korea), and to be even more strongly related to these outcomes than autonomous and controlled forms of motivation (e.g., De Meyer et al., 2016;[START_REF] Kanat-Maymon | The role of basic need fulfillment in academic dishonesty: A self-determination theory perspective[END_REF]. Unfortunately, the bulk of prior research has relied on designs precluding the consideration of how need satisfaction trajectories evolve over time, and how this evolution differs across distinct subgroups of students. Moreover, these need satisfaction trajectories may also differ from one another in terms of outcomes [START_REF] Ratelle | Trajectories of psychological need satisfaction from early to late adolescence as a predictor of adjustment in school[END_REF]. Thus, the present study considers, through longitudinal growth mixture analyses (GMA), how the need satisfaction trajectories of distinct profiles of University students evolve over the course of a semester. In addition, we investigate how these distinct trajectory profiles are related to a broader range of time-varying educational outcomes (i.e., positive and negative affect, effort, expected achievement, and objective achievement) that has typically been taken into account in prior research.

Global Need Satisfaction

SDT proposes that the three psychological needs for autonomy, competence, and relatedness are essential nutriments for individuals' survival, growth, and integrity [START_REF] Deci | The "what" and "why" of goal pursuits: Human needs and the selfdetermination of behavior[END_REF], so that their satisfaction is seen as essential both for wellbeing and positive educational outcomes, an assertion that has been supported in a variety of studies and cultural contexts (e.g., [START_REF] Cordeiro | Factor structure and dimensionality of the balanced measure of psychological needs among Portuguese high school students. Relations to well-being and ill-being[END_REF][START_REF] Sheldon | Manipulating autonomy, competence, and relatedness support in a game-learning context: New evidence that all three needs matter[END_REF]. In contrast, when these needs are not satisfied, maladaptive outcomes Need Satisfaction Trajectories 2 (e.g., dropout intentions, burnout, psychological distress) are expected (e.g., [START_REF] Sulea | Engagement, boredom, and burnout among students: Basic need satisfaction matters more than personality traits[END_REF][START_REF] Taylor | Need satisfaction, work-school Need Satisfaction Trajectories 27 interference and school dropout: An application of self-determination theory[END_REF]. These conclusions appear to hold irrespective whether researchers relied on a total score of need satisfaction encompassing the three psychological needs (e.g., [START_REF] Cheon | A teacher-focused intervention to decrease PE students' amotivation by increasing need satisfaction and decreasing need frustration[END_REF][START_REF] Michou | Social achievement goals, needs satisfaction, and coping among adolescents[END_REF], or on distinct measures of the needs for autonomy, competence, and relatedness (e.g., [START_REF] Johnston | Measuring basic needs satisfaction: Evaluating previous research and conducting new psychometric evaluations of the basic needs satisfaction in general scale[END_REF][START_REF] Niemiec | The path taken: Consequences of attaining intrinsic and extrinsic aspirations in post-college life[END_REF]. The latter studies generally identified relations between need satisfaction and the outcomes of a similar magnitude across the three distinct measures of need satisfaction. This observation suggests that, at least from an outcomes perspective, the global level to which students' psychological needs are satisfied appears to be at least as critical as the extent to which each specific need is itself satisfied. This hypothesis is even consistent with SDT, which underscores the fact that "psychological health requires satisfaction of all three needs; one or two are not enough" (Deci & Ryan, 2000, p. 229).

Bifactor measurement models provide a way (e.g., [START_REF] Morin | A bifactor exploratory structural equation modeling framework for the identification of distinct sources of construct-relevant psychometric multidimensionality[END_REF][START_REF] Reise | The rediscovery of bifactor measurement models[END_REF] to directly assess this hypothesis, by relying on an explicit partition of the covariance observed among need satisfaction ratings into that explained by a global latent factor (the G-factor: global need satisfaction) underlying responses to all indicators, and a series of specific components (the S-factors: satisfaction of the needs for autonomy, competence, and relatedness) exclusive to subsets of indicators but not explained by the global component. As such, bifactor models allow researchers to simultaneously consider students' global levels of need satisfaction, together with their specific needs for autonomy, competence, and relatedness. Indeed, recent research evidence has demonstrated that bifactor measurement models provide a more accurate representation of the complex multidimensionality associated with the measurement of psychological needs when compared to more traditional exploratory or confirmatory factor analytic models (Brunet, Gunnel, Teixeira, Sabiston, & Bélanger, 2016;[START_REF] Myers | Exploratory bifactor analysis in sport, exercise, and performance psychology: A substantive-methodological synergy[END_REF][START_REF] Sánchez-Oliva | A bifactor exploratory structural equation modelling representation of the structure of the Basic Psychological Needs at Work Scale[END_REF][START_REF] Tóth-Király | Investigating the multidimensionality of need fulfillment: A bifactor exploratory structural equation modeling representation[END_REF]. Interestingly, these studies all reported the presence of a well-defined global need satisfaction G-factor underlying responses to all ratings, whereas some of them found that at least one of the S-factors retained only a negligible amount of specificity once global levels of satisfaction were taken into account (e.g., competence: Tóth-Király et Need Satisfaction Trajectories 3 al., 2017;autonomy: Sánchez-Oliva et al., 2017). Perhaps even more importantly, these studies (e.g., [START_REF] Brunet | Should we be looking at the forest or the trees? Overall psychological need satisfaction and individual needs as predictors of physical activity[END_REF][START_REF] Sánchez-Oliva | A bifactor exploratory structural equation modelling representation of the structure of the Basic Psychological Needs at Work Scale[END_REF] also reported that participants' global levels of need satisfaction tended to be the key component responsible for associations between need satisfaction measures and a variety of covariates, underscoring the importance of considering global levels of need satisfaction in future research on the emergence, development, and consequences of need satisfaction.

Changes in Psychological Need Satisfaction

The bulk of research on students' need satisfaction has relied on cross-sectional designs, or short-term longitudinal designs precluding a clear understanding of the developmental trajectories occurring at the individual level [START_REF] Ratelle | Trajectories of psychological need satisfaction from early to late adolescence as a predictor of adjustment in school[END_REF]. Longitudinal research is necessary not only to achieve a better understanding of the longitudinal stability and directionality of associations between variables, but also to achieve a proper understanding of how need satisfaction evolves over time for specific individuals [START_REF] Grimm | Growth Modeling[END_REF]. This last advantage of longitudinal research appears critical in the context of this study as SDT proposes need satisfaction to be partly situational in nature [START_REF] Vallerand | Toward a hierarchical model of intrinsic and extrinsic motivation[END_REF]. In other words, need satisfaction is seen as emerging in part from the changing characteristics of the specific life context to which a person is exposed rather than to be an inherently stable individual characteristics. As such, the ability to study how need satisfaction evolves over time for specific subgroup of individuals would provide a rich window of opportunity into the various developmental mechanisms at play in the emergence of relations between need satisfaction and a variety of important developmental, educational, and professional outcomes. More precisely, the analytical approach (GMA) taken in the present study is specifically designed to examine how the need satisfaction trajectories of distinct profiles of University students evolve over the course of a semester, and to document how these distinct trajectory profiles are related to various educational outcomes. For instance, this study will help to respond to questions such as: Is a large proportion of University students characterized by initially low levels of need satisfaction coupled with a marked decreasing trajectory (Low-Decreasing profile)? What are the educational outcomes associated with this need satisfaction trajectory? Among the few exceptions to this lack of longitudinal research, [START_REF] Wandeler | Hope and self-determination of young adults in the workplace[END_REF] conducted a 3-year longitudinal study of 414 University students. Relying on autoregressive cross Need Satisfaction Trajectories 4 lagged models, these authors were mainly interested in obtaining a clearer picture of the longitudinal rank-order stability in students' levels of need satisfaction, and the directionality of the longitudinal associations between hope and need satisfaction. Interestingly, their results supported the idea that need satisfaction was mainly situational in nature and thus only moderately stable over time (r = .33 to .49), and that hope was only very minimally related to students' levels of need satisfaction over time. [START_REF] Marchand | Motivational dynamics of children's academic help-seeking and concealment[END_REF] reported a higher level of rank-order stability among a sample of children followed up over a 7-month period (r = .49 to .67), consistent with the ideas that children (Grades 3 to 6) tend to be exposed to a more stable environment than adults. Still, they also found significant relations between teachers' report of their own motivational practices, and later levels of need satisfaction among students, a result that has been replicated in a study focusing on the impact of coaching on female adolescents gymnasts measured twice seven months apart [START_REF] Kipp | Social predictors of psychological need satisfaction and well-being among female adolescent gymnasts: A longitudinal analysis[END_REF].

In line with these results, [START_REF] Cheon | A teacher-focused intervention to decrease PE students' amotivation by increasing need satisfaction and decreasing need frustration[END_REF] demonstrated that a teacher-focused intervention could result in a positive increase in middle and high school students' levels of need satisfaction over time, although estimates of rank-order stability obtained across four waves of measurement covering a school year remained of a similar magnitude (r = .55 to .69). [START_REF] Cox | Change in physical education motivation and physical activity behavior during middle school[END_REF] obtained similar estimates of stability (r =.44 to .68) with sixth-and seventh-grade students who completed a survey containing a measure of need satisfaction on two occasions, one year apart.

Need Satisfaction Trajectories

Despite their interest, these studies are limited in their focus on rank-order stability in need satisfaction levels rather than on the estimation of individual trajectories. In particular, rank-order stability does not preclude the presence of normative increases or decreases in need satisfaction over time, and the observed levels of rank-order stability also remain low enough to suggest that substantial change still occurs over time for a substantial proportion of the students. This suggests that individual trajectories of need satisfaction may present substantial inter-individual heterogeneity which has yet to be specifically considered in research. Part of this heterogeneity may be explained by the presence of subpopulations characterized by different need satisfaction trajectories over time, which may be particularly important to identify for intervention purposes. It is noteworthy that previous studies all relied on variable-centered analyses, which rest on the assumption that all students are drawn from a Need Satisfaction Trajectories 5 single population following a similar trajectory. These investigations were thus not designed to test for the presence of developmental heterogeneity in psychological need satisfaction and to verify the extent to which this heterogeneity was related to the presence of unobserved subgroups of students following qualitatively distinct trajectories. Person-centered analyses, such as GMA, are specifically designed to explain longitudinal heterogeneity by separating a general population into profiles of students presenting qualitatively and quantitatively different trajectories [START_REF] Muthén | Beyond SEM: General latent variable modeling[END_REF]. GMA thus represents an alternative to traditional variable-centered analyses in addressing students' changes in psychological need satisfaction over time [START_REF] Morin | School life and adolescents' self-esteem trajectories[END_REF].

To our knowledge, a single study has so far relied on a person-centered approach of students' need satisfaction trajectories [START_REF] Ratelle | Trajectories of psychological need satisfaction from early to late adolescence as a predictor of adjustment in school[END_REF]. In a study of 609 students followed annually from the end of elementary school to the end of secondary school, [START_REF] Ratelle | Trajectories of psychological need satisfaction from early to late adolescence as a predictor of adjustment in school[END_REF] relied on a restricted form of GMA to identify specific developmental trajectories of students' need satisfaction over time and confirmed the presence of substantial developmental heterogeneity. More precisely, these authors revealed that the trajectories depicting how the satisfaction of students' needs for competence, autonomy, and relatedness were each best represented according to five distinct profiles. Although each of these profiles presented differing levels of need satisfaction over time, they all tended to depict mostly stable trajectories, sometimes accompanied by slight increasing or decreasing tendencies with two noteworthy exceptions. These exceptions showed that 12% of the students reported a decrease in autonomy need satisfaction at the beginning of secondary school, whereas 6% of the students reported an increase in competence need satisfaction at the same moment. [START_REF] Ratelle | Trajectories of psychological need satisfaction from early to late adolescence as a predictor of adjustment in school[END_REF] further found that trajectory group membership was closely associated for the three needs, calling once again into question the true necessity of distinguishing among these three specific needs relative to a focus on global levels of need satisfaction. Perhaps even more importantly, they also demonstrated that all trajectory classes differed from one another in terms of academic, social, and emotional adjustment. This is important, since the only way to support a substantive interpretation of latent profiles is through a process of construct validation showing that these profiles present meaningful patterns of associations with theoretically significant covariates [START_REF] Marsh | Latent profile analysis of academic self-concept dimensions: Synergy of person-and variable-centered approaches to the internal/external frame of reference model[END_REF][START_REF] Morin | A multifoci person-centered perspective on workplace affective commitment: A latent profile/factor mixture analysis[END_REF].

Need Satisfaction Trajectories 6

The Present Study

Global Need Satisfaction Trajectories

Unfortunately, despite the pioneering nature of their study, [START_REF] Ratelle | Trajectories of psychological need satisfaction from early to late adolescence as a predictor of adjustment in school[END_REF] relied on a restricted form of GMA in which it was assumed that all students would correspond exactly to the average longitudinal trajectory identified in their own profile (i.e., latent class growth analysis: [START_REF] Nagin | Analyzing developmental trajectories: A semi-parametric, group-based approach[END_REF]. This highly restrictive assumption has been shown to result in the possible over-extraction of latent profiles of students following trajectories of a similar shape but different levels [START_REF] Bauer | The integration of continuous and discrete latent variable models: Potential problems and promising opportunities[END_REF][START_REF] Muthén | Integrating person-centered and variable-centered analyses: Growth mixture modeling with latent trajectory classes[END_REF], to result in drastically different results than more flexible methods allowing for the representation of within-and between-profile heterogeneity [START_REF] Morin | Growth mixture modeling of adolescents trajectories of anxiety: The impact of untested invariance assumptions on substantive interpretations[END_REF], and, more generally, to lead to biased conclusions relative to most alternative GMA specifications [START_REF] Diallo | Impact of misspecifications of the latent variancecovariance and residual matrices on the class enumeration accuracy of growth mixture models[END_REF]. As such, it is perhaps not so surprising to note that many of the profiles identified by [START_REF] Ratelle | Trajectories of psychological need satisfaction from early to late adolescence as a predictor of adjustment in school[END_REF] presented roughly the same shape, and differed from one another mainly in terms of their global levels of need satisfaction over time.

To address this limitation, we rely on a more flexible GMA approach to investigate University students' levels of global need satisfaction over the course of a semester. The decision to rely on a sample of University students is based on three distinct considerations. First, previously reviewed research suggests that need satisfaction levels may become less stable as students get older, thus possibly maximizing our ability to identify meaningfully distinct trajectory profiles. Second, SDT proposes that need satisfaction should be most relevant for self-determined activities (i.e., activities in which individuals engage out of pleasure and/or volition and choice; [START_REF] Vallerand | Toward a hierarchical model of intrinsic and extrinsic motivation[END_REF]. Given that University studies are more typically self-determined, relative to the mandatory nature of primary and secondary education, need satisfaction should thus be most relevant for the prediction of meaningful education outcomes at this level of education. Finally, persistence in University studies is a critically important consideration for educational systems worldwide, as higher education is associated with multiple social, economic, and psychological consequences for the students themselves as well as the society as a whole [START_REF] Voelkle | University dropout: A structural equation approach to discretetime survival analysis[END_REF]. Among the key drivers of educational persistence and motivation, students' level of psychological need satisfaction appears to represent a particularly important mechanism to consider [START_REF] Jang | Can self-determination theory explain what underlies the productive, satisfying learning experiences of collectivistically oriented Korean students[END_REF]). Yet, despite the importance of University education in terms of professional, social, and vocational achievement and success, need satisfaction has received relatively little attention in prior longitudinal research.

Although prior longitudinal studies suggest that need satisfaction trajectories should exhibit some stability, they also suggest that change is possible over the course of a few months, and more likely among this older age group [START_REF] Cheon | A teacher-focused intervention to decrease PE students' amotivation by increasing need satisfaction and decreasing need frustration[END_REF][START_REF] Cox | Change in physical education motivation and physical activity behavior during middle school[END_REF][START_REF] Kipp | Social predictors of psychological need satisfaction and well-being among female adolescent gymnasts: A longitudinal analysis[END_REF][START_REF] Ratelle | Trajectories of psychological need satisfaction from early to late adolescence as a predictor of adjustment in school[END_REF][START_REF] Wandeler | Hope and self-determination of young adults in the workplace[END_REF]. Due to the scarcity of research using a personcentered approach to identify need satisfaction trajectories in the educational domain, it is difficult to propose specific hypotheses about the nature and number of the expected trajectory profiles. Still, based on [START_REF] Ratelle | Trajectories of psychological need satisfaction from early to late adolescence as a predictor of adjustment in school[END_REF], we expect these trajectories to follow distinct longitudinal profiles, although the reliance on a less restrictive GMA approach suggest that fewer than five profiles might be required to fully depict University students' need satisfaction trajectories. As such, we expect that between three and five profiles would be sufficient to adequately depict inter-individual heterogeneity in need satisfaction trajectories. Furthermore, to systematically test the assertion that need satisfaction is a mainly situational variable construct, we assess the extent to which these need satisfaction trajectory profiles depend on stable individual characteristics known to be relevant to the education area (i.e., sex, prior levels of achievement in college, grade repletion in college).

Outcomes of the Need Satisfaction Trajectories

Following from [START_REF] Ratelle | Trajectories of psychological need satisfaction from early to late adolescence as a predictor of adjustment in school[END_REF], and based on the idea that it is critical to demonstrate the criterion-related validity of the extracted trajectory profiles in relation to meaningful external covariates [START_REF] Marsh | Latent profile analysis of academic self-concept dimensions: Synergy of person-and variable-centered approaches to the internal/external frame of reference model[END_REF]Morin, Morizot et al., 2011), we also seek to contrast the extracted latent trajectory profiles in relations to students' levels of positive and negative affect, effort, expected achievement level, and true achievement level at the end of the semester. These covariates were selected based on their documented importance in the educational area, their relevance to SDT, and results from prior research on students' need satisfaction. Thus, we focus on the key educational outcomes of effort and positive affect given mounting research evidence supporting their important role in academic success [START_REF] Gillet | The mediating role of positive and negative affect in the situational motivation-performance relationship[END_REF][START_REF] Trautwein | Students' self-reported effort and time on homework in six school subjects: Between-student differences and within-student variation[END_REF].

We also consider negative affect, as this outcome is known to be a strong predictor of school dropout behavior [START_REF] Fortin | Prediction of risk for secondary school dropout: Personal, family and school factors[END_REF], which is in turn associated with Need Satisfaction Trajectories 8 numerous negative life outcomes in terms of employment and criminality [START_REF] Bjerk | Re-examining the impact of dropping out on criminal and labor outcomes in early adulthood[END_REF].

Consistent with SDT predictions [START_REF] Deci | The "what" and "why" of goal pursuits: Human needs and the selfdetermination of behavior[END_REF], prior studies have generally supported the idea that higher levels of need satisfaction tended to be associated with more adaptive academic outcomes. More specifically, research evidence has shown that psychological need satisfaction is positively related to positive affect, and negatively linked to negative affect [START_REF] Martela | Prosocial behavior increases well-being and vitality even without contact with the beneficiary[END_REF][START_REF] Martela | Meaningfulness as satisfaction of autonomy, competence, relatedness, and beneficence: Comparing the four satisfactions and positive affect as predictors of meaning in life[END_REF][START_REF] Vandercammen | The mediating role of affect in the relationship between need satisfaction and autonomous motivation[END_REF]. In addition, research has also shown that need satisfaction has positive effects on students' effort and achievement levels [START_REF] Cerasoli | Performance, incentives, and needs for autonomy, competence, and relatedness: A meta-analysis[END_REF][START_REF] Taylor | Cultural differences in the relationships among autonomy support, psychological need satisfaction, subjective vitality, and effort in British and Chinese physical education[END_REF]. [START_REF] Ratelle | Trajectories of psychological need satisfaction from early to late adolescence as a predictor of adjustment in school[END_REF] also examined the effects of developmental trajectory membership on academic, social, and emotional adjustment at the end of high school. Overall, the best adjustment scores were observed for students with high levels of need satisfaction and an increasing trajectory, followed by those with moderate levels of need satisfaction, and finally those with low levels of need satisfaction. When we summarize all of the above, it seems that we can expect educational outcomes (i.e., positive and negative affect, effort, expected achievement, and observed achievement) to be differentially related to need satisfaction trajectory profiles. Specifically, we expect that profiles characterized by higher levels and/or by increasing levels of global need satisfaction would be associated with the most adaptive outcomes, whereas profiles characterized by lower and/or by decreasing levels of global need satisfaction would be associated with the least desirable outcomes.

Method

Participants and Procedure

The sample used in this study included a total of 461 first-year undergraduate psychology students (Mean age = 18.52; SD = 0.72), including 83 males and 378 females, enrolled in a French University. The educational context and undergraduate psychology curriculum to which students are exposed in this University are similar to those proposed in other French Universities, as well as to those implemented in most Western Universities. Specifically, the academic program presented in this University is based on an introduction to the field of psychology in general, as well as to its various subfields (clinical, developmental, educational, neuropsychological, etc.) and areas of applications (personality, behaviors, cognitions, etc.) and to quantitative research methods. As for the majority of Need Satisfaction Trajectories 9

French Universities, all the students who have a high-school diploma and want to apply for admission to this University can enroll in the undergraduate psychology program. The proportion of males and females, and age distribution of this sample is aligned with those of French undergraduate psychology students. Admission at the undergraduate level is not as restrictive in France as in other countries, and undergraduate studies are tuition-free, contributing to increase the representativeness of this sample.

Participation was voluntary and all first-year students enrolled in the psychology program of this University were invited to complete a self-reported questionnaire two weeks after the beginning of the fall semester. Among these participants, 421 (91.3%) agreed to complete the questionnaire again at Time 2 (five weeks later) and 379 (82.2%) also completed the questionnaire at Time 3 (ten weeks after Time 1). At each data collection, we explained the general purpose of the study, participants provided informed consent, and then completed a 15 minutes questionnaire in class settings. Participants were ensured that their responses would be kept confidential and would not have any influence on their course grades. They were only required to provide a personal identification code to allow researchers to match their responses at each data collection point. All questionnaires were administered in French and instruments not already available in this language were adapted to French using a standardized back-translation procedure [START_REF] Hambleton | Issues, designs, and technical guidelines for adapting tests to languages and cultures[END_REF]) by a panel of experts.

Measures

Need satisfaction. Participants' need satisfaction was assessed with a questionnaire initially developed and validated by [START_REF] Gillet | Development of a basic need satisfaction scale in the sport context[END_REF] (see also [START_REF] Gillet | Effects of perceived organizational support and job characteristics on job anxiety and burnout: The mediating role of Need Satisfaction Trajectories 23 psychological need satisfaction[END_REF], which was slightly adapted to the educational context for purposes of the present study. This questionnaire includes 9 items scored using a 7-point Likert-type scale ranging from 1 (strongly disagree) to 7 (strongly agree). It assesses three dimensions (3 items each) of students' psychological need satisfaction, including autonomy (e.g., "Generally, I feel free to express my ideas and opinions"; Time 1 α = .55; Time 2 α = .65; Time 3 α = .78), competence (e.g., "Often, I feel that I am very efficient"; Time 1 α = .72; Time 2 α = .81; Time 3 α = .81), and relatedness (e.g., "I have a lot of sympathy for the persons with whom I interact"; Time 1 α = .67; Time 2 α = .75; Time 3 α = .76). As in previous research [START_REF] Michou | Social achievement goals, needs satisfaction, and coping among adolescents[END_REF], we rely in the present study on a score of global need satisfaction (Time 1 α = .75; Time 2 α = .82; Time 3 α = .83).

Positive and negative affect. Participants' levels of positive (3 items; i.e., "active", "determined", and "enthusiastic"; Time 1 α = .72; Time 2 α = .80; Time 3 α = .79) and negative (3 items; i.e., "afraid", "nervous", and "scared"; Time 1 α = .66; Time 2 α = .75; Time 3 α = .79) affect in their studies were assessed with the two relevant subscales from the Short Form of the Positive and Negative Affect Schedule [START_REF] Thompson | Development and validation of an internationally reliable short-form of the Positive and Negative Affect Schedule[END_REF]Watson, Clark, & Tellegen, 1988). Responses were made on a 5-point Likert-type scale (1-not at all to 5-very much).

Effort. Participants' level of effort was assessed using five items (e.g., "I put a lot of effort in my classes"; Time 1 α = .87; Time 2 α = .89; Time 3 α = .90) from the effort/importance subscale of the Intrinsic Motivation Inventory [START_REF] Mcauley | Psychometric properties of the Intrinsic Motivation Inventory in a competitive sport setting: A confirmatory factor analysis[END_REF]. Responses were given on 1 (strongly disagree) to 7 (strongly agree) Likert-type scale.

Expected achievement. At each time point, participants' were asked to report their expected grades (between 0 and 20) at the end of the fall semester on a 0 to 20 scale corresponding to the way class grades were provided in this University.

Observed achievement. At the end of the semester, grade transcripts were received from the administrative office of the University. The French grading system uses grades varying between 0 and 20 for each course.

Analyses Model Estimation and Missing Data

All models estimated in the present study were estimated using Mplus 7.4 [START_REF] Muthén | Mplus user's guide[END_REF] using the robust Maximum Likelihood (MLR) estimator, which provides parameter estimates, standard errors, and goodness-of-fit indices that are robust to the non-normality of the response scales used in the present study. These models were estimated in conjunction with Full Information Maximum Likelihood (FIML; [START_REF] Enders | Applied missing data analysis[END_REF] procedures to account for the relatively limited amount of missing responses present at the item level for participants who completed each specific time point (0% to 2.5%). FIML also allowed us to estimate all longitudinal models using the data from all respondents who completed at least one wave of data rather than using a listwise deletion strategy focusing only on those having answered all, or a subset, of the time waves [START_REF] Enders | Applied missing data analysis[END_REF][START_REF] Graham | Missing data analysis: Making it work in the real world[END_REF]. In total, 461 students provided a total of 1,237 time-specific ratings (M = 2.68 time-specific Need Satisfaction Trajectories 11 ratings per student), with 345 (74.8%) students completing all three time-points, 86 (18.7%) completing 2 time-points, and 30 (6.5%) completing a single time-point. FIML has comparable efficacy to multiple imputation, while being more efficient [START_REF] Enders | Applied missing data analysis[END_REF][START_REF] Graham | Missing data analysis: Making it work in the real world[END_REF][START_REF] Jeličič | Missing data methods in longitudinal studies: The persistence of bad practices[END_REF][START_REF] Larsen | Missing data imputation versus full information maximum likelihood with second level dependencies[END_REF].

Preliminary Analyses

Rather than using scale scores (the mean or sum of the items) to estimate the trajectories and their relations with predictors and outcomes, factor scores (estimated in standardized units with M = 0, SD = 1) from preliminary measurement models were used as inputs for the analyses. The measurement models for the need satisfaction variables were estimated using bifactor confirmatory factor analyses [START_REF] Holzinger | The bi-factor model[END_REF][START_REF] Reise | The rediscovery of bifactor measurement models[END_REF]. This decision is based on recent evidence showing that bifactor measurement models are naturally suited to the representation of need satisfaction (Sánchez-Oliva et al., 2017) based on SDT [START_REF] Deci | The "what" and "why" of goal pursuits: Human needs and the selfdetermination of behavior[END_REF]. [START_REF] Sánchez-Oliva | A bifactor exploratory structural equation modelling representation of the structure of the Basic Psychological Needs at Work Scale[END_REF] showed that a bifactor representation provided a way to obtain a direct and precise estimate of the global level of need satisfaction presented by participants across all three specific needs, which is used in this study to estimate participants' growth trajectories. To ensure comparability in the measures across time waves, these factors scores were saved from longitudinally invariant measurement models [START_REF] Millsap | Statistical approaches to measurement invariance[END_REF].

Although factor scores do not explicitly control for measurement errors the way latent variables do, they do provide a partial control for measurement errors [START_REF] Skrondal | Regression among factor scores[END_REF]) by giving more weight to more reliable items (i.e., items characterized by higher factor loadings and lower uniquenesses). Furthermore, factors scores are able to preserve the nature of the underlying measurement structure (e.g., invariance) better than scale scores (for additional discussions of factor scores, see [START_REF] Morin | Further reflections on disentangling shape and level effects in person-centered analyses: An illustration aimed at exploring the dimensionality of psychological health[END_REF][START_REF] Morin | Multiple-group analysis of similarity in latent profile solutions[END_REF]. Details on these measurement models and their longitudinal invariance are reported in the online supplements. The correlations between all variables used in the main analyses (i.e., the factor scores saved from these final measurement models and single item measures) are reported in Table 1.

Growth Mixture Analyses (GMA)

Need Satisfaction Trajectories 12

In this study, linear GMA models 1 with one to eight latent trajectories of global need satisfaction were estimated and compared. GMA are built from latent curve models [START_REF] Bollen | Latent curve models: A structural equation perspective[END_REF], and aim to identify subgroups of participants following distinct longitudinal trajectories [START_REF] Grimm | Growth Modeling[END_REF][START_REF] Morin | Growth mixture modeling of adolescents trajectories of anxiety: The impact of untested invariance assumptions on substantive interpretations[END_REF]. Linear GMA summarize a series of repeated measures by the estimation of random intercepts and slope factors reflecting, respectively, the initial level of the growth trajectories and the rate of change over time. In the present study, time codes on the slope factors were set to 0 at Time 1 (to allow the intercept factors to reflect global need satisfaction levels at the initial time point), 1 at Time 2, and 2 at Time 3 to reflect the presence of three equally spaced measurement points. To avoid converging on a local maxima, all of these models were estimated using 10,000 random sets of start values, 1000 iterations, and 500 solutions for final stage optimization [START_REF] Hipp | Local solutions in the estimation of growth mixture models[END_REF]. A more technical presentation of GMA is provided in the online supplements.

Current recommendations from the statistical literature are that GMA should, whenever possible, be estimated while allowing all models parameters (intercept and slope means, intercept and slope variances and covariances, and time-specific residuals) to be freely estimated in all profiles [START_REF] Diallo | Impact of misspecifications of the latent variancecovariance and residual matrices on the class enumeration accuracy of growth mixture models[END_REF][START_REF] Morin | Growth mixture modeling of adolescents trajectories of anxiety: The impact of untested invariance assumptions on substantive interpretations[END_REF]. However, this recommendation comes with the recognition that this free estimation of all model parameters is not always possible due to the tendency of these more complex models to converge on improper solutions, or not to converge at all [START_REF] Diallo | Impact of misspecifications of the latent variancecovariance and residual matrices on the class enumeration accuracy of growth mixture models[END_REF], which is typically taken to reflect overparameterization and the need to rely on simpler models [START_REF] Chen | Improper solutions in structural models: Causes, consequences, strategies[END_REF]. This was the case in the present study. In such situations, the recommendation is to implement equality constraints across profiles on model parameters to achieve a more parsimonious representation [START_REF] Diallo | Impact of misspecifications of the latent variancecovariance and residual matrices on the class enumeration accuracy of growth mixture models[END_REF]. Here, we relied on the Mplus default parameterization which sets the latent variance-covariance matrix to be invariant across latent profiles.

We also allowed the time-specific residuals to be freely estimated in each classes, but homoscedastic 1 The decision to rely on linear models is based on the initial comparison of linear versus multibase latent curve models, which revealed no evidence of non-linearity. Multibase models provide a way to test for the presence of non-linearity in models including three time points through the free estimation of the second loading on the slope factor (i.e., time code). In addition, [START_REF] Metha | Putting the individual back into individual growth curves[END_REF] showed that relying on uniform time codes when participants differ in age is proper when: (1) the regression of the intercept of a latent curve model on age is equal to the slope, and (2) the regression of the slope on age is equal to zero. In this study, participants are close in age, of the same grade level, and results supported both conditions, as shown by non-significant χ 2 difference tests (condition 1: ∆χ 2 = 0.101, df = 1; Condition 2: ∆χ 2 = 0.000, df = 1; Conditions 1 and 2: ∆χ 2 = 0.122, df = 2). across time points [START_REF] Li | On inclusion of covariates for class enumeration of growth mixture models[END_REF][START_REF] Enders | The impact of misspecifying class-specific residual variances in growth mixture models[END_REF]. This specification of the residuals is consistent with a multilevel operationalization of growth models, and results in GMA which are assumed to provide an equally efficient representation of the all repeated measures, while allowing this explanatory power to differ across latent profiles.

Controls.

Tests aiming to determine whether demographic control variables needed to be retained for the subsequent analyses were conducted.

Outcomes of Profile Membership. Time-varying outcomes levels measured at each time point (positive affect, negative affect, effort, expected achievement, and observed achievement) were contrasted across profiles using a model-based approach proposed by [START_REF] Lanza | Latent class analysis with distal outcomes: A flexible model-based approach[END_REF] and implemented through the Auxiliary (DCON) function [START_REF] Asparouhov | Auxiliary variables in mixture modeling: Three-step approaches using Mplus[END_REF].

Results

Unconditional Models

The results from the estimation of the alternative GMA solutions converged on a three-profile solution. The rationale supporting this solution is reported in the online supplements. This solution is graphically presented in Figure 1, and specific parameter estimates are reported in Table S5 of the online supplements. It is important to keep in mind that these trajectories were estimated on the basis of invariant factor scores with a mean of 0 and a standard deviation of 1 obtained on the global need satisfaction factor in the context of preliminary analyses reported in the online supplements. This means that 0 corresponds to the average level of global need satisfaction, and that deviations from this mean are expressed in standard deviation units.

In comparing the profiles, it is first informative to note that Profiles 1 and 2 are characterized by an initial level (i.e., the mean on the intercept factor) that are slightly higher than average by about .2 SD, and virtually identical. However, clear differentiations occur between these two profiles as the semester evolves leading to a difference of about .50 SD in global need satisfaction levels at the end of the semester. Profile 1 characterizes 11.50% of students with initially moderate levels of global need satisfaction, which are characterized by a slight decreasing tendency over time (corresponding to -.103 SD units per time point). In contrast, Profile 2 characterizes 61.39% of the students presenting initially moderate levels of global need satisfaction, which are characterized by a slight increasing tendency Need Satisfaction Trajectories 14 over time (.106 SD units per time point). Finally, Profile 3 is the most concerning, and characterizes 27.11% of students presenting an initially low levels of global need satisfaction coupled with a marked decreasing trajectory (-.305 SD units per year).

Statistics regarding the classification accuracy of the students into their most likely profile are reported in Table S6 of the online supplements, and indicate a relatively high level of classification accuracy for members of all three profiles, ranging from 80.2% for the Low-Decreasing profile (3), to 87.8% for members of the Moderate-Decreasing profile (1), and to 90.4% for members of the Moderate-Increasing profile (2), consistent with a relatively high entropy value (.717)2 .

Controls

Once the optimal number of profiles has been selected, we conducted a series of tests aiming to determine whether demographic controls assessed at Time 1 (sex, prior level of achievement, grade repetition) needed to be retained for subsequent analyses as time-invariant predictors (TIP). These controlled variables were included using the start values from the final retained unconditional GMA model [START_REF] Diallo | The impact of total and partial inclusion or exclusion of active and inactive time invariant covariates in growth mixture models[END_REF][START_REF] Morin | Multiple-group analysis of similarity in latent profile solutions[END_REF] and a series of alternative models were contrasted, following recommendations from [START_REF] Diallo | The impact of total and partial inclusion or exclusion of active and inactive time invariant covariates in growth mixture models[END_REF] and previously implemented in applied research by Morin and colleagues [START_REF] Morin | Growth mixture modeling of adolescents trajectories of anxiety: The impact of untested invariance assumptions on substantive interpretations[END_REF]2013;[START_REF] Morin | Academic achievement and adolescent smoking: A general growth mixture model[END_REF]. First, a null effects model was estimated in which the effects of the controls on the probability of membership in all profiles, as well as on the growth factors, were constrained to be zero. Second, a first alternative model was estimated in which the controls were allowed to predict profile membership through a multinomial logistic regression. Tests were then conducted on additional models in which controls were also allowed to influence within-profile variation in the intercepts and slopes of the trajectories (via a multiple regression equation), and in which these effects were allowed to vary from one profile to another. Results from models incorporating controls are reported in the bottom section of Table S4 and support the null effects model. Examination of the detailed parameters estimates from these alternative models supports this conclusion regarding the lack of meaningful associations between the controls and the profiles, and the decision to exclude controls from further analyses.

Time-Varying Outcomes

Results from the comparison of the time-specific outcomes across profiles are reported in Table 2.

These results reveal that the three profiles are very clearly differentiated on the outcomes considered, and that the pattern of associations between profiles and outcomes differs across outcome. Levels of negative affect are undistinguishable between the Moderate-Decreasing and Moderate-Increasing profiles, although both of these profiles present lower levels of negative affect than the Low-Decreasing profile. However, levels of positive affect are highest among the Moderate-Increasing profile, followed by the Moderate-Decreasing profile, and finally by the Low-Decreasing profile, with all pairwise comparisons being significant. Furthermore, levels of effort are undistinguishable between the Moderate-Decreasing and Low-Decreasing profiles, although both of these profiles present lower levels of effort than the Moderate-Increasing profile. Finally, it is interesting to note that expected achievement levels generally follow the same pattern of differences than observed achievement at the end of the semester, at least at the first and last time points where these outcomes are lower among the Low-Decreasing profile than among the other two profiles. It is however interesting to note that in the middle of the semester, students corresponding to the Moderate-Decreasing profile also report higher expected achievement levels than members of the Low-Decreasing profile.

With the sole exception of the aforementioned difference in expected achievement levels observed in the middle of the semester, most of these differences are stable over time. Taken together, these results suggest that, when compared to the other profiles, members of the Moderate-Increasing profile present higher levels of positive affect and invest more effort in their studies. In comparison to this Moderate-Increasing profile, members of the Low-Decreasing profile display less positive affect and more negative affect, invest less effort in their studies and, possibly as a result of these lower effort levels, present lower levels of achievement. Finally, members of the Moderate-Decreasing profile also present lower levels of positive affect and invest less effort in their studies, but display comparable levels of negative affect and achievement as members of the Moderate-Increasing profile.

Discussion

Prior studies have shown that the satisfaction of the three psychological needs for autonomy, Need Satisfaction Trajectories 16 competence, and relatedness tended to be associated with more positive academic outcomes (e.g. [START_REF] Cordeiro | Factor structure and dimensionality of the balanced measure of psychological needs among Portuguese high school students. Relations to well-being and ill-being[END_REF][START_REF] Wang | Will materialism lead to happiness? A longitudinal analysis of the mediating role of psychological needs satisfaction[END_REF]. However, in the educational area, prior research has largely ignored individual developmental trajectories [START_REF] Ratelle | Trajectories of psychological need satisfaction from early to late adolescence as a predictor of adjustment in school[END_REF]. To better understand how need satisfaction evolves over time for specific subgroups of students, the present study was designed to inform how the need satisfaction trajectories of distinct profiles of University students evolved over the course of a semester and related to various educational outcomes (i.e., positive and negative affect, effort, expected achievement, and objective achievement).

Need Satisfaction Trajectories

Recently, [START_REF] Ratelle | Trajectories of psychological need satisfaction from early to late adolescence as a predictor of adjustment in school[END_REF] identified longitudinal trajectories of students' need satisfaction and confirmed the presence of substantial developmental heterogeneity (five distinct profiles for each specific need). However, they relied on a restricted form of GMA (i.e., latent class growth analysis) in which it was assumed that all students would correspond exactly to the average longitudinal trajectory identified in their own profile, and treated each need separately without considering evolution in students' global levels of need satisfaction. In the present study, we relied on a more flexible GMA approach and identified three distinct profiles of students' global need Interestingly, our results also supported the idea that need satisfaction is at least partly situational in nature and thus fluctuates over time [START_REF] Cheon | A teacher-focused intervention to decrease PE students' amotivation by increasing need satisfaction and decreasing need frustration[END_REF][START_REF] Cox | Change in physical education motivation and physical activity behavior during middle school[END_REF][START_REF] Marchand | Motivational dynamics of children's academic help-seeking and concealment[END_REF].

These findings are the first to document longitudinal trajectories of University students' global need satisfaction over the course of a semester. Our participants were first-year undergraduate psychology students who have recently undergone the transition from high school to University, with the accompanying changes in teachers and learning structure, as well as classroom composition that changes over class. Some might even experience more than one simultaneous life transition when, for Need Satisfaction Trajectories 17 instance, they had to move to another neighborhood or city to enter University. More generally, the freshman year is known to represent a challenging life transition accompanied by major changes in students' educational and social environments, incorporating new and unfamiliar academic tasks and learning situations, and evolving social networks (De Clercq, Galand, & Frenay, 2017;[START_REF] Perry | Academic control and action control in the achievement of college students: A longitudinal field study[END_REF]. These changes might influence students' psychological need satisfaction trajectories and explain why groups of students reported changing need satisfaction trajectories over the course of their first University semester. As such, the specific context of the freshman year might have generated slightly more elevated levels of instability in need satisfaction trajectories than what would be observed across more stable life contexts.

Future longitudinal research would be needed to more clearly address this possibility, and to better document the time-invariant (personal characteristics) and time-varying (associated with the academic and social life contexts) characteristics that predict membership into these various trajectory profiles. Interestingly, our results showed that these trajectory profiles were independent from students' demographic characteristics (sex, prior level of achievement, grade repetition), suggesting that changing life contexts might be particularly relevant to consider. For instance, in line with recent studies [START_REF] Taylor | Cultural differences in the relationships among autonomy support, psychological need satisfaction, subjective vitality, and effort in British and Chinese physical education[END_REF] showing that teachers' behaviors relate to students' levels of need satisfaction, these additional investigations might look at the impact of University teachers' autonomysupportive and controlling behaviors. Similarly, following from studies supporting the key role of the social context in supporting students' need satisfaction in the academic area [START_REF] Lu | The associations between perceived maternal psychological control and academic performance and academic self-concept in Chinese adolescents: The mediating role of basic psychological needs[END_REF], it might be interesting to look at the role of changes in students' lives circumstances (moving away from family, integrating new peer groups, etc.) in predicting profile membership.

Outcomes of the Need Satisfaction Trajectories

Still, our findings clearly support the practical importance of the identified need satisfaction trajectories in the prediction of academic outcomes, showing well-differentiated associations between profile membership and the various outcomes considered in this study. First, the Moderate-Increasing profile was associated with the highest levels of positive affect and effort. Second, members of the Low-Decreasing profile tended to display less positive affect and more negative affect, invest less effort in their studies, and present lower levels of achievement than those from the Moderate-

Need Satisfaction Trajectories 18

Increasing profile. Finally, the Moderate-Decreasing profile was associated with lower levels of positive affect and effort than the Moderate-Increasing profile, while these two profiles could not be distinguished from one another on negative affect and achievement. These results support SDT's propositions [START_REF] Deci | The "what" and "why" of goal pursuits: Human needs and the selfdetermination of behavior[END_REF] in demonstrating the positive effects of global levels of need satisfaction. They are also well aligned with those from prior studies conducted in the educational area showing that higher levels of global or specific need satisfaction were particularly beneficial in terms of social, academic, and personal-emotional adjustment [START_REF] Jang | Can self-determination theory explain what underlies the productive, satisfying learning experiences of collectivistically oriented Korean students[END_REF][START_REF] Ratelle | Trajectories of psychological need satisfaction from early to late adolescence as a predictor of adjustment in school[END_REF].

It was interesting to note that the Moderate-Increasing profile tended to be associated with more positive outcomes, at least in terms of positive affect and effort, than the Moderate-Decreasing one. Furthermore, our findings demonstrate that students with comparable initial levels of need satisfaction ended up displaying similar levels of negative affect and achievement irrespective of whether their degree of satisfaction increased or decreased over the course of the semester. Future studies would be needed to verify the possibility that the negative effects of decreases in need satisfaction in terms of achievement and negative affect can be temporally lagged and only emerge in the following semester. Clearly, the adoption of a longer term longitudinal perspective would be useful in helping to unpack the mechanisms at play in these associations.

These results also show that pattern of associations between need satisfaction trajectories and outcomes differs as a function of the type of outcome, and are well aligned with those from [START_REF] Sheldon | Manipulating autonomy, competence, and relatedness support in a game-learning context: New evidence that all three needs matter[END_REF] who similarly found that the effects of need satisfaction on performance and positive affect were not comparable. In this study, we focused on students' achievement (expected performance and objective grades), emotional engagement (positive and negative affect), and behavioral engagement (effort). To confirm the differential effects of need satisfaction trajectories on a wider range of educational outcomes, we encourage researchers to conduct additional research also considering markers of cognitive engagement, such as students' critical thinking. Indeed, mounting research evidence supports the role of students' engagement as a key determinant of academic success that is easier to target in intervention than achievement itself (e.g., [START_REF] Van Rooij | Secondary school students' engagement profiles and their relationship with academic adjustment and achievement in university[END_REF]. Future research may also consider other outcomes representing, for instance, students' emotional (e.g., boredom), cognitive (e.g., disorganization), and behavioral (e.g., dropout intentions) Need Satisfaction Trajectories 19 disengagement from their studies. The importance of boredom and disorganization stems from research identifying these dimensions as negatively related to many desirable academic outcomes, including achievement (e.g., [START_REF] Pekrun | Boredom and academic achievement: Testing a model of reciprocal causation[END_REF]. Moreover, dropout intentions are strongly related to school dropout behavior, which is in turn associated with numerous negative life outcomes such as decreased employment rates, and increased criminal activities [START_REF] Bjerk | Re-examining the impact of dropping out on criminal and labor outcomes in early adulthood[END_REF].

Limitations and Directions for Future Research

The present study has some limitations. First, we relied on self-report measures, with the exception of observed achievement, and such measures can be impacted by social desirability and selfreport biases. We thus encourage researchers to conduct additional research using more objective dropout data as well as informant-reported (e.g., teacher) measures of student engagement as ultimate outcomes. Second, although need satisfaction trajectory profiles were not found to be influenced by sex, prior levels of achievement in college, and grade repetition, it would be interesting for future research to consider a more diversified set of determinants of students' need satisfaction profiles. For instance, it would be interesting to determine the role of social agents within (e.g., teachers, friends) and outside (e.g., parents) the University in explaining the students' need satisfaction trajectories [START_REF] Ratelle | Trajectories of psychological need satisfaction from early to late adolescence as a predictor of adjustment in school[END_REF]. Likewise, it would also be interesting to look more carefully at the possible impact of the course curriculum, learning context, and even pedagogical approach used in a greater diversity of University programs and country so as to be able to identify possible controllable levers for the improvement of students' need satisfaction. Third, the need satisfaction trajectories reported in the present study were observed in first-year undergraduate psychology students enrolled in a French University. Future research should examine whether the same trajectories emerge in samples from different academic levels (primary, secondary, graduate), countries, and cultural backgrounds.

Fourth, it is important to keep in mind that the current results are intimately related to the time lag that was considered in this study: One university semester (10 weeks). We found evidence that need satisfaction was quite stable over a period of five (r = .894 to .889) to ten (r = .691) weeks, which is in line with estimates reported in previous studies (e.g., [START_REF] Cheon | A teacher-focused intervention to decrease PE students' amotivation by increasing need satisfaction and decreasing need frustration[END_REF][START_REF] Marchand | Motivational dynamics of children's academic help-seeking and concealment[END_REF], and supports the idea that studying change in need satisfaction requires relatively long time Need Satisfaction Trajectories 20 lags. Still, longitudinal research always needs to be interpreted in relation to a specific time frame [START_REF] Cole | Testing mediational models with longitudinal data: Questions and tips in the use of structural equation modeling[END_REF]. Thus, relying on a much shorter time frame (e.g., daily diary study) may have allowed us to detect finer associations occurring at the state level of the constructs being studied, whereas relying on longer time frames might have revealed relations occurring at a more fundamental trait level. Conversely, these alternative time frames might have hidden the currently observed relations. Ultimately, longitudinal evidence remains stronger in terms of clarifying the directionality of associations than cross-sectional research, but needs to be built incrementally from an accumulation of studies exploring alternative time frames. Finally, due to limitations posed by the sample size, the current study solely focused on students' trajectories of global need satisfaction, without simultaneously considering their more specific trajectories of competence, relatedness, and autonomy need satisfaction. Despite recent evidence providing tentative support to the key role played by global need satisfaction [START_REF] Brunet | Should we be looking at the forest or the trees? Overall psychological need satisfaction and individual needs as predictors of physical activity[END_REF][START_REF] Sánchez-Oliva | A bifactor exploratory structural equation modelling representation of the structure of the Basic Psychological Needs at Work Scale[END_REF], it remains critical for future studies to more closely consider students more specific need satisfaction trajectories (e.g., [START_REF] Ratelle | Trajectories of psychological need satisfaction from early to late adolescence as a predictor of adjustment in school[END_REF] and how they related to key developmental and educational outcomes.

Practical Implications

From a practical perspective, our findings suggest that teachers should be particularly attentive to students displaying low and decreasing levels of global need satisfaction (Low-Decreasing profile) as these appear to be at risk for a variety of educational difficulties. Such a trajectory might be identified by using a short version of a scale assessing psychological need satisfaction at the beginning of each semester for example, which could be presented as a request for feedback from the teacher. This way, teachers could easily become aware of students, or groups of students, displaying low levels of need satisfaction and attempt to intervene before the emergence of undesirable outcomes. For example, teacher-focused interventions and support systems might be made available for teachers in order to help them increase students' levels of need satisfaction over time [START_REF] Cheon | A teacher-focused intervention to decrease PE students' amotivation by increasing need satisfaction and decreasing need frustration[END_REF]. In the existing literature, numerous studies have shown that autonomy-supportive teaching behaviors were positively related to psychological need satisfaction [START_REF] Jang | Can self-determination theory explain what underlies the productive, satisfying learning experiences of collectivistically oriented Korean students[END_REF][START_REF] Sheldon | Manipulating autonomy, competence, and relatedness support in a game-learning context: New evidence that all three needs matter[END_REF]. Thus, having teachers displaying higher levels of autonomy-supportive behaviors could be associated with a greater likelihood of membership into the most desirable profile (Moderate-Increasing). Incorporating Need Satisfaction Trajectories 21 autonomy-supportive structure into classes may thus be an important pedagogical consideration. [START_REF] Jang | A new autonomy-supportive way of teaching that increases conceptual learning[END_REF] recently tested the educational utility of "teaching in students' preferred ways" as a new autonomy-supportive teaching strategy. Results revealed that students who received a preferred way of teaching (i.e., teachers take their students' perspective and adjust how they deliver a lesson plan so that it aligns with students' preferred ways of teaching) perceived their teacher as more autonomy-supportive and had more positive outcomes. Thus, "teaching in students' preferred ways" represents a way of teaching that may increase students' psychological need satisfaction. Note. Trajectories are estimated on the basis of invariant factor scores with a mean of 0 and a standard deviation of 1 obtained on the global need satisfaction factor in the context of preliminary analyses reported in the online supplements.

  satisfaction: (1) Students with initially moderate levels of global need satisfaction, which are characterized by a slight decreasing tendency (Moderate-Decreasing profile); (2) students with initially moderate levels of global need satisfaction, which are characterized by a slight increasing tendency (Moderate-Increasing profile); and (3) students with initially low levels of global need satisfaction coupled with a marked decreasing trajectory (Low-Decreasing profile). Students' need satisfaction thus fluctuated in a heterogeneous fashion over the course of a University semester.

Figure 1 .

 1 Figure 1. Estimated Growth Trajectories for the Three Need Satisfaction Profiles.

The entropy indicates the precision with which the cases are classified into the various profiles. Higher entropy values (e.g., above .700) indicating a greater level of accuracy, whereas lower levels (e.g., under .500) indicate poorer levels of accuracy.

 Differences, 105, 312-317. Watson, D., Clark, L.A., & Tellegen, A. (1988). Development and validation of brief measures of positive and negative affect: The PANAS scales. Journal of Personality andSocial Psychology, 54, 1063-1070. Need Satisfaction Trajectories 31 11.106 [10.547; 11.665] 11.527 [11.355; 11.699] 10.583 [10.367; 10.799] 1 = 2; 1 = 3; 2 > 3 Objective Grades Time 3 9.766 [8.798;10.734] 10.698 [10.388;11.008] 9.798 [9.357;10.239] 1 = 2; 1 = 3; 2 > 3 Note. * Variables identified by an asterisk are factor scores saved in standardized units from preliminary measurement models.