A Tutorial on Centering in Cross-Sectional Two-Level Models

Measurement in Physical Education and Exercise Science, 14: 275–294, 2010 Copyright © Taylor & Francis Group, LLC ISSN: 1091-367X print / 1532-7841 online DOI: 10.1080/1091367X.2010.520247

A Tutorial on Centering in Cross-Sectional Two-Level Models Nicholas D. Myers Department of Educational and Psychological Studies, University of Miami, Coral Gables, Florida

Ahnalee M. Brincks Department of Epidemiology and Public Health, University of Miami, Coral Gables, Florida

Mark R. Beauchamp The University of British Columbia, School of Human Kinetics, Vancouver, British Columbia, Canada

The primary purpose of this tutorial is to succinctly review some options for, and consequences of, centering Level 1 predictors in commonly applied cross-sectional two-level models. It is geared toward both practitioners and researchers. A general understanding of multilevel modeling is necessary prior to understanding the subtleties of centering decisions. A review of some high-quality journals within the broad discipline of exercise science provides evidence that multilevel modeling is used relatively infrequently in this field. Therefore, a secondary purpose is to introduce Measurement in Physical Education and Exercise Science readers to some core facets of multilevel modeling within the framework of this tutorial. A relevant dataset is used to demonstrate potential consequences of different centering decisions within a multilevel model. Depending on the model and the data, different centering decisions can exert non-trivial influence on the meaning of some model parameters, results from fitting the model, and subsequent conclusions. Key words: grand mean centering, group mean centering, raw score, centering within cluster, multilevel modeling, hierarchical linear modeling

INTRODUCTION The primary purpose of this tutorial is to review some options for, and consequences of, centering Level 1 predictors in commonly applied cross-sectional two-level multilevel models (MLMs). For the purpose of this tutorial, centering is defined as subtracting a number (e.g., a mean) from a raw score (RAS). Centering can be used to more clearly investigate research questions of interest Correspondence should be sent to Nicholas D. Myers, Department of Educational and Psychological Studies, University of Miami, 311-E Merrick Bldg., Coral Gables, FL, 33124-2040. E-mail: [email protected]

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within MLMs. A review of some high-quality journals within the broad discipline of exercise science provides evidence that multilevel modeling (MLM) is used relatively infrequently in this field. Therefore a secondary purpose is to introduce Measurement in Physical Education and Exercise Science (MPEES) readers to some core facets of MLM within the framework of this tutorial. Cross-sectional data in exercise science are often nested (an equivalent expression is “clustered”). Examples of such clusters might include athletes nested within teams or students nested within physical education classes. Nested data can also be described by hierarchical level (e.g., when athletes are nested within teams, this data structure can be described as athletes/individuals at Level 1 and teams/clusters at Level 2). When a non-trivial percentage (e.g., 5%; Julian, 2001) of variance in a dependent variable (e.g., satisfaction with a coach or teacher) is attributable to at least one cluster variable (e.g., a variable that identifies what team each athlete is nested within), imposing a linear regression model is often problematic from both an empirical perspective (e.g., increased Type I error—incorrectly rejecting a true null hypothesis) and a theory-building perspective (e.g., inability to model relationships at different levels). MLMs partition the variance of the dependent variable(s) into levels (e.g., athlete level and team level) and allow for the possibility of modeling at each level (Raudenbush & Bryk, 2002). Centering predictors is an important aspect of MLM that has long been misunderstood (Enders & Tofighi, 2007). Examples of such misunderstanding cited in Enders and Tofighi include complete omission of centering decisions (rendering regression coefficients extremely difficult or impossible to interpret) and basing centering decisions on only empirical information instead of making such decisions based on substantive research questions. Imposing MLMs, irrespective of centering decisions, is a relatively recent development and can be difficult to implement well. Such difficulty may partially explain why this review of fulllength articles published in the Journal of Exercise and Sport Psychology (JSEP), Journal of Sports Sciences (JSS), Psychology of Sport and Exercise (PSE), and Research Quarterly for Exercise and Sport (RQES) over the last decade, 1998–2008 (except for PSE, which began publishing in 2000), identified a total of 25 articles (JSEP = 6, JSS = 3, PSE = 11, RQES = 5) that imposed MLMs. Thus, MLM appears to be used infrequently in exercise science. Within the majority (approximately 75%) of the relatively small number of identified studies, the centering decision(s) was/were not made explicit. Thus, it appears that when MLM is used in exercise science, information regarding centering decision(s) often is not provided. (The table from this review is not provided due to space limitations but is available upon request.) The focus of this tutorial, therefore, is toward interpretation of MLMs under varying centering decisions through use of a familiar dataset. A full review of the mathematical details explaining why differences can occur in both parameter estimates and the stability of estimation due to centering decision(s) is beyond the scope of this tutorial. This tutorial is organized as follows. First, an example from a practitioner’s perspective is provided. Second, a historical research outline is provided, which identifies key citations upon which the more technical information of this tutorial is based (the technical material is geared toward researchers). Within this outline, some key concepts (e.g., common centering options) are introduced, and the specific focus of this tutorial is made explicit. Third, relevant MLMs are put in a familiar context under varying centering decisions and interpreted more fully for both researchers and practitioners. The final section, relevant recommendations and examples, is written without any statistical symbols and with minimal abbreviations.

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As with any methodological area with more than one set of contributors across time, notation, terminology, and software for MLMs vary. The notation and terminology used in this tutorial are most consistent with Raudenbush and Bryk (2002). The software used in the subsequent examples was Hierarchical Linear and Non-linear Modeling version 6 (HLM; Raudenbush, Bryk, Cheong, Congdon, & du Toit, 2004), with maximum likelihood utilized as the estimation method. Data used in the subsequent examples are available on request from the first author.

THE EXAMPLE FROM A PRACTITIONER’S PERSPECTIVE Suppose that an athletic director (AD) supervised a large number of team sports that were each led by a different head coach. Over time, the AD noticed that athletes who were not satisfied with their head coach often quit participating in the sport and that teams that were not satisfied with their head coach often failed to develop into a cohesive unit. The AD further noticed that athlete satisfaction with a head coach varied within teams (i.e., different athletes within the same team sometimes differed dramatically in how satisfied they were with the same head coach) and between teams (i.e., despite some athlete-level differences within teams, on average, some teams were much more satisfied with their head coach than other teams were with their head coach). Suppose that the AD believed that a key to athlete/team satisfaction with the head coach was athlete/team perception of the head coach’s ability to positively affect the psychological mood and psychological skills of his/her athletes/team (i.e., athlete/team motivation competency). The AD wanted to conduct a study that explored if, and to what degree, athlete/team satisfaction with the head coach was predicted by athlete/team perceptions of their head coach’s ability to affect the psychological mood and psychological skills of his/her athletes. The AD also wanted to make sure that the study accounted for the team context in which these data were collected. Key questions in the two previous paragraphs in relation to this tutorial include: (a) why might MLM be an appropriate methodological framework for the AD’s study and (b) how can centering decisions within MLM be used to help answer key research questions in the AD’s study? Very general answers to these questions are: (a) because MLM would allow the team context to be represented in the model in a more realistic way (e.g., perhaps the relationship between motivation competency and satisfaction with the head coach is different within different teams) than a typical linear regression model would reveal and (b) MLM would separate the relationship between motivation competency and satisfaction with the head coach at the athlete level and at the team level within a single model (e.g., perhaps the relationship is different at the athlete level than it is at the team level). In the paragraphs that follow, a more thorough treatment of the consequences of centering decisions in MLMs is provided from both a researcher’s perspective and a practitioner’s perspective by utilizing the example provided in this section.

A HISTORICAL RESEARCH OUTLINE Seminal MLM Texts and Some Common Notation In the late 1980s and early 1990s, several MLM texts were presented (e.g., Bock, 1989; Bryk & Raudenbush, 1992; Goldstein, 1987). In each of these texts, a primary focus was given to

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cross-sectional two-level MLMs with a normally distributed continuously observed dependent variable. Application of this type of MLM is most common within the exercise science literature. Therefore, this tutorial focuses on this particular application of MLM. A paragraph on common notation for independent variables in the MLM literature is warranted prior to encountering this notation within the complete, more complex MLM notation. Independent variables at Level 1 often are generically denoted as Xqij (or, more specifically, as Motivation Competencyij , for example), where q allows different predictors to be generally labeled as X 1ij , X 2ij , . . ., Xkij ; i denotes a particular observation at Level 1 (e.g., a particular athlete), and j denotes a particular cluster at Level 2 (e.g., a particular team). Independent variables at Level 2 often are denoted in one of two ways. When a variable is measured at Level 2, it is often denoted as Wqsj , where q communicates which Level 1 coefficient(s), β qj (e.g., β 0j and/or β 1j and/or β kj ), a given variable was specified to predict; s allows different predictors to be uniquely identified (e.g., Wq1j , Wq2j , etc.). When a variable is measured at Level 1 and then aggregated at Level 2, it is often denoted as Xq·j (or, more specifically, as Team Motivation Competency.j , for example) to make clear that the variable was aggregated across individuals within each group (subscript s is commonly omitted). Subscripts are dropped in this tutorial, and often in practice, when their function is redundant and/or otherwise handled. Seminal Scholarship on Centering within MLM Approximately parallel in time to the publication of the MLM texts described in the previous section, literature has also emerged on the consequences of centering decisions for Level 1 predictors within MLMs (e.g., Longford, 1989a, 1989b; Plewis, 1989; Raudenbush, 1989a, 1989b). The primary options debated were the use of RAS, grand mean centering (GMC or ∗ ), and centering within cluster (CWC or ∗∗ ). The instigating article for this sequence of brief communications (Raudenbush, 1989a) extended the work of Cronbach and Webb (1975), who utilized “a forerunner of today’s multilevel methods” (Raudenbush, 1989a, p. 10). Centering issues have been, and likely will continue to be, integral to the use of MLMs. Centering options for Level 2 predictors do not include CWC because there are no upperlevel units for these observations to be nested within. Centering decisions at Level 2 should be informed by the same considerations as in fixed effects regression. From this point forward, all Level 2 predictors are GMC when data are modeled. Subsequent Influential Scholarship on Centering in MLM The literature on centering in MLMs was expanded and addressed formally by Kreft, de Leeuw, and Aiken (1995), where the primary focus was to mathematically define conditions under which imposing different centering options (RAS, GMC, and CWC) for Level 1 predictors in two common specifications of MLMs resulted in theoretically equivalent models. Kreft et al.’s (1995) “theoretically” equivalent disclaimer allowed for the possibility of empirical inequalities that can result from the actual fitting of theoretically equivalent models to data (e.g., γ 00 represents a population value whereas γˆ00 represents an estimated value). The Kreft et al. (1995) disclaimer can explain cases where the results for theoretically equivalent models that appear later in this tutorial did not yield exactly the same results (i.e., ≈ instead of =).

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Kreft et al. (1995) defined equivalence as equality in both (a) the model implied expected values of the outcome, Yˆ ij (i.e., expectations) and (b) the model implied variance of the outcome, Var(Yij ) (i.e., dispersions). Model equivalence did not require equality in point estimates with ∗ ∗∗ = γ00 ). Given that applications of MLM typically matched non-superscript notation (e.g., γ00 focus more on the fixed effects, γ (which plays a central role in model-based expectations), rather than on the Level 2 variance-covariance matrix, T (which plays a central role in modelbased dispersions), this tutorial focuses more heavily on γ than on T. The accompanying tables, however, provide a broader focus as a resource. Adopting the classifications of Raudenbush and Bryk (2002), the two models of primary focus in Kreft et al. (1995) were (a) random coefficient regression (RCR) with a single Level 1 predictor, X1ij , and (b) intercepts and slopes as outcomes (ISO1 ) with the group means X 1·j as a predictor of the intercepts β 0j , added to RCR. Table 1 formally presents both of the models (under the three centering options) and summarizes key results of centering decisions and model equivalence. In sum, within both models, RAS and GMC are equivalent to each other and nonequivalent to CWC. All of the models briefly described in this section will be introduced more fully from both a researcher’s and a practitioner’s perspective in the next section. Kreft et al. (1995) extended ISO1 by specifying the group means X 1·j as a predictor of the slopes β 1j , which yielded ISO2 . This model added γ 11 , a cross-level interaction term; Table 1 formally presents the model, under the three centering options, and summarizes key results of centering decisions and model equivalence. In sum, RAS and GMC are equivalent to each other and non-equivalent to CWC. Due to space limitations, ISO2 is covered only briefly (and from a researcher’s perspective) in this tutorial, but key references will be provided.

MLMS AND CENTERING DECISIONS WITHIN A FAMILIAR CONTEXT Data Data were taken from Myers, Wolfe, Maier, Feltz, and Reckase (2006) to put broader methodological concepts into a familiar context. Most of the specifics of the previous study are outside the scope of this tutorial; however, a brief summary of how the measures were created and the nesting structure of the data is provided. Satisfaction data were calibrated to the Rasch rating scale model (Andrich, 1978) using Winsteps (Wright & Linacre, 1998). Competency data were calibrated to the multidimensional random coefficients multinomial logit model (Adams, Wilson, & Wang, 1997) using Conquest (Wu, Adams, & Wilson, 1998). Athletes (N = 585) were nested within 32 teams (J = 32): intercollegiate men’s (j = 8) and women’s (j = 13) soccer teams and women’s ice hockey teams (j = 11). The dependent variable was satisfaction with the head coach (i.e., Yij = SATij ). The independent variable at Level 1 was each athlete’s judgment of his/her head coach’s motivational competency (i.e., Xij = MOTij ). The Level 2 independent variable was an aggregate of each individual team member’s score on MOTij to form a team mean judgment of MOTij (i.e., X.j = MOT.j ). The specific relationships proposed in subsequent models of this tutorial are consistent with relevant coaching effectiveness theory (Horn, 2002). The statistical significance of individual parameters, however, is often of secondary interest within this tutorial. Table 2 provides key descriptive statistics within the Myers et al. (2006) study. Note that an estimate of the intraclass correlation coefficient (ICC = τ00 /(τ00 + σ 2 ), where τ 00 is the variance

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Under CWC: X 1ij = (X 1ij – X 1 .j )

Yij = γ 00 + γ 01 X 1 .j + γ 10 X 1ij + γ 11 X 1 .j X1ij + u0j + u1j X1ij + rij Under GMC: X 1ij = (X 1ij – X 1 ..)

Under CWC: X 1ij = (X 1ij – X 1 .j )

Yij = γ 00 + γ 01 X 1 .j + γ 10 X 1ij + u0j + u1j X 1ij + rij Under GMC: X 1ij = (X 1ij – X 1 ..)

Under CWC: X 1ij = (X 1ij – X 1 .j )

Yij = γ 00 + γ 10 X 1ij + u0j + u1j X 1ij + rij Under GMC: X 1ij = (X 1ij – X 1 ..)

Specifications

Note: E = expected value; V = variance-covariance; σ 2 = within-group variance.

ISO2

ISO1

RCR

Model

[V(Yij ) – σ 2 ]

[V(Yij ) – σ 2 ] E(Yij )

E(Yij )

[V(Yij ) – σ 2 ]

E(Yij )

Model Equivalence

γ 00 = γ 00 ∗ – γ 10 ∗ X 1 .. (Eq. (9)) γ 01 = γ 01 ∗ – γ 11 ∗∗ X 1 .. (Eq. (10)) see Eq. (2) γ 11 = γ 11 ∗ (Eq. (11)) RAS = GMC (see Eqs. (3)–(5))

γ 00 = γ 00 ∗ – γ 10 ∗ X 1 .. = γ 00 ∗∗ (Eq. (6))a γ 01 = γ 01 ∗ = γ 01 ∗∗ – γ 10 ∗∗ (Eq. (7)) γ 10 = γ 10 ∗ = γ 10 ∗∗ (Eq. (8)) RAS = GMC (see Eqs. (3)–(5)) RAS = GMC

RAS = GMC τ 00 = τ 00 ∗ – 2τ 10 ∗ X 1 .. + τ 11 ∗ X 2 1 .. (Eq. (3)) τ 10 = τ 10 ∗ – τ 11 ∗ X 1 .. (Eq. (4)) τ 11 = τ 11 ∗ (Eq. (5)) RAS = GMC = CWC

γ 00 = γ 00 ∗ – γ 10 ∗ X 1 .. (Eq. (1)) γ 10 = γ 10 ∗ (Eq. (2))

Grand Mean (GMC) γ 00 Denoted γ 00 ∗ , etc.

RAS = GMC

Raw Score (RAS) γ 00 Denoted γ 00 , etc.

Centering Decisions

TABLE 1 Centering Decisions and Model Equivalence: A Summary of Kreft et al. (1995)

— — CWC  = RAS = GMC

—

—

CWC  = RAS = GMC CWC  = RAS = GMC

—

—

— CWC  = RAS = GMC —

—

CWC  = RAS = GMC

Group Mean (CWC) γ 00 Denoted γ 00 ∗∗ , etc.

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TABLE 2 Correlations, Key Means and Dispersion Indices, and Proportion of Variance due to Team (η2 ) Variable 1. Satisfaction (SATij ) 2. Motivation (MOTij ) 3. MOTij gmc 4. MOTij cwc 5. Team satisfaction (SAT.j ) 6. Team motivation (MOT.j ) M (athlete level) SD (athlete level) Min, Max (athlete level) η2 M (team level) SD (team level) Min, Max (team level)

1

2

3

4

5

6

— 0.72 0.72 0.54 0.54 0.48 1.58 1.31 −2.19, 3.70 0.29 — — —

— 1.00 0.81 0.53 0.59 −0.07 1.55 −4.15, 3.16 0.35 −0.11 0.96 −3.33, 1.63

— 0.81 0.53 0.59 0.00 1.55 −4.08, 3.23 0.35 −0.05 0.96 −3.27, 1.70

— 0.00 0.00 0.00 1.25 −4.49, 4.27 0.00 0.00 0.00 0.00, 0.00

— 0.88 — — — — 1.56 0.72 −0.57, 2.79

— — — — — −0.11 0.96 −3.33, 1.63

Note: — = value was omitted because it was judged to introduce unhelpful clutter.

in satisfaction that is attributable to team, and (τ 00 + σ 2 ) is the total variance in satisfaction) for SATij equals .29. This value suggests that 29% of the variance in satisfaction with the head coach is attributable to between-team differences and that 71% of the variance in satisfaction with the head coach is attributable to within-team differences. From a statistical perspective, this value provides evidence against an assumption of independence of observations (by team) and evidence for fitting the data to MLM (Raudenbush & Bryk, 2002). Practitioner’s Perspective Suppose the data described above were collected by the AD. The information collected thus far is consistent with the AD’s experience-based observation that different athletes within the same team sometimes differ dramatically in how satisfied they are with the same head coach (i.e., most of the variance in satisfaction with the head coach is due to differences within teams). The information also is consistent with the AD’s experience-based observation that, despite some athlete-level differences within teams, on average, some teams are much more satisfied with their head coach than other teams are with their head coach (i.e., almost one-third of the variance in satisfaction with the coach was attributable to differences between teams). The information also is consistent with the AD’s concern that the team context should be accounted for when evaluating the satisfaction with the head coach data. Key Characteristics of the Data Proportion of Variance Due to Between-Team Differences η2 MOTij gmc is derived by subtracting the grand mean (i.e., average motivation competency score across athletes and teams), MOT ·· = −.067, from each athlete observation, MOTij . Because

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MOT ·· is a constant, moving from MOTij to MOTij gmc results in a linear transformation that changes the mean only. Mean MOTij gmc is .00, but η2 for MOTij gmc is the same as η2 for MOTij , η2 RAS and GMC = .35. Simply, team-level differences in MOTij are unchanged, though rescaled, under GMC. MOTij cwc is derived by subtracting the relevant group mean (i.e., MOT.j ) from each MOTij within each team (i.e., where j = j). Note that MOT.j varies across teams prior to CWC (i.e., η2 RAS and GMC = .35). Thus, moving from MOTij to MOTij cwc results in a more complex transformation as compared to moving from MOTij to MOTij gmc . Mean MOTij cwc was .00 and η2 for MOTij cwc = .00. Cells within a 7 × 3 sub-matrix within Table 2, defined by rows M (athlete level) to Min, Max (team level) and under columns labeled 3, 4, and 5, summarize key information under different centering decisions. Total, within, and Between Relationships Relations between SATij and MOTij are decomposed consistent with Raudenbush and Bryk (2002). As can be viewed in Table 2, the relationship between the two variables occurs at two levels. There are three estimates of the correlation at the athlete level: MOTij with SATij = .72, MOTij gmc with SATij = .72, and MOTij cwc with SATij = .54. The first two values differ from the third value for the same reason. Recall that some of the variance in both MOTij and MOTij gmc is attributable to team-level differences, while team-level differences are removed from MOTij cwc . The correlation between MOTij gmc (or MOTij ) and SATij = .72, therefore, describes the total association between the two variables and results in a “blended” estimate of the within- and between-team association. The correlation between MOTij cwc and SATij = .54 describes only the pooled within-team association between the two variables. The between-team correlation between MOT.j and SAT.j is r = .88. The three unique associations are depicted within separate fixed-effects regression equations (see Figure 1) consistent with Enders and Tofighi (2007). The first equation, SATi = gmc β0 + β1 MOTij + ri , yields an estimate of the total effect of MOTij gmc on SATij , βˆt = .61 (see dashed line) by imposing a model that ignores the clustering of the data. The second equation, SATi = β0 + β1 MOTijcwc + ri , yields an estimate of the pooled within-team effect of MOTij cwc on SATij , βˆw = .57 (see dotted line) by imposing a model after between-team variance is removed from motivation competency. The third equation, SAT.j = β0 + β1 MOT.j + rj , yields an estimate of the between-team effect of MOT.j on SAT.j ,βˆb = .66 (see solid line) by imposing a model after within-team variance is removed from both variables. Raudenbush and Bryk (2002, p. 138) described βˆt as “generally an uninterruptible blend of β b and β w ” given by βˆt = η2 βˆb + (1 − η2 )βˆw , where η2 was defined as previously for MOTij or MOTij gmc . Simply, βˆt represents two potentially different relationships, βˆb and βˆw . This results in a blended coefficient unless βˆb = βˆw or η2 = 0 or 1. Note that for the data used in this tutorial, all three estimates are very similar (see Figure 1). There is not a one-to-one correspondence among all of the information contained in the previous paragraph, which is based on fixed Level 1 coefficients and a MLM that allows Level 1 coefficients to vary (e.g., the weights for γˆt are likely to differ from the weights for βˆt ). The ideas, however, translate to MLM (Raudenbush & Bryk, 2002). Therefore,

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4.00

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dashed line = total regression dotted line =vwithin-teams regression solid line = between-teams regression

satisfaction

2.00

0.00

–2.00

–6.00

–4.00

–2.00 0.00 Motivatiob competency

2.00

4.00

FIGURE 1 Total, within-teams, and between-teams regressions of satisfaction on motivation competency.

parallel notation is adopted in the MLM framework when the same idea is expressed (e.g., βt = γt , βb = γb , βw = γw ). Practitioner’s Perspective The information reviewed under the “Key Characteristics of the Data Section” suggests that a key to both athlete/team satisfaction with the head coach may well be athlete/team perception of the head coach’s motivation competency. At both the athlete level and the team level, there is preliminary evidence that motivation competency is a positive predictor of satisfaction with the head coach. More specifically, as athletes/teams perceive their head coach to be more able to positively affect the psychological mood and psychological skills of his/her athletes/team, athletes/teams are more satisfied with their head coach, on average. Interestingly, the magnitude of this positive relationship may be approximately equal at both the athlete and team levels. MLM can be used to simultaneously estimate this relationship at both the athlete and team levels.

Overview of Model Presentations The type of MLM specified in the “Historical Outline” section is interpreted in a more familiar context via a four-step didactic-driven process. Step 1 introduces each new model by expressing

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it in uncombined form with Xqij scaled under RAS or GMC. The uncombined model is a way to write the equation that emphasizes the meaning of the parameters, which can change based on centering decisions (Raudenbush & Bryk, 2002). The combined model is an equivalent, compact expression (see Tables 1 and 3) that will be referred to when pedagogically appropriate. Step 2 provides a detailed interpretation of key terms and parameters. The ordering of interpretations highlights key relations. Step 3 reinterprets selected terms and parameters under the assumption of GMC and/or CWC. Within both Step 2 and Step 3, occasionally a clarification and/or an alternative interpretation, signified by •, is provided. Step 4 refers to and explains key results. In all cases, the term “key” is meant in relation to the purpose of this tutorial only. For the first model, RCR, an interpretation of most terms and parameters is provided. Key connections between centering decisions and interpretations are made explicit. After the RCR, interpretations of only the γ are provided and knowledge of the connection between centering decision and related interpretations is generally assumed. To keep within page limitations, RAS is not fully interpreted after RCR (Tables 1 and 3 include some information). RCR In this RCR model, athlete-level motivation is specified as a predictor of athlete-level satisfaction, while the team-level outcomes (i.e., intercepts and slopes) are free to vary randomly. Step 1: The Uncombined Model Level 1 : SATij = β0j + β1j MOTij + rij Level 2 : β0j = γ00 + u0j β1j = γ10 + u1j Step 2: An Interpretation of Each Term and Each Parameter Under RAS β 0j = expected satisfaction for athletes who judge their head coach to have “0” motivational competency in the jth team γ 00 = average expected satisfaction across teams for athletes who judge their head coach to have “0” motivational competency u0j = unique effect (or residual) of the jth team on the average expected satisfaction across teams for athletes who judge their head coach to have “0” motivational competency τ 00 = unconditional team-level variance around the average expected satisfaction across teams for athletes who judge their head coach to have 0 motivational competency • variance of the intercepts (i.e., Var(β0 ) which in this case = Var(u0 )) • unconditional is meant to communicate the absence of predictors, W sj , of the intercepts β 1j = change in expected athlete satisfaction given a one-unit increase in motivational competency in the jth team

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SATij = γ 00 + γ 01 (MOT.j – MOT..) + γ 10 MOTij + u0j + u1j MOTij + rij Under GMC: MOTij = (MOTij – MOT..) Under CWC: MOTij = (MOTij – MOT.j )

SATij = γ 00 + γ 01 (MOT.j –MOT..) + γ 10 MOTij + γ 11 (MOT.j – MOT..)MOTij + u0j + u1j MOTij + rij Under GMC: MOTij = (MOTij – MOT..) Under CWC: MOTij = (MOTij – MOT.j )

ISO1

ISO2

1.635 (.06) 0.577 (.03) 0.0705 (.03) 0.0036 (.006) −0.0154 (.009) 0.745 (.05) 1.626 (.06) 0.066 (.07) 0.567 (.03) 0.0657 (.03) 0.0026 (.006) −0.0127 (.009) 0.746 (.05) 1.655 (.06)

0.062 (.07) 0.570 (.03) −0.038 (.03) 0.0656 (.03) 0.0016 (.005) −0.0100 (.008) 0.744 (.05)

γ 01 (SE) γ 10 (SE) τ 00 (SE) τ 11 (SE) τ 10 (SE) σ 2 (SE) γ 00 (SE) γ 01 (SE) γ 10 (SE) γ 11 (SE) τ 00 (SE) τ 11 (SE) τ 10 (SE) σ 2 (SE)

Raw Score (RAS)

γ 00 (SE) γ 10 (SE) τ 00 (SE) τ 11 (SE) τ 10 (SE) σ 2 (SE) γ 00 (SE)

Parameter

0.064 (.07) 0.570 (.03) −0.038 (.03) 0.0669 (.03) 0.0016 (.005) −0.0101 (.009) 0.744 (.05)

0.066 (.07) 0.567 (.03) 0.0675 (.03) 0.0026 (.006) −0.0129 (.009) 0.746 (.05) 1.617 (.06)

1.596 (.06) 0.577 (.03) 0.0726 (.03) 0.0036 (.006) −0.0157 (.010) 0.745 (.05) 1.588 (.06)

Grand Mean (GMC or ∗ )

0.669 (.06) 0.570 (.03) −0.053 (.03) 0.0661 (.03) 0.0032 (.007) −0.0111 (.01) 0.742 (.05)

0.646 (.06) 0.567 (.03) 0.0663 (.03) 0.0038 (.007) −0.0128 (.01) 0.745 (.05) 1.555 (.06)

1.560 (.125) 0.570 (.03) 0.4606 (.126) 0.0056 (.008) −0.0421 (.024) 0.741 (.05) 1.555 (.06)

Group Mean (CWC or ∗∗ )

Note: SATij = ith athlete’s satisfaction with the jth head coach; MOTij = ith athlete’s motivational competency judgment of the jth head coach; MOT.. = average motivation competency judgment across all athletes; MOT.j = average motivation competency judgment across athletes in the jth team.

SATij = γ 00 + γ 10 MOTij + u0j + u1j MOT ij + rij Under GMC: MOTij = (MOTij – MOT..) Under CWC: MOTij = (MOTij – MOT.j )

Specifications

RCR

Model

Centering Decisions

TABLE 3 Centering Decisions and Model Equivalence: An Application of Kreft et al. (1995)

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γ 10 = γ t = average change in expected athlete satisfaction across teams given a one-unit increase in motivational competency • coefficient that blends (or contains both) the within-and between-team effect u1j = unique effect (or residual) of the jth team on the average change in expected athlete satisfaction across teams given a one-unit increase in motivational competency τ 11 = unconditional team-level variance around the average change in expected athlete satisfaction across teams given a one-unit increase in motivational competency • variance of the slopes (i.e., Var(β1 ) which in this case = Var(u1 )) • unconditional is meant to communicate the absence of predictors, W sj , of the slopes rij = residual satisfaction for the ith athlete in the jth team, after adjusting for the effect of motivational competency σ 2 = conditional athlete-level variance in satisfaction within teams, after controlling for the effect of motivational competency

Note the connection between β0j , γ00 , u0j , τ00 and the shared dependence of each interpretation on MOTij = 0. Placement of MOTij = 0, then, has direct consequences for the meaning of each of these terms. Note the connection between β1j , γ10 , u1j and τ11 and the apparent independence of interpretation from MOTij = 0. It should be noted that MOTij cwc can guard against negatively biased estimates of τ 11 or more generally, τ qq where q > 0 (Raudenbush & Bryk, 2002). Step 3: Different Centering Decisions GMC. Suppose that MOT.. is subtracted from each MOTij . Then interpretation of key terms could be as follows: β 0j = expected satisfaction for athletes in the jth team who judge their head coach to have motivational competency equal to the grand mean across athletes γ 00 = average expected satisfaction across teams for athletes who judge their head coach to have motivational competency equal to the grand mean across athletes τ 00 = unconditional team-level variance around the average expected satisfaction across teams for athletes who judge their head coach to have motivational competency equal to the the grand mean across athletes • variance of the intercepts (i.e., Var(β0 ) which in this case = Var(u0 )) β 1j = change in expected athlete satisfaction given a one-unit increase in motivational competency in the jth team γ 10 = γ t = average change in expected athlete satisfaction across teams given a one-unit increase in motivational competency • coefficient that blends the within-and between-team effect τ 11 = unconditional team-level variance around the average change in expected athlete satisfaction across teams given a one-unit increase in motivational competency • variance of the slopes (i.e., Var(β1 ) which in this case = Var(u1 ))

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CWC. Suppose that each MOT.j is subtracted from each of the relevant MOTij . Then interpretation of key terms could be as follows: β 0j = mean satisfaction for athletes in the jth team γ 00 = average mean satisfaction across teams τ 00 = unconditional team-level variance around the average mean satisfaction across teams • variance of the intercepts (i.e., Var(β0 ) which in this case = Var(u0 )) β 1j = change in expected athlete satisfaction given a one-unit increase in motivational competency in the jth team γ 10 = γ w = average change in expected athlete satisfaction across teams given a one-unit increase in motivational competency • pooled within-team coefficient τ 11 = unconditional team-level variance around the average change in expected athlete satisfaction across teams given a one-unit increase in motivational competency • variance of the slopes (i.e., Var(β1 ) which in this case = Var(u1 ))

Step 4: Relevant Results Table 3 provides relevant results. The equivalences summarized under RCR in Table 1 are ∗ − observed within the parallel section of Table 3. A key question is: Why was γˆ00 = γˆ00 ∗ ∗∗ X1 .. = γˆ00 ? Recall that the Level 1 intercept within any particular team can be written as γˆ10 β0j = μYj − β1j MOT.j . Recall from the “Key Characteristics” section that η2 for MOTij is the same as η2 for MOTij gmc , specifically η2 RAS and GMC = .35. Therefore β 0j can be interpreted as an adjusted mean under both RAS and GMC, whereas β 0j can be interpreted as an unadjusted mean under CWC because all team means = .00 and, thus, β0j = μYj . Both results extend naturally to β0j = μYj − β1j MOT.j − β2j X2 .j . . . − βkj Xk .j , whether the Xqij are continuous and/or dichotomous (Enders & Tofighi, 2007). ∗ ∗∗ = γˆ10 ? Recall from the “Key Characteristics of A second key question is: Why was γˆ10 = γˆ10 the Data” section that there are three different regression equations for predicting satisfaction by motivation:βt , βb , and βw . Recall from Step 1 that MOT.j is not explicitly entered into this model and that identical team-level differences are observed in both MOTij and MOTij gmc but not in ∗ ∗∗ = γˆt and γˆ10 = γˆw . The two unique estimates are similar in this MOTij cwc . Therefore, γˆ10 = γˆ10 case, γˆt ≈ γˆw , for two reasons. First, recall (see Figure 1) that both the magnitude and the sign of the pooled within effect, βˆw = .57, are similar to the between effect, βˆb = .66. Second, the blended coefficient , βˆt = .61, is pulled slightly more toward βˆw than βˆb because of the approx2 2 imate weights, 1−ηRAS and GMC = .65 and ηRAS and GMC = .35. In cases where the two effects are more divergent, the difference between γˆt and γˆw will likely be greater. This leads directly to the next model where MOT.j was explicitly modeled as a predictor of the β 0j . ∗ = .004, χ 2 (31) = 41.85, p = .092 and Unique results in this example include τˆ11 ∗∗ 2 τˆ11 = .006, χ (31) = 41.94, p = .091. Under both centering approaches, a case could be made to drop the relevant random effect u1j , because τˆ11 was not significantly different than

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zero. Dropping u1j would fix the relevant Level 1 coefficients, β 1j , to a constant, γ 10 . Doing so implies that the researcher believes that the change in expected athlete satisfaction given a one-unit increase in motivational competency does not truly vary beyond chance across teams. Given the purpose of this tutorial, u1j is retained. It should be noted, however, that prior to dropping a Level 2 random effect uqj in a model with at least one Xqij , a likelihood ratio test that compares model deviances of the nested models should be conducted to consider the impact of removing multiple parameter estimates τˆ11 and τˆ10 in this case. Practitioner’s Perspective The RCR model focuses on estimating the relationship between motivation competency and satisfaction with the head coach at the athlete level. Whether or not this athlete-level relationship was confounded with the team-level relationship (between motivation competency and satisfaction with the coach) depended on the centering decision for motivation competency. Centering motivation competency by subtracting the relevant team mean for motivation competency from each athlete’s motivation score (i.e., CWC) assisted in isolating the athlete-level relationship between motivation competency and satisfaction with the head coach. Interestingly, the positive relationship between motivation competency and satisfaction with the head coach at the athlete level does not appear to differ drastically within different teams. ISO1 In the ISO1 model, the previous RCR model is extended by adding team-level motivation as a predictor of the team-level intercepts. Step 1: The Uncombined Model Level 1 : SATij = β0j + β1j (MOTij − MOT..) + rij Level 2 : β0j = γ00 + γ01 (MOT.j − MOT..) + u0j β1j = γ10 + u1j Step 2: Interpretation of Key Terms and Parameters Under GMC γ 00 = average expected satisfaction (for athletes who judge their head coach to have motivational competency equal to the grand mean across athletes) across teams for teams that have average team motivation γ 01 = γ c = change in expected satisfaction (for athletes who judge their head coach to have motivational competency equal to the grand mean across athletes), given a one-unit increase in team motivation • contextual coefficient (i.e., between-team effect less the within-team effect) γ 10 = γ w = average change in expected athlete satisfaction across teams given a one-unit increase in motivational competency • pooled within-team coefficient

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Step 3: Different Centering Decision CWC. Suppose that each MOTj is subtracted from each of the relevant MOTij . Then interpretation of key terms could be as follows: γ 00 = average mean satisfaction across for teams that have average team motivation γ 01 = γ b = change in expected satisfaction given a one-unit increase in team motivation • between-team coefficient γ 10 = γ w = average change in expected athlete satisfaction across team given a one-unit increase in motivational competency • pooled within-team coefficient

Step 4: Relevant Results Table 3 provides relevant results. Note that the equivalences summarized under ISO1 in Table 1 are approximately observed within the parallel section of Table 3. A key question is: Why ∗ ∗ ∗∗ − γˆ10 X1 .. ≈ γˆ00 ? Simply, unlike in the RCR model, MOT.j is explicitly modeled was γˆ00 = γˆ00 as a predictor of the β 0j in ISO1 . The impact MOT.j exerted in RCR (under RAS and GMC but not CWC) is directly modeled in ISO1 under all centering decisions. The interpretation of the effect of MOT.j on the β 0j , however, differs under RAS and GMC versus CWC. Why the interpretation of the effect of MOT.j differs under RAS and GMC versus CWC ∗ ∗∗ ∗∗ ≈ γˆ01 − γˆ10 and can be explained through a linked pair of questions: Why was (a) γˆ01 = γˆ01 ∗ ∗∗ ∗ ∗∗ − (b) γˆ10 = γˆ10 = γˆ10 ? In both cases, the answer has to do with why (c) γˆ01 = γˆ01 ≈ γˆ01 ∗∗ ∗∗ ∗ ∗∗ = γˆc , where c = contextual; (d) γˆ01 = γˆb ,; and (e) γˆ10 = γˆ10 = γˆ10 = γˆw . With respect to γˆ10 (a), recall the adjustment to each β 0j in the RCR under RAS and GMC and the lack of adjustment to each β 0j under CWC. The same pattern occurs in ISO1 and for the same reasons. Therefore, MOT.j is specified to predict an adjusted set of β 0j under RAS and GMC (adjusted for differences in MOT.j and MOT.j – MOT..j , respectively) and an unadjusted set of β 0j under CWC. Therefore, the effect of MOT.j is interpreted as γˆc (i.e., γˆb − γˆw ) under RAS and GMC and γˆb under CWC. ∗ ∗∗ = γˆ10 , recall that in the RCR under RAS and GMC the parallel With respect to (b) γˆ10 = γˆ10 coefficient is interpreted as γˆt , because, in part, MOT.j varies within both MOTij and MOTij gmc but is interpreted as γˆw under CWC because MOT.j does not vary within MOTij cwc . In ISO1 , γ 10 is estimated while controlling for the effect of MOT.j under all centering decisions and, therefore, ∗ ∗∗ = γˆ10 = γˆw . This difference is more easily observed in the combined model for ISO1 γˆ10 = γˆ10 in Table 3. As can be viewed in Table 3, opposite conclusions would likely be reached regarding the statistical significance of the effect of MOT.j under RAS or GMC (not significant) versus CWC (significant)—even though the models are equivalent. From an empirical perspective, it would be most parsimonious to drop MOT.j under GMC because γˆc does not significantly differ from 0 (i.e., γˆb ≈ γˆw ) and thereby return to RCR. Simply, γˆt effectively does the work of both γˆw and γˆb . It is easy to imagine how the (non-)equivalence of γˆb and γˆw in a variety of substantive relationships could advance this theory. That is, some phenomena work in a similar way within teams (i.e., the athlete level) as they do across teams (i.e., at the team level) and some do not.

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It is also easy to imagine that various team-level aggregates (e.g., MOT.j ) may predict variation in various Level 1 slopes (e.g., β 1j ) in addition to variation in Level 1 means (β 0j ). Practitioner’s Perspective The ISO1 model focuses on directly estimating two of the three relationships between motivation competency and satisfaction with the head coach. The three relationships are the effect at the athlete level (i.e., motivation competency is specified as a predictor at Level 1), the effect at the team level (i.e., team motivation competency is specified as a predictor of the intercepts at Level 2), and the difference between the team-level effect as compared to the athlete-level effect (i.e., the contextual effect). The two effects that are estimated (and the third that is not estimated) depend on the centering decision. If the AD wishes to estimate the athlete-level effect and the contextual effect, then subtracting mean motivation competency across athletes from each athlete’s motivation score (i.e., GMC) is the optimal centering decision. If the AD wishes to estimate the athlete-level effect and the team-level effect, then subtracting the relevant team mean for motivation competency from each athlete’s motivation score (i.e., CWC) is the optimal centering decision. Given the AD’s initial research interests, to investigate whether athlete/team satisfaction with the head coach was predicted by athlete/team motivation competency, CWC may be preferred.

ISO2 In the ISO2 model, the first ISO model is extended by adding team-level motivation as a predictor of the team-level slopes.

Step 1: The Uncombined Model

Level 1 : SATij = β0j + β1j (MOTij − MOT..) + rij Level 2 : β0j = γ00 + γ01 (MOT.j − MOT..) + u0j β1j = γ10 + γ11 (MOT.j − MOT..) + u1j Step 2: Interpretation of Key Terms and Parameters Under GMC γ 00 = average expected satisfaction (for athletes who judge their head coach to have motivational competency equal to the grand mean across athletes) across teams for teams that have average team motivation γ 01 = change in expected satisfaction (for athletes who judge their head coach to have motivational competency equal to the grand mean across athletes) given a one-unit increase in team motivation γ 10 = average change in expected athlete satisfaction across teams given a one-unit increase in motivational competency for teams that have average team motivation

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γ 11 = change in the change of expected athlete satisfaction given a one-unit increase in motivational competency, given a one-unit increase in team motivation

Step 3: Different Centering Decision CWC. Suppose that each MOT.j is subtracted from each of the relevant MOTij . Then interpretation of key terms could be as follows: γ 00 = average mean satisfaction across teams for teams that have average team motivation γ 01 = change in expected satisfaction given a one-unit increase in team motivation γ 10 = average change in expected athlete satisfaction across teams given a one-unit increase in motivational competency for teams that have average team motivation γ 11 = change in the change of expected athlete satisfaction given a one-unit increase in motivational competency, given a one-unit increase in team motivation

Step 4: Relevant Results Table 3 provides relevant results. Note that the equivalences summarized under ISO2 in Table 1 are observed within the parallel section of Table 3. Note that many of the patterns that occur under RAS and ISO1 also occur under ISO2 , and they occur for the same reasons. The non-equivalence of CWC can be observed by expanding γ11 MOTij MOT.j under both centering decisions. Under GMC, this term can be re-expressed as γ11 MOTij MOT.j − γ11 MOT..MOT.j , and under CWC, this term can be re-expressed as γ11 MOTij MOT.j − γ11 MOT.2j , which provides a sufficient reason for the non-equivalence of CWC to GMC within ISO2 . Enders and Tofighi (2007) extended the literature on (non-)equivalences for the expectations (Yˆ ij ) in models where the Level 2 predictor in the cross-level interaction is not an aggregate of the Level 1 predictor. The presence of γ11 MOT.j MOTij under either centering decision complicates the interpretation of either main effect in isolation (see combined model for ISO2 in Table 3). For simplicity, allow the following designations: MOT·j = moderator and MOTij = focal predictor. Then the ∗ = decomposition simplicities observed for particular point estimates in ISO1 (e.g., γˆ10 = γˆ10 ∗∗ γˆ10 = γˆw ) are no longer static because the magnitude of the effect of MOTij depends on the level of MOT·j . Note that literature has recently emerged (e.g., Bauer & Curran, 2005) that views ISO2 as containing a simple intercept, (γ00 + γ01 MOT.j ), and a simple slope, (γ10 + γ11 MOT.j )MOTij , whatever the centering decision.

RELEVANT RECOMMENDATIONS AND EXAMPLES FOR PRACTITIONERS AND RESEARCHERS The summative recommendation for centering Level 1 predictors within MLM is that such decisions should be informed by the research questions, substantive theory, and empirical information. That two parameterizations of MLMs, which differ only by centering decision(s), are statistically equivalent does not necessarily mean that both models do an equally good job of

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representing specific research questions and/or theory-driven hypotheses. Therefore, informed a priori specification of MLMs requires precise research questions, multilevel substantive theory, and a thorough understanding of the terms that define the model. Thoughtful consideration of relevant empirical information requires knowledge of key characteristics of the sample data prior to imposing MLMs and accurate interpretation of output. The process of integrating experiencebased observations, substantive theory, and empirical information within a study may well be iterative, given the relative infancy of multilevel theorizing and MLM. Examples are sketched below (without statistical symbols and minimal use of previously defined abbreviations) for the purpose of demonstrating reasonable processes of making a priori decisions for centering Level 1 predictors and then integrating empirical information. For example 1, suppose there is reason to believe that (a) a student-level predictor (e.g., student gymnastics skill) relates to a dependent variable of interest (e.g., student gymnastics selfconcept), both at the student level and at the classroom level (e.g., classroom mean gymnastics self-concept) and (b) that these two relationships are not equal to each other (see Chanal, Marsh, Sarrazin, & Bois, 2005, for a similar example). A reasonable a priori model specification could include centering within classroom at Level 1 (e.g., student gymnastics skill minus classroom mean gymnastics skill) to provide a pure student-level effect. Further specification of classroom mean gymnastics skill as a predictor of classroom mean gymnastics self-concept would provide a pure classroom-level effect. Suppose that key characteristics of the multilevel data (completed as a part of describing the data) are determined to be consistent with a priori expectations and that the results from fitting MLMs are consistent with theoretical expectations. The only remaining task is to explain how the results support and extend the substantive theory. For example 2, change example 1 by assuming that the student-level and the classroom-level relationships are theorized to be approximately equal in size (see Myers et al., 2006, for a similar example). A reasonable a priori model specification could include GMC at Level 1 (e.g., individual gymnastics skill minus average gymnastics skill across students) to provide a coefficient that blends the student- and classroom-level effect. Suppose that key characteristics of the multilevel data are inconsistent with theoretical expectations (e.g., the student-level effect of gymnastics skills on gymnastics self-concept appears to be quite a bit smaller than the classroom-level effect of gymnastics skills on gymnastics self-concept). The a priori specification may still be imposed initially when fitting MLMs, but a post hoc re-specification may impose classroom mean gymnastics skill as a predictor of classroom mean (technically, intercept) gymnastics self-concept. A significant effect at the classroom level provides evidence for an unexpected contextual effect (e.g., the classroom-level effect is significantly larger than the student-level effect). The only remaining task is to explain how the results support (e.g., gymnastics skills is a significant predictor of gymnastics self-concept at the student-level) and provide evidence against (e.g., the difference in the magnitude of the relationship between gymnastics skills and gymnastics selfconcept at the student-level and the classroom level is significant) the relevant a priori substantive theory. Specifically, why does the researcher/practitioner believe that evidence was observed for the effect at the classroom level being larger than the effect at the student-level? For example 3, suppose that (a) the predictor of primary interest is measured at the schoollevel (e.g., communal environment) and (b) that there is reason to believe that this variable is a predictor of physical education teacher efficacy at the school level, even though previous research has provided evidence against this effect (see Bryk & Driscoll, 1988, for a similar example). Suppose further that (c) there is a previously unstudied teacher-level covariate (e.g., average

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student physical fitness level for each teacher) that the practitioner/researcher believes may serve as a control variable at Level 1 (i.e., adjusting mean teacher efficacy for differences in average student physical fitness level at the teacher level prior to modeling average teacher efficacy at the school level). Also suppose that (d) the effect of the average student physical fitness level on teacher efficacy at the teacher level is believed to be similar across schools. A reasonable a priori model specification could include GMC at Level 1 and dropping the random term for the effect of average student physical fitness level on teacher efficacy at Level 2 (i.e., this teacher-level effect would be constant across schools). Suppose that the key characteristics of the multilevel data are consistent with a priori expectations (e.g., a fixed-effects linear regression model is imposed in each school, and the slopes are very similar across schools). Suppose further that the results from fitting MLMs are also consistent with a priori expectations (e.g., the effect of communal environment on average teacher efficacy is statistically significant, after controlling for the effect of average student physical fitness level at the teacher level). The only remaining task is to explain why the results may contradict previous findings. Quality applications of MLM are gradually emerging in exercise science (e.g., Beauchamp, Bray, Fielding, & Eys, 2005; Gaudreau, Fecteau, & Perreault, 2010; Papaioannou, Marsh, & Theodorakis, 2004). The primary purpose of this tutorial was to attempt to further assist this process in relation to centering decisions for Level 1 predictors and, to a lesser extent, to introduce MPEES readers to some core facets of MLMs. This tutorial, however, has limitations. As can be discerned from the “Historical Research Outline” section, both MLM and centering decisions for Level 1 predictors are vibrant areas of methodological research. This tutorial cannot cover the full breath of either topic. It is hoped, however, that the information presented in this tutorial will help to expedite the rate of quality applications within the broad discipline of exercise science and provide salient direction when further information is needed.

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