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Daniel F. Gucciardi, Chun-Qing Zhang, Vellapandian Ponnusamy, Gangyan Si, and Andreas Stenling

The aims of this study were to assess the cross-cultural invariance of athletes’ self-reports of mental toughness and to introduce and illustrate the application of approximate measurement invariance using Bayesian estimation for sport and exercise psychology scholars. Athletes from Australia (n = 353, M age = 19.13, SD = 3.27, men = 161), China (n = 254, M age = 17.82, SD = 2.28, men = 138), and Malaysia (n = 341, M age = 19.13, SD = 3.27, men = 200) provided a cross-sectional snapshot of their mental toughness. The cross-cultural invariance of the mental toughness inventory in terms of (a) the factor structure (configural invariance), (b) factor loadings (metric invariance), and (c) item intercepts (scalar invariance) was tested using an approximate measurement framework with Bayesian estimation. Results indicated that approximate metric and scalar invariance was established. From a methodological standpoint, this study demonstrated the usefulness and flexibility of Bayesian estimation for single-sample and multigroup analyses of measurement instruments. Substantively, the current findings suggest that the measurement of mental toughness requires cultural adjustments to better capture the contextually salient (emic) aspects of this concept.

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Matthew S. Tenan, Andrew J. Tweedell, and Courtney A. Haynes

individual data points iteratively and processes the series until a changepoint is detected. The ‘bcp’ package 17 performs a form of Bayesian changepoint analysis in a time series as originally described by Barry and Hartigan. 18 In contrast to the other methods (both established and novel), the

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Matthew Ellis, Mark Noon, Tony Myers, and Neil Clarke

fitting models with both the lowest out-of-sample prediction error (Watanabe–Akaike information criterion) and the highest variance explained ( R 2 ) were multilevel models allowing intercepts to vary for each participant. Bayesian multilevel model was fitted with postscore as the dependent variable and

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Anthony Koschmann

research makes several other contributions. It makes use of a novel and comprehensive data set of player performance from the 2008 NFL season. Measuring player quality is difficult, and even key scoring metrics are inadequate. This study uses an empirical Bayesian hierarchical linear model (HLM) that

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Andreas Stenling, Andreas Ivarsson, Urban Johnson, and Magnus Lindwall

Bayesian statistics is on the rise in mainstream psychology, but applications in sport and exercise psychology research are scarce. In this article, the foundations of Bayesian analysis are introduced, and we will illustrate how to apply Bayesian structural equation modeling in a sport and exercise psychology setting. More specifically, we contrasted a confirmatory factor analysis on the Sport Motivation Scale II estimated with the most commonly used estimator, maximum likelihood, and a Bayesian approach with weakly informative priors for cross-loadings and correlated residuals. The results indicated that the model with Bayesian estimation and weakly informative priors provided a good fit to the data, whereas the model estimated with a maximum likelihood estimator did not produce a well-fitting model. The reasons for this discrepancy between maximum likelihood and Bayesian estimation are discussed as well as potential advantages and caveats with the Bayesian approach.

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Rosemary A. Arthur, Nichola Callow, Ross Roberts, and Freya Glendinning

understanding of the nature of coaching of PS. Study 2 involved questionnaire development using the qualitative findings, and then testing of the questionnaires’ factor structure via a Bayesian structural equation modeling (BSEM) approach to confirmatory factor analysis. Finally, in Study 3, we confirmed the

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Yin-Hua Chen, Isabella Verdinelli, and Paola Cesari

This paper carries out a full Bayesian analysis for a data set examined in Chen & Cesari (2015). These data were collected for assessing people’s ability in evaluating short intervals of time. Chen & Cesari (2015) showed evidence of the existence of two independent internal clocks for evaluating time intervals below and above the second. We reexamine here, the same question by performing a complete statistical Bayesian analysis of the data. The Bayesian approach can be used to analyze these data thanks to the specific trial design. Data were obtained from evaluation of time ranges from two groups of individuals. More specifically, information gathered from a nontrained group (considered as baseline) allowed us to build a prior distribution for the parameter(s) of interest, and data from the trained group determined the likelihood function. This paper’s main goals are (i) showing how the Bayesian inferential method can be used in statistical analyses and (ii) showing that the Bayesian methodology gives additional support to the findings presented in Chen & Cesari (2015) regarding the existence of two internal clocks in assessing duration of time intervals.

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Richard J. Barker and Matthew R. Schofield

In a recent commentary on statistical inference, Batterham and Hopkins1 advocated an approach to statistical inference centered on expressions of uncertainty in parameters. After criticizing an approach to statistical inference driven by null hypothesis testing, they proposed a method of “magnitude-based” inference and then claimed that this approach is essentially Bayesian but with no prior assumption about the true value of the parameter. In this commentary, after we address the issues raised by Batterham and Hopkins, we show that their method is “approximately” Bayesian and rather than assuming no prior information their approach has a very specific, but hidden, joint prior on parameters. To correctly adopt the type of inference advocated by Batterham and Hopkins, sport scientists need to use fully Bayesian methods of analysis.

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Katherine A. Tamminen, Patrick Gaudreau, Carolyn E. McEwen, and Peter R.E. Crocker

Efforts to regulate emotions can influence others, and interpersonal emotion regulation within teams may affect athletes’ own affective and motivational outcomes. We examined adolescent athletes’ (N = 451, N teams = 38) self- and interpersonal emotion regulation, as well as associations with peer climate, sport enjoyment, and sport commitment within a multilevel model of emotion regulation in teams. Results of multilevel Bayesian structural equation modeling showed that athletes’ self-worsening emotion regulation strategies were negatively associated with enjoyment while other-improving emotion regulation strategies were positively associated enjoyment and commitment. The team-level interpersonal emotion regulation climate and peer motivational climates were also associated with enjoyment and commitment. Team-level factors moderated some of the relationships between athletes’ emotion regulation with enjoyment and commitment. These findings extend previous research by examining interpersonal emotion regulation within teams using a multilevel approach, and they demonstrate the importance of person- and team-level factors for athletes’ enjoyment and commitment.

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Will G. Hopkins and Alan M. Batterham