Cross-cultural invariance of the MTI 1 Running head: Cross-cultural invariance of the MTI Cross-Cultural Invariance of the Mental Toughness Inventory among Australian, Chinese, and Malaysian Athletes: A Bayesian Estimation Approach

1 The aims of this study were to assess the cross-cultural invariance of athletes’ self-reports of mental 2 toughness, and introduce and illustrate the application of approximate measurement invariance 3 using Bayesian estimation for sport and exercise psychology scholars. Athletes from Australia (n = 4 353, Mage = 19.13, SD = 3.27, males = 161), China (n = 254, Mage = 17.82, SD = 2.28, males = 138), 5 and Malaysia (n = 341, Mage = 19.13, SD = 3.27, males = 200) provided a cross-sectional snapshot 6 of their mental toughness. The cross-cultural invariance of the mental toughness inventory in terms 7 of (i) the factor structure (configural invariance), (ii) factor loadings (metric invariance), and (iii) 8 item intercepts (scalar invariance) was tested using an approximate measurement framework with 9 Bayesian estimation. Results indicated that approximate metric and scalar invariance was 10 established. From a methodological standpoint, this study demonstrated the usefulness and 11 flexibility of Bayesian estimation for single-sample and multi-group analyses of measurement 12 instruments. Substantively, the current findings suggest that the measurement of mental toughness 13 requires cultural adjustments to better capture the contextually-salient (emic) aspects of this 14 concept. 15 16