Testing Measurement Invariance in Physical Education and Exercise Science: A Tutorial Using the Well-Being Self-Efficacy Scale

ABSTRACT

In physical education and exercise science, it is common to examine mean differences between groups or to assess change across time. However, before group differences or change can be confidently examined, measurement invariance can be tested. Measurement invariance tests the equivalence of a construct across groups or across time. If measurement invariance is supported, then differences in latent means can more confidently be attributed to individuals’ different standings on a construct. Though an important first step to confidently examine group differences and change across time, this technique is sometimes not used in the field. Thus, the purpose of this manuscript was to provide a didactic review and illustration of measurement invariance within the field. We review a methodological approach to measurement invariance, the sequential steps used in this approach, assessing model-data fit, and testing partial invariance. We provide an illustration of the technique and conclude with practical considerations.

KEYWORDS:

• Measurement equivalence
• confirmatory factor analysis
• multi-group analysis
• exploratory structural equation modeling

Notes

¹ The superscript, g, is used in the expansion of equation 2(2) ${{\mathbf{x}}^g}_{px1}=\hspace{0.17em}{{\tau }^g}_{px1}+\hspace{0.17em}{{\mathrm{\Lambda }}^g}_{pxm}{{\xi }^g}_{mx1}+\hspace{0.17em}{{\delta }^g}_{px1}$(2) to denote group or time-point.

² We focus on testing measurement invariance between groups when describing the key steps. Readers should note that the same steps are taken when examining invariance across time points.

³ We note that prior to assessing configural invariance, an omnibus test can be conducted which assesses the equality of the variance-covariance matrices and mean vector across groups (Jöreskog, 1971; Jung & Yoon, 2016; Vandenberg & Lance, 2000). The omnibus test is seldom implemented in practice, thus, for our tutorial, we follow common practice and do not report an omnibus test.

⁴ Researchers are encouraged to always report (a) the chi-square and fit indices for each model tested and (b) the results of the chi-square difference test and the change in multiple fit indices as this provides the reader salient information to interpret the fit of the model(s).

5 If full metric invariance is supported and beginning partial invariance testing with strong invariance, then the baseline model is the full metric invariant model. If full strong invariance is supported and beginning partial invariance testing with strict invariance, then the baseline model is the full strong invariant model.

6 If reference variables are used, then a new parameter is not created for the constrained items.

7 In order for the configural model to be identified, the residual variances were constrained to be positive.