# Confirmatory Factor Analysis (CFA) for nested data

I would like to conduct CFA to examine support for a 3-factor model with team_cohesion, team_trust, and team_performance variables. I have the data in the following format with individuals rating the team:

Can I perform the CFA analysis on this data ignoring the nested/hierarchical nature of the data (i.e., individual nested within teams)? OR should I be first aggregating items corresponding to these factors (i.e., compute mean score) and then run CFA analysis? If I need to aggregate, should I first justify aggregation using r-wg scores?

These sound like team-level constructs (what Stapleton et al., 2016, called "shared constructs"), so hopefully the individuals responded to items that were worded that way. In this context, I would consider team member as "raters" of team-level constructs, and you will see me refer to interrater reliability (IRR) below.

• You should not "ignore nesting" (e.g., by running a person-level model with cluster-robust SEs and tests) because the constructs are characteristics of the team, not of individuals.
• In principle, you could calculate cluster aggregates (team means) from individual responses, but those are likely to contain a LOT of measurement error. It is usually not easy to measure cluster-level constructs with high reliability.

I would run a multilevel CFA and check the assumption of measurement invariance (see Jak & Jorgensen, 2017, for details), and follow Lai (2021, p. 95, Eq. 17) to estimate reliability of team composites calculated by average individual scores across both items AND group members (i.e., simultaneously quantifying IRR and scale reliability).

You may notice Stapleton et al. (2016, 2019, also referenced by Lai, 2021, in his Eq. 20) advocating for a CFA with saturated within-level structure to model shared constructs, but I disagree with this (Jak et al., 2021). However, Lai's (2021) Eq. 20 could still be interpreted similar to Eq. 17, and Eq. 20 would be more robust to any misspecification of the Level-1 model.

References

Jak, S., & Jorgensen, T. D. (2017). Relating measurement invariance, cross-level invariance, and multilevel reliability. Frontiers in Psychology, 8(1640). https://doi.org/10.3389/fpsyg.2017.01640

Jak, S., Jorgensen, T. D., & Rosseel, Y. (2021). Evaluating Cluster-Level Factor Models with lavaan and Mplus. Psych, 3(2), 134–152. https://doi.org/10.3390/psych3020012

Lai, M. H. (2021). Composite reliability of multilevel data: It’s about observed scores and construct meanings. Psychological Methods, 26(1), 90–102. https://doi.org/10.1037/met0000287

Stapleton, L. M., & Johnson, T. L. (2019). Models to examine the validity of cluster-level factor structure using individual-level data. Advances in Methods and Practices in Psychological Science, 2(3), 312–329. https://doi.org/10.1177/2515245919855039

Stapleton, L. M., Yang, J. S., & Hancock, G. R. (2016). Construct meaning in multilevel settings. Journal of Educational and Behavioral Statistics, 41(5), 481–520. https://doi.org/10.3102/1076998616646200