Wrap up

Markus Steiner

Institute for Mental and Organisational Health, FHNW

06 June, 2025

Alternative and Equivalent Models

  • If multiple theories exist, compare alternative models.
  • Equivalent models
    • Special case of alternative models: Exact same fit, \(\Sigma\), etc.
    • For details, see, e.g., Lee & Hershberger (1990); Hershberger & Marcoulides (2013).

Figure 1.6 from Hershberger & Marcoulides (2013)

EFA vs. CFA revisited

Nájera et al. (2025) suggest that CFA might be overly strict

  • in presence of cross-loadings
  • in presence of slight misspecifications
  • leading to biased parameter estimates.

Alternatively, they suggest the usage of

  • EFA (best in one sample and then CFA in another sample)
  • ECFA (first EFA, determine substantial loadings acc. to spec. rules, then fit a CFA)

Some Things to Consider

  • Tradeoff between internal consistency and predictive validity (Revelle, 2024; White, 2025)
    • The more similar (correlated) items are, the higher internal consistency, but the lower the increase in predictive validity.
  • Formative (descriptive) vs. reflective (causal) factors.
    • Higher-order factors need not be causal: Overlapping processes can lead to higher-order structure (Kovacs & Conway, 2016, 2019; Revelle, 2024)
    • For a discussion of formative indicators, see Bollen & Diamantopoulos (2017).

Further Ressources

  • EFA:
    • Fabrigar & Wegener (2012)
  • CFA:

(CB-)SEM

  • Factor Analysis focuses on measurement models and scoring.
  • SEM allow us to model directed relationships (i.e., regressions) between latent constructs.
  • Resources:

Example: Industrialization and political democracy in developing countries.

model <- '
  # measurement model
    ind60 =~ x1 + x2 + x3
    dem60 =~ y1 + y2 + y3 + y4
    dem65 =~ y5 + y6 + y7 + y8
  # regressions
    dem60 ~ ind60
    dem65 ~ ind60 + dem60
  # residual correlations
    y1 ~~ y5
    y2 ~~ y4 + y6
    y3 ~~ y7
    y4 ~~ y8
    y6 ~~ y8
'
fit <- sem(model, data = PoliticalDemocracy)

PLS-SEM

  • Constructs as weighted composites (vs. explaining covariation between indicators).
  • Prediction and explanation of target constructs.
  • Non-parametric method: No distributional assumptions
  • Can easily model formative constructs.
  • Higher power than CB-SEM
  • Cannot model error terms (e.g. covariation of errors)
  • No overall GOF
  • No easy model comparisons to alternative models

Example: Corporate reputation model

# measurement model
corp_rep_mm <- constructs(
  composite("COMP", multi_items("comp_", 1:3)),
  composite("LIKE", multi_items("like_", 1:3)),
  composite("CUSA", single_item("cusa")),
  composite("CUSL", multi_items("cusl_", 1:3)))

# structural model
corp_rep_sm <- relationships(
  paths(from = c("COMP", "LIKE"),
        to = c("CUSA", "CUSL")),
  paths(from = c("CUSA"), to = c("CUSL")))

# model estimation
estimate_pls(data = seminr::corp_rep_data,
             measurement_model = corp_rep_mm,
             structural_model  = corp_rep_sm,
             inner_weights = path_weighting,
             missing = mean_replacement,
             missing_value = "-99")

PLS-SEM

Example: Corporate reputation model

References

Bollen, K. A., & Diamantopoulos, A. (2017). In Defense of Causal-Formative Indicators: A Minority Report. Psychological Methods, 22(3), 581–596. https://doi.org/10.1037/met0000056
Brown, T. A. (2015). Confirmatory Factor Analysis for Applied Research (2nd ed.). The Guilford Press.
Fabrigar, L. R., & Wegener, D. T. (2012). Exploratory Factor Analysis. Oxford University Press.
Hair, J. F., Hult, G. T. M., Ringle, C. M., & Sarstedt, M. (2017). A primer on partial least squares structural equation modeling (PLS-SEM) (Second edition). SAGE.
Hair, J. F., Hult, G. T. M., Ringle, C. M., Sarstedt, M., Danks, N. P., & Ray, S. (2021). Partial Least Squares Structural Equation Modeling (PLS-SEM) Using R: A Workbook. Springer International Publishing. https://doi.org/10.1007/978-3-030-80519-7
Hancock, G. R., & Mueller, R. O. (2013). Structural equation modeling. A second course. Information Age Publishing Inc.
Hershberger, S., & Marcoulides, G. A. (2013). The problem of equivalent structural models. In G. R. Hancock & R. O. Mueller (Eds.), Structural Equation Modeling. A second course (2nd ed.). Information Age Publishing Inc.
Kline, R. B. (2023). Principles and practice of structural equation modeling (5th ed.). The Guilford Press.
Kovacs, K., & Conway, A. R. A. (2016). Process Overlap Theory: A Unified Account of the General Factor of Intelligence. Psychological Inquiry, 27(3), 151–177. https://doi.org/10.1080/1047840X.2016.1153946
Kovacs, K., & Conway, A. R. A. (2019). A unified cognitive/differential approach to human intelligence: Implications for IQ Testing. Journal of Applied Research in Memory and Cognition, 8(3), 255–272. https://doi.org/10.1016/j.jarmac.2019.05.003
Lee, S., & Hershberger, S. (1990). A Simple Rule for Generating Equivalent Models in Covariance Structure Modeling. Multivariate Behavioral Research, 25(3), 313–334. https://doi.org/10.1207/s15327906mbr2503_4
Nájera, P., Abad, F. J., & Sorrel, M. A. (2025). Is exploratory factor analysis always to be preferred? A systematic comparison of factor analytic techniques throughout the confirmatory–exploratory continuum. Psychological Methods, 30(1), 16–39. https://doi.org/10.1037/met0000579
Ray, S., Danks, N. P., & Calero Valdez, A. (2024). Seminr: Building and Estimating Structural Equation Models (p. 2.3.4). Comprehensive R Archive Network.
Revelle, W. (2024). The seductive beauty of latent variable models: Or why I don’t believe in the Easter Bunny. Personality and Individual Differences, 221, 112552. https://doi.org/10.1016/j.paid.2024.112552
White, M. (2025). A peculiarity in psychological measurement practices. Psychological Methods. https://doi.org/10.1037/met0000731