Activity One
The observed variables (kft_v1, kft_v3, kft_n1, kft_n3, kft_q1, kft_q3) are endogenous as they are indicators of the latent construct KFT_g rather than latent endogenous variables, confirming alignment with the graphic as measures reflecting the latent variable (Brown, 2015, pp. 167-178). The term endogenous applies to the latent variable KFT_g in this model because it is the construct measured by the observed variables.
Unique measurement errors (e.g., q3_u, v1_u) are exogenous variables. They are independent of other variables in the model, and their presence in the graphic should be distinct from the latent factor and its observed indicators.
The significant unstandardised factor loadings, indicated by the regression weights table with associated p-values, affirm that the observed variables significantly influence the latent construct KFT_g (Kline, 2023, pp. 229-250). These loadings are foundational for establishing the relevance of each observed variable to the latent construct.
Standardised factor loadings provide insight into the relative impact of each observed variable on the latent construct. The significance of these loadings is derived from the unstandardised loadings. Notably, the standardised loadings’ significance is based on the unstandardised loadings and their p-values provided in the additional information (Schreiber et al., 2006).
Variances for the unique measurement errors are free from Heywood cases—implausible values exceeding 1.0 or negative estimates—suggesting a well-specified model (Kenny & McCoach, 2003).
The R² values, representing the variance explained by the latent factor, match the graphic and confirm the predictive power of the latent variable KFT_g over its indicators.
The Comparative Fit Index (CFI) and Tucker-Lewis Index (TLI) indicate a good fit for the model, with values suggesting that the model aligns well with observed data. These indices help compare non-nested models and assess fit and complexity. However, the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) provide a more detailed comparison by penalising model complexity, though they are not included in the provided data (Hu & Bentler, 1999).
References
- Brown, T.A. (2015). Confirmatory factor analysis (2nd ed.). Guilford.
- Hu, L., & Bentler, P.M. (1999). Cutoff criteria for fit indexes. Structural Equation Modelling, 6(1), 1-55. https://doi.org/10.1080/10705519909540118
- Kenny, D.A., & McCoach, D.B. (2003). Effect of the number of variables on measures of fit. Structural Equation Modelling, 10(3), 333-351. https://doi.org/10.1207/S15328007SEM1003_1
- Kline, R.B. (2023). Principles and practice (5th ed.). Guilford.
- Schreiber, J.B., Nora, A., Stage, F.K., Barlow, E.A., & King, J. (2006). Reporting structural equation modelling. Journal of Educational Research, 99(6), 323-338. https://doi.org/10.3200/JOER.99.6.323-338
Activity Two
In structural equation modelling (SEM), observed variables are measurable indicators of latent constructs (Byrne, 2016, pp. 3-15). For example, despite being truncated in the graphic, CONSULT and SICK represent measures related to Physical Health, an exogenous variable. In SEM, endogenous variables like Functional Health (FH) and Subjective Health (SH) are influenced by others, while exogenous variables exert influence without being affected (Kline, 2023, pp. 414-424).
The R² for FH is 52%, indicating that Physical Health (PH) explains 52% of its variance. SH has an R² of 75%, showing that PH and FH explain 75% of its variance. FH has one predictor, PH, and SH has two, PH and FH. The non-significant chi-square (p > .05) suggests that the model fits the data well (Schreiber et al., 2006). Fit indices (CFI, TLI, IFI) near 1 support this, but values strictly at 1 may indicate overfitting.
All regression weights are significant (p < .05), confirming the robustness of the model's relationships. The first three standardised beta coefficients are crucial: FH on PH (-.722), SH on FH (.200), and SH on PH (-.709), which indicate strong negative effects of PH on both FH and SH and a positive impact of FH on SH. These betas are central to the research question, hence their importance (Hu & Bentler, 1999).
The absence of Heywood cases, as the variances are within acceptable limits, indicates a plausible model (Mercoulides et al., 2023, pp. 523-525). The 'U' variables represent unique variances of observed measures, and the 'Res' variables are residuals for endogenous latent variables. This model correctly identifies all observed variables and their relations, confirming the theoretical framework.
The SEM analysis shows a valid model with significant relationships, acceptable goodness-of-fit, and no Heywood anomalies. The model's predictors effectively account for a substantial portion of the variance in key health constructs, supporting the theoretical assumptions.
References
- Byrne, B.M. (2016). Structural equation modelling with AMOS (3rd ed.). Routledge.
- Hu, L., & Bentler, P.M. (1999). Cutoff criteria for fit indexes. Structural Equation Modelling, 6(1), 1-55. https://doi.org/10.1080/10705519909540118
- Kline, R.B. (2023). Principles and practice (5th ed.). Guilford.
- Mercoulides, K.M., Yuan, K., & Deng, L. (2023). Structural equation modelling with small samples and many variables. In R.H. Hoyle (Ed.), Handbook of structural equation modelling (2nd ed.). Guilford.
- Schreiber, J.B., Nora, A., Stage, F.K., Barlow, E.A., & King, J. (2006). Reporting structural equation modelling. Educational Research, 99(6), 323-338. https://doi.org/10.3200/JOER.99.6.323-338
