# FAQ

Below you can find a list of frequently asked questions, organized by topic, that reach us via email. Click the question to see our response.

## RI-CLPM extensions

How do I run a power analysis for an RI-CLPM with multiple indicators?

Unfortunately, the `powRICLPM` package does not have the functionality (yet) to include multiple indicator extensions of the RI-CLPM. However, one can use Mplus’s Monte Carlo functionalities to specify multiple indicator RI-CLPMs (MI-RICLPM) and simulate their power. Furthermore, there are some general comments to be made about the effect of including multiple indicators on power. For example, Oertzen et al. (2010) claim that the the separation of measurement error from true score variance increases power, and that this principle generalizes to all structural equation models, regardless of the regression model linking the latent variables. This implies that MI-RICLPM’s have more power than the basic RI-CLPM.

To verify this, a small simulation study was performed using Mplus (model syntax for these simulations can be found on GitHub). 10,000 datasets were generated from an MI-RICLPM with a latent factor per occasion and with three indicators. These factors contain a trait-like part that is captured by the higher-order random intercepts, and a state-like part that is used to capture the dynamics over time (see bottom panel of Figure 3 in Mulder and Hamaker (2021)). The factor loadings linking the indicators with the latent factor were set to 1, measurement error variances were set to 0.5, and the cross-lagged effects were fixed at 0.2. All other population parameter values were kept the same as in the illustrative example in Mulder (under review), implying that the proportion of measurement error in the observed variables was \(20\%\). The simulated data sets were analyzed using the MI-RICLPM that generated the data (and separates out measurement error variance), and a basic RI-CLPM (assuming no measurement error, using only the first indicator). Results show that the power to detect non-zero lagged effects is indeed larger in the MI-RICLPM (ranging from 0.89 to 0.93 for the standardized cross-lagged effects of 0.2, and from 0.927 to 0.939 for the standardized autoregressive effects of 0.3) than in the basic RI-CLPM (ranging from 0.70 to 0.74 for the same standardized cross-lagged effects of 0.2, and from 0.807 to 0.827 for the standardized autoregressive effects of 0.3).

All in all, while the inclusion of multiple indicators complicates the RI-CLPM power analysis, it can be generally stated that the separation of measurement error and true score variance through the extension with multiple indicators is beneficial for the power to detect lagged effects. As measurements, especially in the social and behavioral sciences are prone to include measurement error, this extension may be well-worth considering from a power point-of-view.

How do I run a power analysis for a multiple group RI-CLPM?

The multiple group RI-CLPM is based on fitting a multiple group version of the RI-CLPM both with and without constraints across groups (e.g., the constraint of equal lagged effects), and comparing the model fit to determine whether the imposed constraints are tenable. Power thus refers to the probability of rejecting a bad-fitting model due to untenable across-group constraints in this context, rather than rejecting the null-hypothesis for a specific parameter. The effect size then refers to how much worse the constrained model fits the data compared to the more general model (with less, or no across-group constraints). Analytic solutions, like the likelihood ratio test by Satorra and Saris (1985) or power analyses based on the RMSEA by MacCallum, Browne, and Sugawara (1996), are more efficient to use for these types of power analyses than computationally intensive Monte Carlo simulation studies. See, for example, the `SSpower()` function from the R-package `semTools` for multiple group SEM power analysis by Jak et al. (2020), and Jorgensen et al. (2021).

Can I run a power analysis for a tri- or more-variate RI-CLPM?

Unfortunately, `powRICLPM` currently only performs power analyses for the bivariate RI-CLPM. There are plans to extend the functionality of the package to include tri- or more-variate models, but work on this is likely to only start sometime in summer 2023.

One of the issues that needs to be tackled when going to a more-variate version of the model, is that it becomes increasingly difficult to specify population parameter values that combined allow you to simulate data from it. This is related the first limitation that is mentioned in Mulder (2023), see Section 6.1.

## Post hoc power analysis

How can I do posthoc power analysis using `powRICLPM`?

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## References

Jak, Suzanne, Terrence D. Jorgensen, Mathilde G. E. Verdam, Frans J. Oort, and Louise Elffers. 2020. “Analytical power calculations for structural equation modeling: A tutorial and Shiny app.” Behavior Research Methods 53 (4): 1385–1406. https://doi.org/10.3758/s13428-020-01479-0.
Jorgensen, Terrence D., Sunthud Pornprasertmanit, Alexander M. Schoeman, and Yves Rosseel. 2021. “semTools: Useful tools for structural equation modeling.” https://cran.r-project.org/package=semTools.
MacCallum, Robert C., Michael W. Browne, and Hazuki M. Sugawara. 1996. “Power analysis and determination of sample size for covariance structure modeling.” Psychological Methods 1 (2): 130–49. https://doi.org/10.1037/1082-989X.1.2.130.
Mulder, Jeroen D. 2023. “Power Analysis for the Random Intercept Cross-Lagged Panel Model Using the powRICLPM r-Package.” Structural Equation Modeling: A Multidisciplinary Journal 30 (4): 645–58. https://doi.org/10.1080/10705511.2022.2122467.
Mulder, Jeroen D., and Ellen L. Hamaker. 2021. “Three extensions of the random intercept cross-lagged panel model.” Structural Equation Modeling: A Multidisciplinary Journal 28 (4): 638–48. https://doi.org/10.1080/10705511.2020.1784738.
Oertzen, Timo, Christopher Hertzog, Ulman Lindenberger, and Paolo Ghisletta. 2010. “The effect of multiple indicators on the power to detect inter-individual differences in change.” British Journal of Mathematical and Statistical Psychology 63 (3): 627–46. https://doi.org/10.1348/000711010X486633.
Satorra, A., and W. E. Saris. 1985. “Power of the likelihood ratio test in covariance structure analysis.” Psychometrika 50 (1): 83–90.