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The 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, Hertzog,
Lindenberger, and Ghisletta (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.

The multiple indicator 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, Jorgensen,
Verdam, Oort, and Elffers (2020), and Jorgensen,
Pornprasertmanit, Schoeman, and Rosseel (2021).