The basic power analysis as described in the vignette Getting Started can be extended to using bounded estimation, estimation with constraints over time, inclusion of measurement error in the generated data and estimation model (i.e., the STARTS model), and generating data with skewness and kurtosis. These extensions are described below.

`lavaan`

To prevent non-convergence for small sample sizes (say, less than
100), bounds can be imposed on the parameter space during estimation of
the model using `bounds = TRUE`

(De
Jonckere and Rosseel 2022). This can aid the optimization
algorithm to find unique solutions and prevents it from searching in the
completely wrong direction for one, or multiple parameters. Sensible
lower bounds involve those on the (residual) variances of latent
variables (e.g., the random intercept variances), as negative variances
are theoretically not possible. Upper bound for variances are determined
based on the observed variances for variable. In the context of the
RI-CLPM, the factor loadings are (usually) fixed, and hence these
parameters are not estimated. The lagged effects are theoretically
infinite, and hence there are no sensible bounds we can place à priori
on these parameters.

The use of bounded estimation is theoretically appealing for models that are known to have convergence issues, such as the stable trait autoregressive trait state (STARTS) model. The Special Topics vignette Measurement Error uses the powRICLPM package to explore the impact of unmodeled measurement error in the RI-CLPM, and the use of bounded estimation to aid the convergence of the STARTS model.

`powRICLPM()`

offers users the option to impose various
constraints over time on the estimation model through the
`constraints`

argument. This has statistical advantages as
constraints over time reduce model complexity, thereby potentially
reducing convergence issues and increasing power. Moreover, some
researchers are interested in so called ‘stationarity’ constraints for
theoretical reasons. A disadvantage of such constraints is that they
assume certain parameters to be time-invariant. This might not be an
assumption researchers are willing to make, especially in developmental
contexts where you expect lagged effects might change over time (e.g.,
the variable `wA`

gets more important in driving
`wB`

as one gets older). Therefore, by default
`constraints = "none"`

, implying that all lagged effects, and
within-components (residual) variances and covariances are freely
estimated over time.

Constraint options include:

`constraints = "lagged"`

: Autoregressive and cross-lagged effects are constrained to be equal over time.`constraints = "residuals`

: Within-unit residual variances and covariance (from wave 2 onward) are constrained to be equal over time.`constraints = "within"`

: Both lagged effects and residual variances and covariances are constrained to be equal over time.`constraints = "stationary"`

: Constraints are imposed on the variances of the within-components at the first wave, and residual variances at wave 2 and further, such that the variances of the within-components themselves are all 1. This implies that the variances at the first wave are fixed to 1, and that the residual variances are a function of the lagged effects, and correlation between within-components at the same wave. These constraints are deduced in the supplementary materials of Mulder and Hamaker (2021), see the FAQ “How can I constrain the*standardized*parameters to be invariant over time?”.`constraints = "ME"`

: Constraints are imposed on the measurement error variances across time (separately for each variable). This constraint is only possible when`estimate_ME = TRUE`

.

While it is generally advisable to include measurement error when
analyzing psychological data, the RI-CLPM does not include it. Adding
measurement error to the model would result in the bivariate STARTS
model by Kenny and Zautra (2001), and
requires at least 4 waves of data to be identified. Users can add
measurement error variances to the estimation model by specifying
`estimate_ME = TRUE`

. Measurement error can be added to the
simulated data using the `reliability`

argument.

Note, however, that the STARTS model has been shown to be prone to empirical under-identification, often requiring upwards of 8 waves of data and sample sizes larger than 500. The Special Topics vignette Measurement Error uses the powRICLPM package to explore the impact of unmodeled measurement error in the RI-CLPM, and the use of bounded estimation to aid the convergence of the STARTS model.

Asymmetry in the empirical distributions of psychometric measurements
is rather common (Micceri 1989), and this
can impact the power of SEM models that assume normally distributed
variables. The powRICLPM package allows researchers to investigate power
under asymmetrically distributed data by setting values for the
`skewness`

and `kurtosis`

arguments (default: 0).
For example, suppose we have reason to believe the \(A\) and \(B\) variables are positively skewed, and
have heavy tails (i.e., a higher kurtosis) we can include the arguments
`skewness = 1`

and `kurtosis = 0.5`

. When the
`skewness`

and `kurtosis`

arguments are set to
values other than 0, the powRICLPM package defaults to using robust
maximum likelihood (`estimator = "MLR"`

).

De Jonckere, Julie, and Yves Rosseel. 2022. “Using bounded estimation to avoid nonconvergence in small
sample structural equation modeling.” *Structural
Equation Modeling* 29 (3): 412–27. https://doi.org/10.1080/10705511.2021.1982716.

Kenny, David A., and Alex Zautra. 2001. “Trait–state models for longitudinal data.”
In *New Methods for the Analysis of Change*, 243–63. Washington:
American Psychological Association. https://doi.org/10.1037/10409-008.

Micceri, Theodore. 1989. “The unicorn, the
normal curve, and other improbable creatures.”
*Psychological Bulletin* 105 (1): 156–66.

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.