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This vignette details how you can set up and execute a basic power analysis for a bivariate random intercept cross-lagged panel model (RI-CLPM) using the powRICLPM package. Throughout, an illustrating example will be used in which we wish to detect a small cross-lagged effect \(\beta_{2}\) (defined here as the effect of \(a_{1}^{*}\) to \(b_{2}^{*}\), where \(a_{1}^{*}\) and \(b_{2}^{*}\) denote the latent within-unit components of \(a_{1}\) and \(b_{2}\), respectively) of 0.2 (standardized). For the design of our power analysis we follow the steps in the strategy as described in Mulder (2022). Various extensions are available for this basic power analysis, and are described in the Vignette Extensions.

Step 1: Determine experimental conditions

Before performing the power analysis, you must first determine the experimental conditions of interest. Experimental conditions (or: simulation conditions) are defined by characteristics of the study design that can impact statistical power. This includes, among others, characteristics like the sample size and the number of repeated measures. Decide on the number of repeated measures that will be used in the simulations, as well as the range of sample sizes over which you want to simulate the power.

For this example, we take a sample size range from 100 to 1000 first, increasing with steps of 100. Let the numbers of repeated measures range from 3 to 5. If these experimental conditions do not lead to the desired amount of power for detecting the small cross-lagged effect, the ranges can be extended later.

Step 2: Choose population parameter values

Next, determine population parameter values for generating data from the RI-CLPM. This requires the specification of:

Phi: Standardized autoregressive and cross-lagged effects for the within-unit components of the model. These values are collected in a matrix, with columns representing predictors and rows representing outcomes.
within_cor: A correlation for the within-unit components.
ICC: The proportion of variance at the between-unit level (relative to the total variance).
RI_cor: The correlation between the random intercepts.

For our example, the parameter values are set to:

Phi <- matrix(c(.4, .1, .2, .3), ncol = 2, byrow = T)
# The .2 refers to our standardized cross-lagged effect of interest
within_cor <- 0.3
ICC <- 0.5
RI_cor <- 0.3

If you are unsure if you have specified the Phi matrix as intended, you can use the check_Phi() function to give you a summary of how the effects in your Phi are interpreted.

Steps 3-5: Perform the power analysis

Steps 3 to 5 are automated by the powRICLPM() function. As input, you must provide:

the desired power level using the target_power argument,
the range of sample sizes to simulate the power for using the search_lower, search_upper, and search_step arguments (alternatively, you can specify this directly by providing a vector of sample sizes to the sample_size argument),
the number of time points for the simulated data using the time_points argument,
the population values Phi, within_cor, ICC, and RI_cor, and
the number of Monte Carlo replications we want to perform per experimental condition in the reps argument.

Optionally, you can specify:

skewness and kurtosis: An integer (vector) that determines the skewness and kurtosis for the simulated observed variables, respectively. 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 (default: 0).
alpha: A numeric value denoting the significance criterion (default: 0.05).
seed: An integer to control the starting point of the random number generator. This is important to use if you want to replicate the results.
reliability: A numeric value representing the reliability of the indicators (i.e., the proportion of true score variance) (default: 1).
bootstrap_reps: A numeric value, denoting the number of bootstrap replications are used to simulate the uncertainty around the power analysis results (default: 1000).
estimator: A character string, denoting the estimator that is used for estimating models (default: “ML”; when skewness and/or kurtosis values are set to nonzero values, it defaults to “MLR”).

The constraints, bounds, and estimate_ME arguments can be set as well to extend the basic power analysis setup. These options are further described in the Vignette Extensions.

For our example, we can perform the power analysis by running:

output <- powRICLPM(
  target_power = 0.8,
  search_lower = 100,
  search_upper = 1000,
  search_step = 50,
  time_points = c(3, 4, 5),
  ICC = ICC,
  RI_cor = RI_cor, 
  Phi = Phi,
  within_cor = 0.3,
  reps = 1000
)

Parallel processing using `furrr`

Performing a Monte Carlo power analysis with a large number of replications, and across multiple simulation conditions can be time-consuming. To speed up the process, it is recommended to perform the power analysis across simulation conditions in parallel (i.e., on multiple cores). The powRICLPM() function has implemented future’s parallel processing capabilities using the furrr package.

Load the furrr package, and use its plan() function to change the power analysis execution from sequential (i.e., single-core, the default), to multisession (i.e., multicore). Use the workers argument to specify how many cores you want to use. Next, run the powRICLPM analysis, and the power analysis will run on the specified number of cores. This can result in a significant reduction of computing time. For more information on other parallel execution strategies, see ?furrr::plan().

Progress bar using `progressr`

It can be useful to get an approximation of the progress of the powRICLPM analysis while running the code, especially when running the analysis in parallel. powRICLPM() has implemented progress notifications using the progressr package. Simply put, there are two options through which you can get progress notifications:

You can subscribe to progress updates from a specific expression by wrapping this expression with with_progress({...}).
You can subscribe to progress updates from everywhere by running handlers(global = T).

The second option is not fully developed yet for the furrr package, so instead I focus on the first. Implementing the with_progress({...}) option, as well as parallel execution of the powRICLPM analysis, results in the below code for the example:

# Load the furrr and progressr packages
library(furrr)
library(progressr)

# Check how many cores are available
future::availableCores()

# Plan powRICLPM analysis to run on 1 core less than number of available cores
plan(multisession, workers = 7) # For the case of 8 available cores

# Run the powRICLPM analysis 
with_progress({ # Subscribe to progress updates
  output <- powRICLPM(
    target_power = 0.8,
    search_lower = 100,
    search_upper = 1000,
    search_step = 50,
    time_points = c(3, 4, 5),
    ICC = ICC,
    RI_cor = RI_cor, 
    Phi = Phi,
    within_cor = 0.3,
    reps = 1000
  )
})

# Revert back to sequential execution of code 
plan(sequential)

For more information about progress notification options using progressr for end-users, including auditory and email updates, see https://progressr.futureverse.org.

Step 6: Summarize results

The powRICLPM() function creates a powRICLPM object: A list with results, upon which we can call print(), summary(), give(), and plot() functions to print, summarize, extract results, and visualize the results, respectively.

print() outputs a textual summary of the power analysis design contained within the object it was called upon. It does not output any performance metrics computed by the power analysis.

summary() can be used in one of four ways. First, summary can be used simply like print() to get information about the design of the power analysis (the different experimental conditions), as well as the number of problems the occurred per condition (e.g., non-convergence, fatal estimation errors, or inadmissible results). Second, by specifying the parameter = "..." argument in summary(), the function will print the results specifically for that parameter across all experimental conditions. Third, if you specify a specific experimental condition using summary()’s sample_size, time_points, and ICC arguments, performance measures are outputted for all parameters in that experimental condition.

# Summary of study design
summary(output)

# Summary of results for a specific parameter, across simulation conditions
summary(output, parameter = "wB2~wA1")

# Summary of all parameter for a specific simulation condition
summary(output, sample_size = 400, time_points = 4, ICC = 0.5)

give() extracts various bits of information from an powRICLPM object. The exact information to be extracted is given by the what = "..." argument:

what = "conditions" gives the different experimental conditions per row, where each condition is defined by a unique combination of sample size, number of time points and ICC.
what = "estimation_problems" gives the proportion of fatal errors, inadmissible values, or non-converged estimations (columns) per experimental conditions (row).
what = "results" gives the average estimate average, minimum estimate minimum, standard deviation of parameter estimates SD, the average standard error SEavg, the mean square error MSE, the average width of the confidence interval accuracy, the coverage rate coverage, and the proportion of times the p-value was lower than the significance criterion power. It requires setting the parameter = "..." argument.
what = "names" gives the parameter names contained within the powRICLPM object.

# Extract experimental conditions
give(output, what = "conditions")

# Extract estimation problems
give(output, what = "estimation_problems")

# Extract results for cross-lagged effect "wB2~wA1" 
give(output, what = "results", parameter = "wB2~wA1")

# Extract parameter names
give(output, what = "names")

Finally, plot() creates a ggplot2-plot for a specific parameter (specified using the parameter = "..." argument) with sample size on the x-axis, the simulated power on the y-axis, lines grouped by number of time-points, and plots wrapped by proportion of between-unit variance. plot() returns a ggplot2 object that can be fully customized using ggplot2 functionality. For example, you can change the scales, add titles, change geoms, etc. More information about options in the ggplot2 framework can be found at https://ggplot2-book.org/index.html. In the below example, I add a title and change the labels on the x-axis:

# Create basic plot of powRICLPM object
p <- plot(output, parameter = "wB2~wA1")
p

# Adjust plot aesthetics
p2 <- p + 
  ggplot2::labs(
    title = "Power analysis for RI-CLPM",
    caption = "Based on 1000 replications."
  ) +
  ggplot2::scale_x_continuous(
    name = "Sample size",
    breaks = seq(100, 1000, 100),
    guide = ggplot2::guide_axis(n.dodge = 2)
  )
p2