A Priori Monte Carlo Simulation for Sample Size Planning for SEM Targeted Effects

Conduct a priori Monte Carlo simulation to empirically study the effects of (mis)specifications of input information on the calculated sample size. Random data are generated from the true covariance matrix but fit to the proposed model, whereas sample size is calculated based on the input covariance matrix and proposed model.

Usage

ss_aipe_sem_path_sensitivity(
  model,
  est_Sigma,
  true_Sigma = est_Sigma,
  which_path,
  desired_width,
  N = NULL,
  conf_level = 0.95,
  assurance = NULL,
  G = 100,
  save = FALSE,
  filename = "ss_aipe_sem_path_sensitivity_result.csv",
  ...
)

Arguments

model: A single character string giving the free analysis model in lavaan model syntax (see model.syntax), the model that would be fit to the data. The target path must carry a parameter label so it can be referred to by name, for example "f2 ~ b*f1" labels the structural path b. The model may or may not be the true data generating model
est_Sigma: the covariance matrix used to calculate sample size, may or may not be the true covariance matrix. The row names and column names of est_Sigma should be the same as the observed variables in model
true_Sigma: the true population covariance matrix, which will be used to generate random data for the simulation study. The row names and column names of true_Sigma should be the same as the observed variables in model
which_path: the parameter label of the targeted path, given as a character string, for example "b" for the path labeled f2 ~ b*f1 in model
desired_width: desired confidence interval width for the model parameter of interest
N: the sample size of random data. If it is NULL, it will be determined by the sample size planning method
conf_level: confidence level (i.e., 1- Type I error rate)
assurance: the assurance that the confidence interval obtained in a particular study will be no wider than desired (must be NULL or a value between 0.50 and 1)
G: number of replications in the Monte Carlo simulation
save: option to save simulation results. It can be saved with save = TRUE outside of the printed results
filename: the name of the file that simulation results will be saved to
...: allows one to potentially include parameter values for inner functions

Value

A data.frame with columns term and value summarizing the a priori Monte Carlo study. The term entries are: "mean_path", "median_path", "sd_path" (summaries of the realized estimates of the targeted path across the converged replications); "mean_ci_width", "median_ci_width", "sd_ci_width" (summaries of the realized interval widths); "pct_ci_less_w" (proportion of realized widths at or below desired_width); "pct_ci_miss_low" and "pct_ci_miss_high" (tail-specific empirical non-coverage of the population path); "total_type_I_error" (overall empirical non-coverage, the sum of the two tails); and the echoes "suc_rep" (number of converged replications), "total_N" (the N evaluated), "true_path" (the population value of the targeted path under true_Sigma), "width", "conf_level", and "assurance" (present only when an assurance was supplied). The proportion rows are on the 0 to 1 scale, not percentages.

Details

This function implements the sample size planning methods proposed in Lai and Kelley (2011). It calls ss_aipe_sem_path to plan the sample size and to identify the targeted path, then fits the analysis model to data simulated from true_Sigma with sem. The analysis model is written in lavaan model syntax with the targeted path given a parameter label; see model.syntax for the syntax and sem for the fitting machinery. The population covariance matrices are most naturally produced by cov_sem from a fully fixed population model. This function requires lavaan and MASS to be installed.

Note

Occasionally a replication fails to converge when the analysis model is fit to a simulated data set. Such replications are not counted toward the G converged replications; the simulation draws fresh data and continues. A safety cap stops the loop after 20 * G attempts, and a single warning is issued if fewer than G replications converged.

References

Lai, K., & Kelley, K. (2011). Accuracy in parameter estimation for targeted effects in structural equation modeling: Sample size planning for narrow confidence intervals. Psychological Methods, 16(2), 127–148. doi:10.1037/a0021764

Maxwell, S. E., Delaney, H. D., & Kelley, K. (2027). Designing experiments and analyzing data: A model comparison perspective (4th ed.). Routledge.

Rosseel, Y. (2012). lavaan: An R package for structural equation modeling. Journal of Statistical Software, 48(2), 1–36. doi:10.18637/jss.v048.i02

Author

Ken Kelley kkelley@nd.edu

Examples

# This function is itself a Monte Carlo study: it plans a sample size and
# then fits the analysis model to G freshly simulated data sets, so even a
# modest G takes long enough that the worked example below is shown here
# rather than run.
#
# The planning values a researcher would bring to ss_aipe_sem_path():
# each factor is measured by three indicators, each with a residual
# variance of 0.5.
#   planning_model <- "
#     f1 =~ 1*y1 + 0.8*y2 + 0.8*y3
#     f2 =~ 1*y4 + 0.8*y5 + 0.8*y6
#     f2 ~ 0.5*f1
#     f1 ~~ 1*f1
#     f2 ~~ 0.75*f2
#     y1 ~~ 0.5*y1; y2 ~~ 0.5*y2; y3 ~~ 0.5*y3
#     y4 ~~ 0.5*y4; y5 ~~ 0.5*y5; y6 ~~ 0.5*y6
#   "
#
# The population the study will actually sample from: the same structural
# path of 0.5, but noisier indicators than the planning values assumed,
# with residual variances of 0.8.
#   true_model <- "
#     f1 =~ 1*y1 + 0.8*y2 + 0.8*y3
#     f2 =~ 1*y4 + 0.8*y5 + 0.8*y6
#     f2 ~ 0.5*f1
#     f1 ~~ 1*f1
#     f2 ~~ 0.75*f2
#     y1 ~~ 0.8*y1; y2 ~~ 0.8*y2; y3 ~~ 0.8*y3
#     y4 ~~ 0.8*y4; y5 ~~ 0.8*y5; y6 ~~ 0.8*y6
#   "
#
#   analysis_model <- "
#     f1 =~ y1 + y2 + y3
#     f2 =~ y4 + y5 + y6
#     f2 ~ b*f1
#   "
#
#   est_Sigma <- cov_sem(planning_model)$sigma_theta
#   true_Sigma <- cov_sem(true_model)$sigma_theta
#
# The sample size planned from the optimistic measurement quality is
# evaluated against the population that actually holds: the realized
# intervals are wider than desired, and few of them meet the target.
#   set.seed(113)
#   ss_aipe_sem_path_sensitivity(model = analysis_model,
#                                est_Sigma = est_Sigma,
#                                true_Sigma = true_Sigma, which_path = "b",
#                                desired_width = 0.30, G = 1000)