Sample Size Planning for SEM Targeted Effects

Plan sample size for structural equation models so that the confidence interval for the targeted model parameter is sufficiently narrow

Usage

ss_aipe_sem_path(
  model,
  Sigma,
  desired_width,
  which_path,
  conf_level = 0.95,
  assurance = NULL,
  detail = FALSE,
  internal = FALSE,
  ...
)

Arguments

model: A single character string giving the free analysis model in lavaan model syntax (see model.syntax). 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. This is the model that would be fit to the data; its parameters are free, not fixed to population values
Sigma: Estimated population covariance matrix of the observed variables, with row and column names matching the observed variables in model. It is typically obtained from a fully fixed population model via cov_sem
desired_width: Desired confidence interval width for the model parameter of interest
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
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)
detail: if TRUE, additionally print the model parameter names and the observed variable names (the returned table is unchanged)
internal: option to output a list for internal use (for ss_aipe_sem_path_sensitivity)
...: Allows one to potentially pass additional arguments to sem

Value

A data.frame (a dmar_tbl) with term and value columns whose rows are necessary_N (the planned sample size), path_index (the position of the target path among the model parameters), and var_theta_j (the population sampling variance of the target path at the planned sample size). The returned table is the same whether or not detail = TRUE. When internal = TRUE a list is returned for use by ss_aipe_sem_path_sensitivity.

Details

This function implements the sample size planning methods proposed in Lai and Kelley (2011). It requires lavaan to be installed and uses sem to obtain the expected information, that is the asymptotic covariance matrix of the parameter estimates, by fitting the free analysis model to the population covariance matrix Sigma at a very large sample size. 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 matrix Sigma is most naturally produced by cov_sem from a fully fixed population model.

When assurance is supplied, the assurance adjustment is based on a chi square approximation to the sampling variability of the confidence interval width and can undershoot the nominal assurance in finite samples; use ss_aipe_sem_path_sensitivity to check the realized width and coverage at the planned sample size.

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

# Population covariance from a fully fixed model (see cov_sem()).
pop_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
"
Sigma <- cov_sem(pop_model)$sigma_theta

# Free analysis model with the target structural path labeled "b".
analysis_model <- "
  f1 =~ y1 + y2 + y3
  f2 =~ y4 + y5 + y6
  f2 ~ b*f1
"
ss_aipe_sem_path(model = analysis_model, Sigma = Sigma,
                 desired_width = 0.30, which_path = "b")
#>  term        value  
#>  necessary_N 264    
#>  path_index  5      
#>  var_theta_j 0.00584
#> 
#> Confidence level: 95%

# That sample size holds the interval to the desired width on average, so
# about half of the studies it plans return a wider one. Adding assurance
# plans for the width to be met in 90 percent of studies instead, at the
# cost of a larger sample size.
ss_aipe_sem_path(model = analysis_model, Sigma = Sigma,
                 desired_width = 0.30, which_path = "b",
                 assurance = 0.90)
#>  term        value  
#>  necessary_N 293    
#>  path_index  5      
#>  var_theta_j 0.00526
#> 
#> Confidence level: 95%