Sensitivity Analysis for the Sample Size Planning Method for Standardized ANCOVA Contrast

Sensitivity analysis for the sample size planning method with the goal to obtain sufficiently narrow confidence intervals for standardized ANCOVA complex contrasts.

Usage

ss_aipe_sc_ancova_sensitivity(
  true_psi = NULL,
  estimated_psi = NULL,
  c_weights,
  desired_width = NULL,
  n_per_group = NULL,
  mu_x = 0,
  sigma_x = 1,
  rho,
  divisor = "s_ancova",
  assurance = NULL,
  conf_level = 0.95,
  G = 10000,
  print_iter = TRUE,
  save = FALSE,
  filename = "ss_aipe_sc_ancova_sensitivity_result.csv",
  ...
)

Arguments

true_psi: the population standardized ANCOVA contrast
estimated_psi: the estimated standardized ANCOVA contrast
c_weights: the contrast weights
desired_width: the desired full width of the obtained confidence interval
n_per_group: selected sample size to use in order to determine distributional properties of a given value of sample size
mu_x: the population mean for the covariate
sigma_x: the population standard deviation of the covariate
rho: the population correlation coefficient between the response and the covariate
divisor: which error standard deviation to be used in standardizing the contrast; the value can be either "s_ancova" or "s_anova"
assurance: parameter to ensure that the obtained confidence interval width is narrower than the desired width with a specified degree of certainty (must be NULL or between zero and unity)
conf_level: the desired confidence interval coverage, (i.e., 1 - Type I error rate)
G: number of generations (i.e., replications) of the simulation
print_iter: to print the current value of the iterations
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 Monte Carlo sensitivity analysis across G replications. The term entries are: mean_psi, median_psi, sd_psi (summaries of the realized standardized ANCOVA contrast); mean_ci_width, median_ci_width, sd_ci_width (summaries of the full interval widths); mean_ci_width_lower and mean_ci_width_upper (mean one-sided widths, measured from the observed contrast to each limit); pct_ci_less_w (proportion of intervals at or below the target width); pct_ci_miss_low and pct_ci_miss_high (tail-specific empirical non-coverage of true_psi); total_type_I_error (overall empirical non-coverage, the sum of the two tails); and the input echoes n_per_group, total_N, true_psi, estimated_psi (NA when n_per_group was supplied instead), rho, width, conf_level, and assurance (present only when an assurance was supplied). The proportion and Type I error rows are proportions on the 0 to 1 scale, not percentages.

Details

The sample size planning method this function is based on is developed in the context of simple (i.e., one-response-one-covariate) ANCOVA model and randomized design (i.e., same population covariate mean across groups).

An ANCOVA contrast can be standardized in at least two ways: (a) divided by the error standard deviation of the ANOVA model, (b) divided by the error standard deviation of the ANCOVA model. This function can be used to analyze both types of standardized ANCOVA contrasts.

The population mean and standard deviation of the covariate does not affect the sample size planning procedure; they can be specified as any values that are considered as reasonable by the user.

References

Kelley, K. (2007). Confidence intervals for standardized effect sizes: Theory, application, and implementation. Journal of Statistical Software, 20(8), 1–24. doi:10.18637/jss.v020.i08

Kelley, K., & Rausch, J. R. (2006). Sample size planning for the standardized mean difference: Accuracy in parameter estimation via narrow confidence intervals. Psychological Methods, 11(4), 363–385. doi:10.1037/1082-989X.11.4.363

Lai, K., & Kelley, K. (2012). Accuracy in parameter estimation for ANCOVA and ANOVA contrasts: Sample size planning via narrow confidence intervals. British Journal of Mathematical and Statistical Psychology, 65, 350–370. doi:10.1111/j.2044-8317.2011.02029.x

Maxwell, S. E., Delaney, H. D., & Kelley, K. (2027). Designing experiments and analyzing data: A model comparison perspective (4th ed.). Routledge. (See Chapter 9.)

Steiger, J. H., & Fouladi, R. T. (1997). Noncentrality interval estimation and the evaluation of statistical methods. In L. L. Harlow, S. A. Mulaik, & J. H. Steiger (Eds.), What if there were no significance tests? (pp. 221–257). Mahwah, NJ: Lawrence Erlbaum.

Author

Ken Kelley kkelley@nd.edu

Examples

# Sensitivity analysis for a standardized ANCOVA contrast across
# three groups, contrast (-1, 0, 1), a covariate-outcome correlation
# of 0.4, and a planning target width of 0.5. Sizes are kept small
# here so the Monte Carlo sweep runs quickly; raise G for a stable
# estimate in practice.
set.seed(113)
ss_aipe_sc_ancova_sensitivity(
  true_psi = 0.5, estimated_psi = 0.5,
  c_weights = c(-1, 0, 1),
  desired_width = 0.5, rho = 0.4,
  conf_level = 0.95, G = 50, print_iter = FALSE
)
#>  term                value  
#>  mean_psi            0.481  
#>  median_psi          0.467  
#>  sd_psi              0.119  
#>  mean_ci_width       0.5    
#>  median_ci_width     0.499  
#>  sd_ci_width         0.00249
#>  mean_ci_width_lower 0.25   
#>  mean_ci_width_upper 0.249  
#>  pct_ci_less_w       0.58   
#>  pct_ci_miss_low     0      
#>  pct_ci_miss_high    0.02   
#>  total_type_I_error  0.02   
#>  n_per_group         126    
#>  total_N             378    
#>  true_psi            0.5    
#>  estimated_psi       0.5    
#>  rho                 0.4    
#>  width               0.5    
#>  conf_level          0.95   
#> 
#> Confidence level: 95%