Sample Size Planning From the AIPE Perspective for Standardized ANCOVA Contrasts

Sample size planning from the accuracy in parameter estimation (AIPE) perspective for standardized ANCOVA contrasts.

Usage

ss_aipe_sc_ancova(
  psi = NULL,
  sigma_anova = NULL,
  sigma_ancova = NULL,
  psi_standardized = NULL,
  ratio = NULL,
  rho = NULL,
  divisor = "s_ancova",
  c_weights,
  width,
  conf_level = 0.95,
  alpha_lower = NULL,
  alpha_upper = NULL,
  assurance = NULL,
  ...
)

Arguments

psi: The population unstandardized ANCOVA (adjusted) contrast
sigma_anova: The population error standard deviation of the ANOVA model
sigma_ancova: The population error standard deviation of the ANCOVA model
psi_standardized: The population standardized ANCOVA (adjusted) contrast
ratio: The ratio of sigma_ancova over sigma_anova
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"
c_weights: Contrast weights
width: The desired full width of the obtained confidence interval
conf_level: The desired confidence interval coverage (i.e., 1 - Type I error rate). Default is .95, which gives a symmetric two-sided interval. Specify either conf_level or both of alpha_lower and alpha_upper, not both.
alpha_lower: Lower-tail Type I error rate, used to plan an asymmetric confidence interval. When supplied together with alpha_upper, the planned interval has lower-tail probability alpha_lower and upper-tail probability alpha_upper. Set conf_level = NULL when supplying these.
alpha_upper: Upper-tail Type I error rate, used together with alpha_lower to plan an asymmetric confidence interval.
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)
...: Allows one to potentially include parameter values for inner functions

Value

A 1-row data.frame with columns term and value. The term is "necessary_n_per_group" and value is the per-group sample size needed for the planned ANCOVA contrast.

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.

Not all of the effect size arguments need to be specified. When divisor="s_ancova" the input is either (a) psi_standardized, or (b) psi (the unstandardized ANCOVA contrast) and sigma_ancova. When divisor="s_anova", the valid input combinations are (a) psi_standardized and ratio; (b) psi_standardized and rho; or (c) psi, sigma_anova, and sigma_ancova.

Note

When divisor="s_anova" and the argument assurance is specified, the necessary sample size per group returned by the function with assurance specified is slightly underestimated. The method to obtain exact sample size in the above situation has not been developed yet. A practical solution is to use the sample size returned as the starting value to conduct a priori Monte Carlo simulations with function ss_aipe_sc_ancova_sensitivity, as discussed in Lai & Kelley (2012).

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

ss_aipe_sc_ancova(psi_standardized = .8, width = .5, c_weights = c(.5, .5, 0, -1))
#>  term                  value
#>  necessary_n_per_group 98   
#> 
#> Confidence level: 95%

ss_aipe_sc_ancova(psi_standardized = .8, ratio = .6, width = .5,
                  c_weights = c(.5, .5, 0, -1), divisor = "s_anova")
#>  term                  value
#>  necessary_n_per_group 39   
#> 
#> Confidence level: 95%

ss_aipe_sc_ancova(psi_standardized = .5, rho = .4, width = .3,
               c_weights = c(.5, .5, 0, -1), divisor = "s_anova")
#>  term                  value
#>  necessary_n_per_group 221  
#> 
#> Confidence level: 95%