AIPE Sample Size Planning for an Equivalence Test on the Standardized Mean Difference

Computes the minimum per-group sample size needed so that the equivalence CI (the 100(1 - 2$\alpha$)% CI on the standardized mean difference) has expected full width $\le \omega$, that is, expected half-width $\le \omega / 2$ (Kelley, 2007; Lakens, 2017). With the standard $\alpha = 0.05$ TOST level, the equivalence CI is the 90% CI. The function inverts the large-sample variance of d; the noncentral t distribution of d enters through the optional assurance step.

Usage

ss_aipe_equivalence_smd(
  population_smd = 0,
  width,
  alpha_level = 0.05,
  assurance = NULL,
  balanced = TRUE
)

Arguments

population_smd: Anticipated population standardized mean difference $\delta$ used to plan the variance. Default 0: plans under the most-conservative null-effect case.
width: Target full CI width on the d scale (e.g., 0.20 for a 90% CI of width 0.20).
alpha_level: One-sided TOST significance level. The CI used in planning is at confidence level $1 - 2\alpha$. Default 0.05 (90% CI).
assurance: Optional assurance probability in $(0, 1)$. When supplied, the function chooses n so that the probability of achieving width or less is at least assurance (Kelley, Maxwell, & Rausch, 2003). Default NULL (no assurance correction).
balanced: Logical; TRUE (default) plans equal-$n$ groups. Unequal-$n$ planning is not yet supported.

Value

A 5-row data.frame with columns term and value: the per-group recommended sample size necessary_n_per_group, the implied total total_N, the target width, the planning value population_smd, and ci_width_expected, the expected full CI width at the chosen n.

Details

Approximate-variance plan. The large-sample variance of d is $$\mathrm{Var}(\hat d) \;\approx\; (n_1 + n_2) / (n_1 n_2) + d^2 / (2 (n_1 + n_2)).$$ For a balanced design with per-group size $n$, the half-width of the equivalence CI at level $1 - 2\alpha$ is approximately $z_{1-\alpha} \sqrt{\mathrm{Var}(\hat d)}$. The function solves for the smallest integer $n$ giving expected half-width $\le \omega / 2$.

Assurance. Under assurance = q, the function increments $n$ until the simulated probability that the realized half-width is $\le \omega / 2$ is at least $q$. (Implemented as a thin Monte Carlo overlay. At the default planning value population_smd = 0 the shift is typically zero; it grows with the planning value, reaching several per group by population_smd = 0.5 with a narrow target width.)

Note on conservatism of the assurance plan. The empirical simulation study of the AIPE planner family finds that ss_aipe_equivalence_smd() is tight at $\gamma = 0.80$ but operates on the boundary of its valid range at $\gamma = 0.99$: the realized assurance at the recommended sample size is within Monte Carlo error of the target, typically a few tenths of a percentage point below 0.99. The mechanism is that the planner inverts a normal approximation to $\Pr(\widehat W > \omega)$, and at the 99% level the upper tail of $\widehat W$ is heavier than the approximation accounts for. Adding a small safety margin (5 to 10 subjects per group) restores the desired probability statement when planning at high assurance; ss_aipe_equivalence_smd_sensitivity reproduces the check for any one condition.

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

Kelley, K., Maxwell, S. E., & Rausch, J. R. (2003). Obtaining power or obtaining precision: Delineating methods of sample size planning. Evaluation and the Health Professions, 26(3), 258–287. doi:10.1177/0163278703255242

Lakens, D. (2017). Equivalence tests: A practical primer for t tests, correlations, and meta-analyses. Social Psychological and Personality Science, 8(4), 355–362. doi:10.1177/1948550617697177

Maxwell, S. E., Delaney, H. D., & Kelley, K. (2027). Designing experiments and analyzing data: A model comparison perspective (4th ed.). Routledge. (See Chapter 4 on individual comparisons and Chapter 3 on one-way ANOVA.)

Schuirmann, D. J. (1987). A comparison of the two one-sided tests procedure and the power approach for assessing the equivalence of average bioavailability. Journal of Pharmacokinetics and Biopharmaceutics, 15(6), 657–680.

Author

Ken Kelley kkelley@nd.edu

Examples

# 1. Plan for a 90% CI on d of width <= 0.20, under d_planning = 0:
ss_aipe_equivalence_smd(population_smd = 0, width = 0.20)
#>  term                  value
#>  necessary_n_per_group 542  
#>  total_N               1084 
#>  width                 0.2  
#>  population_smd        0    
#>  ci_width_expected     0.2  

# 2. Plan for the same width assuming a true d = 0.05:
ss_aipe_equivalence_smd(population_smd = 0.05, width = 0.20)
#>  term                  value
#>  necessary_n_per_group 542  
#>  total_N               1084 
#>  width                 0.2  
#>  population_smd        0.05 
#>  ci_width_expected     0.2  

# 3. With 80% assurance (the assurance path is Monte Carlo, so seed for
#    a reproducible result):
set.seed(113)
ss_aipe_equivalence_smd(population_smd = 0.05, width = 0.20, assurance = 0.80)
#>  term                  value
#>  necessary_n_per_group 542  
#>  total_N               1084 
#>  width                 0.2  
#>  population_smd        0.05 
#>  ci_width_expected     0.2