AIPE Sample Size Planning for an Equivalence Test on the Pearson Correlation

Computes the minimum sample size needed so that the equivalence CI (the 100(1 - 2$\alpha$)% CI on the Pearson correlation $\rho$) has expected full width $\le \omega$ (Kelley, 2007; Lakens, 2017). With the standard $\alpha = 0.05$ TOST level, the equivalence CI is the 90% CI. The interval is the Fisher's $Z$ construction that equivalence_r and ci_r use, so the plan and the analysis invert the same interval.

Usage

ss_aipe_equivalence_r(
  population_r = 0,
  width,
  alpha_level = 0.05,
  assurance = NULL
)

Arguments

population_r: Anticipated population correlation $\rho$ used to plan the width. Default 0: the confidence interval for a correlation is widest at $\rho = 0$, so the default plans under the widest-interval case and is conservative for any other value.
width: Target full CI width on the correlation scale (e.g., 0.20 for a 90% CI of width 0.20). Must be in $(0, 2)$, the width of the correlation scale itself.
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).

Value

A 4-row data.frame with columns term and value: the recommended sample size necessary_N, the target width, the planning value population_r, and the resulting ci_width_expected at the chosen N.

Details

Closed form on the Fisher's $Z$ scale. The equivalence CI has half-width $h = z_{1-\alpha} / \sqrt{N - 3}$ on the Fisher's $Z$ scale, and its width on the correlation scale is $$w(N) \;=\; \tanh(Z_\rho + h) - \tanh(Z_\rho - h),$$ where $Z_\rho = \tanh^{-1}(\rho)$. The function returns the smallest integer $N \ge 4$ with $w(N) \le \omega$. At $\rho = 0$ this is available in closed form, $N = \lceil 3 + (z_{1-\alpha} / \tanh^{-1}(\omega / 2))^2 \rceil, $ and away from zero the back-transform shortens the interval, so the required N can only decrease as $|\rho|$ grows.

Choosing the width from equivalence bounds. To leave room for an equivalence verdict inside bounds $(-b, b)$, the interval must at minimum fit inside the bounds when centered at the anticipated $\rho$, so a width somewhat below $2 b$ (for $\rho$ near 0) is the natural target; the Monte Carlo sensitivity sibling ss_aipe_equivalence_r_sensitivity reports the realized proportion of equivalence verdicts at the planned N.

Assurance. Under assurance = q, the function increments $N$ until the Monte Carlo probability that the realized width is $\le \omega$ is at least $q$, drawing the sampling distribution of $\widehat Z$ as normal with mean $Z_\rho$ and variance $1 / (N - 3)$.

References

Counsell, A., & Cribbie, R. A. (2015). Equivalence tests for comparing correlation and regression coefficients. British Journal of Mathematical and Statistical Psychology, 68(2), 292–309. doi:10.1111/bmsp.12045

Goertzen, J. R., & Cribbie, R. A. (2010). Detecting a lack of association: An equivalence testing approach. British Journal of Mathematical and Statistical Psychology, 63(3), 527–537. doi:10.1348/000711009X475853

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., 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

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 the correlation of width <= 0.20, under
#    the widest-interval planning value rho = 0:
ss_aipe_equivalence_r(population_r = 0, width = 0.20)
#>  term              value
#>  necessary_N       272  
#>  width             0.2  
#>  population_r      0    
#>  ci_width_expected 0.2  

# 2. The same width assuming a true correlation of 0.30 requires
#    fewer participants, since the interval narrows away from zero:
ss_aipe_equivalence_r(population_r = 0.30, width = 0.20)
#>  term              value
#>  necessary_N       226  
#>  width             0.2  
#>  population_r      0.3  
#>  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_r(population_r = 0.30, width = 0.20, assurance = 0.80)
#>  term              value
#>  necessary_N       240  
#>  width             0.2  
#>  population_r      0.3  
#>  ci_width_expected 0.194