DMAR (version 1.0.0)

ss_aipe_r: Sample Size for AIPE on a Pearson Correlation

Description

Determines the sample size needed for a confidence interval on a population Pearson correlation \(\rho\) to have a desired width (accuracy in parameter estimation; Kelley & Maxwell, 2003). The interval planned for is the Fisher's Z interval that correlations_test reports for method = "pearson": the correlation is transformed as \(z(r) = \mathrm{atanh}(r)\), an interval with standard error \(1/\sqrt{n - 3}\) is formed on the z scale, and the limits are back-transformed through \(\tanh(\cdot)\) (Fisher, 1921; Bonett & Wright, 2000, Equation 2). Because the plan targets the same interval the analysis will report, the planned width and the analyzed width agree.

Usage

ss_aipe_r(rho, width, conf_level = 0.95, assurance = NULL)

Value

A data.frame with the rows necessary_N

(the recommended total sample size, rounded up),

expected_width at that sample size, and the inputs echoed back.

Arguments

rho

Anticipated population Pearson correlation, in \((-1, 1)\).

width

Desired full width of the confidence interval on the correlation.

conf_level

Desired confidence level (default 0.95).

assurance

Optional. Probability that the realized CI is no wider than width (\(1 - \gamma\)). When supplied, the sample size is inflated using the standard chi squared correction (Kelley, 2008); when NULL, the assurance is fixed at 0.5.

Author

Ken Kelley [email protected]

Details

Closed-form first pass. On the Fisher's Z scale the interval has half-width \(z_{1 - \alpha/2} / \sqrt{n - 3}\), and the delta method maps it back to the correlation scale as approximately $$w \;\approx\; 2\, z_{1 - \alpha/2} \, \frac{1 - \rho^2}{\sqrt{n - 3}},$$ which solves to the first-stage approximation of Bonett and Wright (2000), $$n_0 \;=\; 3 + \Big\lceil 4\, (z_{1 - \alpha/2})^2 (1 - \rho^2)^2 / w^2 \Big\rceil.$$

Exact iteration. The back-transformed width depends on \(\rho\) through \(\tanh(\cdot)\), so the delta method approximation can land a few observations off in either direction. Starting from \(n_0\), the function evaluates the exact back-transformed width \(\tanh(z_\rho + z_{1 - \alpha/2}/\sqrt{n - 3}) - \tanh(z_\rho - z_{1 - \alpha/2}/\sqrt{n - 3})\) and steps the integer \(n\) until it is the smallest sample size whose width is at or below width. Where Bonett and Wright (2000) stop after a single second-stage adjustment, this search is exact.

The planning value matters least near zero. At a fixed sample size the back-transformed width is largest at \(\rho = 0\) and shrinks as \(|\rho|\) grows, so a planning value closer to zero yields a larger, more conservative sample size. When little is known about the population correlation, rho = 0 gives the sample size that suffices for any population value.

When to use simple vs. partial correlation planning. Use this function when the inferential target is the correlation between two variables with nothing partialed out. When the target is the correlation after statistically controlling for other variables, see ss_aipe_partial_r.

The Monte Carlo companion ss_aipe_r_sensitivity evaluates how the plan behaves when the population correlation differs from the planning value.

References

Bonett, D. G., & Wright, T. A. (2000). Sample size requirements for estimating Pearson, Kendall and Spearman correlations. Psychometrika, 65(1), 23--28. tools:::Rd_expr_doi("10.1007/BF02294183")

Fisher, R. A. (1921). On the "probable error" of a coefficient of correlation deduced from a small sample. Metron, 1, 3--32.

Kelley, K. (2008). Sample size planning for the squared multiple correlation coefficient: Accuracy in parameter estimation via narrow confidence intervals. Multivariate Behavioral Research, 43(4), 524--555. tools:::Rd_expr_doi("10.1080/00273170802490632")

Kelley, K., & Maxwell, S. E. (2003). Sample size for multiple regression: Obtaining regression coefficients that are accurate, not simply significant. Psychological Methods, 8(3), 305--321. tools:::Rd_expr_doi("10.1037/1082-989X.8.3.305")

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

See Also

ss_aipe_r_sensitivity, correlations_test, ss_aipe_partial_r, ss_power_r, convert_r_Z

design_consequences for what a chosen design delivers: power, the Type S (sign) and Type M (exaggeration) errors of the significance filter, and the expected confidence interval width.

Other AIPE sample size planning: ss_aipe_c_sensitivity(), ss_aipe_cliff_delta(), ss_aipe_cliff_delta_sensitivity(), ss_aipe_composite_sem(), ss_aipe_equivalence_r(), ss_aipe_equivalence_r_sensitivity(), ss_aipe_equivalence_smd(), ss_aipe_equivalence_smd_sensitivity(), ss_aipe_icc(), ss_aipe_icc_sensitivity(), ss_aipe_indirect_effect(), ss_aipe_indirect_effect_sensitivity(), ss_aipe_mixed_effects_sensitivity(), ss_aipe_omega_squared(), ss_aipe_omega_squared_sensitivity(), ss_aipe_partial_r(), ss_aipe_partial_r_sensitivity(), ss_aipe_pcm_sensitivity(), ss_aipe_r_sensitivity(), ss_aipe_reliability_sensitivity(), ss_aipe_semipartial_r(), ss_aipe_semipartial_r_sensitivity()

Examples

# Plan n so the 95% CI on the Pearson correlation has full width
# at most 0.20, when the anticipated correlation is 0.30.
ss_aipe_r(rho = 0.30, width = 0.20)

# A narrower target width requires a larger sample size.
ss_aipe_r(rho = 0.30, width = 0.10)

# With 80% assurance that the realized interval is no wider than
# the target (Kelley, 2008):
ss_aipe_r(rho = 0.30, width = 0.20, assurance = 0.80)

# Planning at rho = 0 gives the sample size that suffices for any
# population correlation, since the interval is widest there.
ss_aipe_r(rho = 0, width = 0.20)