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.
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); whenNULL, the assurance is fixed at 0.5.
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.
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. 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. 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. 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()
Author
Ken Kelley kkelley@nd.edu
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)
#> term value
#> necessary_N 320
#> expected_width 0.2
#> rho 0.3
#> width_target 0.2
#> conf_level 0.95
#>
#> Confidence level: 95%
# A narrower target width requires a larger sample size.
ss_aipe_r(rho = 0.30, width = 0.10)
#> term value
#> necessary_N 1274
#> expected_width 0.1
#> rho 0.3
#> width_target 0.1
#> conf_level 0.95
#>
#> Confidence level: 95%
# 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)
#> term value
#> necessary_N 342
#> expected_width 0.193
#> rho 0.3
#> width_target 0.2
#> conf_level 0.95
#>
#> Confidence level: 95%
# 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)
#> term value
#> necessary_N 385
#> expected_width 0.2
#> rho 0
#> width_target 0.2
#> conf_level 0.95
#>
#> Confidence level: 95%