ss_aipe_partial_r: Sample Size for AIPE on a Partial Correlation

Description

Determines the sample size needed for a confidence interval on a population partial correlation $\rho_{XY \cdot Z_1 \cdots Z_J}$ to have a desired width (Accuracy in Parameter Estimation; Kelley, 2008). The function inverts the asymptotic variance of the partial Pearson correlation, either on the raw scale (Olkin & Finn, 1995) or on the Fisher's Z-transformed scale (Fisher, 1921, 1924; Bonett, 2008), and solves for the smallest $n$ that achieves the target half-width (or full width).

Usage

ss_aipe_partial_r(
  rho,
  J,
  width,
  which_width = c("Full", "Lower", "Upper"),
  conf_level = 0.95,
  fisher_z = FALSE,
  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 partial correlation, in $(-1, 1)$.
J: Number of variables partialled out (count of $Z_1, \ldots, Z_J$); must be at least 1.
width: Desired full width of the confidence interval on the partial correlation.
which_width: Whether width is the "Full" width of the interval (default) or a half-width: "Lower" and "Upper" both interpret width as half the full width, so they plan for a full width of twice width and return the same sample size. Because the interval is not generally symmetric about the estimate (with fisher_z = TRUE in particular), its realized lower and upper half-widths can differ from each other and from half the full width; the planner does not target them separately. A genuinely one-sided width target is not currently offered.
conf_level: Desired confidence level (default 0.95).
fisher_z: Logical. If TRUE, the half-width target is applied on the Fisher-z scale (variance $1/(n - J - 3)$; Bonett, 2008) and the resulting CI is back-transformed via $\tanh$. If FALSE (default), uses the raw-scale Olkin-Finn (1995) asymptotic variance.
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

Raw-scale Olkin-Finn asymptotic variance. The half-width of a $100(1 - \alpha)\%$ CI on the partial Pearson correlation is approximately $$w_{1/2} \;\approx\; z_{1 - \alpha/2} \cdot \sqrt{\,\frac{(1 - \rho_{XY \cdot Z}^{\,2})^2}{n - J - 1}\,}.$$ Solving for $n$: $$n \;=\; J + 1 + \Big\lceil (z_{1 - \alpha/2})^2 \cdot (1 - \rho_{XY \cdot Z}^{\,2})^2 / w_{1/2}^{2} \Big\rceil.$$ This is the planning analog of the half-width of ci_r applied to a partial correlation.

Fisher-z scale (recommended for small $\rho$, near boundary, or small $n - J$). Bonett (2008) advocates planning on the variance-stabilized Fisher-z scale and back-transforming the bounds. On the Fisher's $Z$ scale, the asymptotic half-width is $$w^{(z)}_{1/2} \;\approx\; z_{1 - \alpha/2} / \sqrt{n - J - 3}.$$ Solving for the $n$ that achieves a given back-transformed $w_{1/2}$ is done by a 1-D search; this is generally the more accurate route when $n$ is small or $|\rho|$ is large.

When to use partial vs. simple correlation planning. Use this function when the inferential target is the population correlation between $X$ and $Y$ after statistically controlling for $Z_1, \ldots, Z_J$. For the simple Pearson correlation, see ss_aipe_r.

Note on conservatism of the assurance plan. The empirical simulation study of the AIPE planner family finds that ss_aipe_partial_r() is tight (zero overshoot) at 80% assurance but modestly conservative at 99% assurance, with an empirical ideal sample size of about 5 to 10 subjects smaller than the recommended sample size. The mechanism is the usual one for AIPE assurance plans: the Olkin-Finn (1995) Wald- style upper bound on $\Pr(\widehat W > \omega)$ that the planner inverts is not tight at the recommended sample size, especially at the 99% level where the inversion has to push further into the upper tail of $\widehat W$. The recommended sample size is a sufficient sample size rather than the smallest possible sample size. ss_aipe_partial_r_sensitivity quantifies the overshoot for any one condition.

References

Algina, J., & Olejnik, S. (2003). Sample size tables for correlation analysis with applications in partial correlation and multiple regression analysis. Multivariate Behavioral Research, 38(3), 309--323. tools:::Rd_expr_doi("10.1207/s15327906mbr3803_02")

Bonett, D. G. (2008). Confidence intervals for standardized linear contrasts of means. Psychological Methods, 13(2), 99--109. tools:::Rd_expr_doi("10.1037/1082-989X.13.2.99")

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

Fisher, R. A. (1924). The distribution of the partial correlation coefficient. Metron, 3, 329--332.

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 3 on the one-way ANOVA and Chapter 4 on contrasts.)

Olkin, I., & Finn, J. D. (1995). Correlations redux. Psychological Bulletin, 118(1), 155--164. tools:::Rd_expr_doi("10.1037/0033-2909.118.1.155")

Examples

Run this code

# 1. Plan n so the 95% CI on rho_XY.Z (J = 2 controls) has
#        full width <= 0.20, when the anticipated partial r is 0.30.
ss_aipe_partial_r(rho = 0.30, J = 2, width = 0.20)

# 2. Same problem on the Fisher's Z scale (Bonett 2008):
ss_aipe_partial_r(rho = 0.30, J = 2, width = 0.20, fisher_z = TRUE)

# 3. With 80% assurance (Kelley 2008):
ss_aipe_partial_r(rho = 0.30, J = 2, width = 0.20, assurance = 0.80)

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