Sample Size for AIPE on a Partial Correlation

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
)

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.

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

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. doi:10.1207/s15327906mbr3803_02

Bonett, D. G. (2008). Confidence intervals for standardized linear contrasts of means. Psychological Methods, 13(2), 99–109. 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. 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 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. doi:10.1037/0033-2909.118.1.155

Author

Ken Kelley kkelley@nd.edu

Examples

# 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)
#>  term           value
#>  necessary_N    322  
#>  expected_width 0.2  
#>  rho            0.3  
#>  J              2    
#>  width_target   0.2  
#>  conf_level     0.95 
#> 
#> Confidence level: 95%

# 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)
#>  term           value
#>  necessary_N    322  
#>  expected_width 0.2  
#>  rho            0.3  
#>  J              2    
#>  width_target   0.2  
#>  conf_level     0.95 
#> 
#> Confidence level: 95%

# 3. With 80% assurance (Kelley 2008):
ss_aipe_partial_r(rho = 0.30, J = 2, width = 0.20, assurance = 0.80)
#>  term           value
#>  necessary_N    344  
#>  expected_width 0.193
#>  rho            0.3  
#>  J              2    
#>  width_target   0.2  
#>  conf_level     0.95 
#> 
#> Confidence level: 95%