ss_aipe_semipartial_r: Sample Size for AIPE on a Semipartial (Part) Correlation

Description

Determines the sample size needed for a confidence interval on a population semipartial correlation $r_{Y(X \cdot Z_1 \cdots Z_J)}$ (the unique contribution of $X$ to $Y$ after controlling for $Z_1, \ldots, Z_J$, with $Y$ not residualized) to have a desired width, using the Olkin-Finn (1995) / Algina-Olejnik (2003) asymptotic variance and the AIPE framework of Kelley & Maxwell (2003).

Usage

ss_aipe_semipartial_r(
  r_sp,
  J,
  width,
  which_width = c("Full", "Lower", "Upper"),
  conf_level = 0.95,
  assurance = NULL
)

Value

A data.frame with rows for the recommended sample size, the expected CI width at that sample size, and the inputs echoed back.

Arguments

r_sp: Anticipated population semipartial correlation, in $(-1, 1)$.
J: Number of variables partialled out of $X$ (count of $Z_1, \ldots, Z_J$); must be at least 1.
width: Desired full width of the confidence interval on the semipartial correlation.
which_width: Whether width refers to the "Full" width (default) or the "Lower"/"Upper" half-width.
conf_level: Desired confidence level (default 0.95).
assurance: Optional. Probability that the realized CI is no wider than width. When supplied, the sample size is inflated using the standard chi squared correction (Kelley & Maxwell, 2003).

Author

Ken Kelley [email protected]

Details

Asymptotic variance of the semipartial. Under multivariate normality, the sample semipartial correlation $r_{Y(X \cdot Z)}$ has asymptotic variance $$\mathrm{Var}(\hat r_{Y(X \cdot Z)}) \;\approx\; \frac{(1 - r_{Y(X \cdot Z)}^2)^2}{n - J - 1}$$ (Olkin & Finn, 1995, with the partial-correlation degrees-of-freedom correction). Inverting for the sample size needed to achieve a target half-width $w_{1/2}$ at confidence level $1 - \alpha$: $$n \;=\; J + 1 + \Big\lceil z_{1 - \alpha/2}^{2} \cdot (1 - r_{Y(X \cdot Z)}^{2})^2 / w_{1/2}^{2} \Big\rceil.$$

Comparison with partial-r planning. The partial correlation $r_{XY \cdot Z}$ divides the covariance after residualizing both $X$ and $Y$ on $Z$; the semipartial divides after residualizing only $X$. The semipartial is the natural effect size companion to a standardized regression coefficient: its square equals the $\Delta R^2$ contributed by $X$ above and beyond the controls. See var_semipartial_r for the asymptotic variance, and ss_aipe_partial_r for the partial-correlation analog of this function.

Note on conservatism of the assurance plan. The empirical simulation study of the AIPE planner family finds that ss_aipe_semipartial_r() is on the boundary of its valid range at 80% assurance and modestly conservative at 99% assurance. At $\gamma = 0.80$, the realized assurance at the recommended sample size is within Monte Carlo error of the target, that is, the bound is operating at the edge of its validity. At $\gamma = 0.99$, the ideal sample size is about 15 to 20 subjects smaller than the recommended sample size, reflecting the looser upper-tail bound at the 99% level. The recommended sample size is therefore a sufficient sample size rather than the smallest possible sample size. A small safety margin (5 to 10 subjects) is advisable when planning at $\gamma = 0.80$. ss_aipe_semipartial_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")

Cohen, J., Cohen, P., West, S. G., & Aiken, L. S. (2003). Applied multiple regression/correlation analysis for the behavioral sciences (3rd ed.). Lawrence Erlbaum.

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 r_sp (J = 3) has full width <= 0.15
#        when the anticipated semipartial is 0.25.
ss_aipe_semipartial_r(r_sp = 0.25, J = 3, width = 0.15)

# 2. With 80% assurance:
ss_aipe_semipartial_r(r_sp = 0.25, J = 3, width = 0.15,
                      assurance = 0.80)

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