Sample Size for AIPE on a Semipartial (Part) Correlation

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
)

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).

Value

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

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. 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. 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 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)
#>  term           value
#>  necessary_N    605  
#>  expected_width 0.15 
#>  r_sp           0.25 
#>  J              3    
#>  width_target   0.15 
#>  conf_level     0.95 
#> 
#> Confidence level: 95%

# 2. With 80% assurance:
ss_aipe_semipartial_r(r_sp = 0.25, J = 3, width = 0.15,
                      assurance = 0.80)
#>  term           value
#>  necessary_N    635  
#>  expected_width 0.146
#>  r_sp           0.25 
#>  J              3    
#>  width_target   0.15 
#>  conf_level     0.95 
#> 
#> Confidence level: 95%