Determine the necessary sample size for the omnibus test of the squared multiple correlation coefficient (\(R^2\)), or the realized statistical power given a specified sample size, under either fixed or random predictors. The fixed-predictors path uses Cohen's (1988) noncentral F formulation; the random-predictors path uses the Lee (1971) two-moment approximation to the sampling distribution of the sample \(R^2\) under joint multivariate normality.
ss_power_R2(
population_R2 = NULL,
alpha_level = 0.05,
desired_power = 0.85,
p,
specified_N = NULL,
cohen_f2 = NULL,
null_R2 = 0,
random_predictors = TRUE,
print_progress = FALSE,
...
)A data.frame with columns term and value.
For an \(N\) search the rows are necessary_N,
actual_power, noncentral_f_parm (only meaningful for
random_predictors = FALSE; NA otherwise), and
effect_size (Cohen's \(f^2\)). For a power-at-specified-N
computation the first row is specified_N instead of
necessary_N.
Population squared multiple correlation coefficient
Type I error rate
Desired degree of statistical power
The number of predictor variables
The sample size used to calculate power (rather than determine necessary sample size). This is the total sample size across all groups or observations.
Cohen's (1988) effect size for multiple regression: population_R2/(1-population_R2)
Value of the null hypothesis that the squared multiple correlation will be evaluated against (this will typically be zero)
Whether the predictor variables are treated as random (TRUE, the default) or fixed (FALSE). See Details.
If the progress of the iterative procedure is printed to the screen as the iterations are occurring
Possible additional parameters for internal functions
Ken Kelley [email protected]
Determine the necessary sample size given a particular
population_R2, alpha_level, p, and
desired_power. Alternatively, given population_R2,
alpha_level, p, and specified_N, the function can
be used to determine the statistical power.
Fixed vs.\ random predictors. The two regression models give
different sampling distributions for the omnibus \(F\)-statistic,
and so different power. Under fixed predictors the design matrix is
treated as constant in hypothetical replications of the study, and
\(F\) follows a noncentral F with \(p\) and \(N-p-1\)
degrees of freedom and noncentrality \(\lambda = N \cdot f^2\),
where \(f^2 = \rho^2 / (1 - \rho^2)\) (Cohen, 1988). Under random
predictors the design matrix is itself a draw from a joint multivariate
normal distribution, and the unconditional distribution of the sample
\(R^2\) is given by Lee (1971); ss_power_R2() uses Lee's
two-moment Patnaik (1949) approximation to that distribution, the same
approximation ci_R2() uses for random-predictor confidence
intervals. Gatsonis and Sampson (1989) document the comparison and
show that Cohen's fixed-predictor formula tends to over-state power
(and so under-state required \(N\)) relative to the random model;
the discrepancy is modest at moderate to large \(N\) but non-trivial
for small \(N\) with moderate-to-large effects. In the behavioral,
educational, and social sciences predictor variables are almost always
random, so the default is random_predictors = TRUE; pass
random_predictors = FALSE for designs in which the predictor
variables are fixed by design (for example, planned dosing levels).
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum.
Gatsonis, C., & Sampson, A. R. (1989). Multiple correlation: Exact power and sample size calculations. Psychological Bulletin, 106(3), 516--524.
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")
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. (2008). Sample size planning with applications to multiple regression: Power and accuracy for omnibus and targeted effects. In P. Alasuutari, L. Bickman, & J. Brannen (Eds.), The Sage handbook of social research methods (pp. 166--192). Sage.
Lee, Y. S. (1971). Some results on the sampling distribution of the multiple correlation coefficient. Journal of the Royal Statistical Society, Series B, 33(1), 117--130.
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 \(R^2\) as a model comparison effect size.)
Maxwell, S. E., Kelley, K., & Rausch, J. R. (2008). Sample size planning for statistical power and accuracy in parameter estimation. Annual Review of Psychology, 59, 537--563. tools:::Rd_expr_doi("10.1146/annurev.psych.59.103006.093735")
Patnaik, P. B. (1949). The non-central \(\chi^2\)- and F-distributions and their applications. Biometrika, 36(1--2), 202--232. tools:::Rd_expr_doi("10.1093/biomet/36.1-2.202")
Anderson, S. F., Kelley, K., & Maxwell, S. E. (2017). Sample-size planning for more accurate statistical power: A method adjusting sample effect sizes for publication bias and uncertainty. Psychological Science, 28(11), 1547--1562. tools:::Rd_expr_doi("10.1177/0956797617723724")
ss_aipe_R2, ss_power_R2_sensitivity,
ss_power_reg_coef, ci_nc_F,
ci_R2
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 sample size for power:
power_fisher_exact(),
ss_aipe_mixed_effects(),
ss_power_R2_sensitivity(),
ss_power_c(),
ss_power_c_ancova(),
ss_power_composite_ancova(),
ss_power_composite_ancova_2group(),
ss_power_composite_anova(),
ss_power_composite_factorial_ancova(),
ss_power_composite_factorial_ancova_het(),
ss_power_composite_factorial_anova(),
ss_power_composite_sem(),
ss_power_contrast(),
ss_power_equivalence_c(),
ss_power_factorial_ancova(),
ss_power_factorial_anova(),
ss_power_indirect_effect(),
ss_power_mixed_effects(),
ss_power_one_way_anova(),
ss_power_pcm(),
ss_power_r(),
ss_power_rc(),
ss_power_reg_coef(),
ss_power_reg_coef_sensitivity(),
ss_power_rm_anova(),
ss_power_sc(),
ss_power_sem(),
ss_power_smd(),
ss_power_split_plot_anova()
# Random predictors (default; appropriate for most behavioral / social
# science applications).
ss_power_R2(population_R2 = .5, alpha_level = .05, desired_power = .85, p = 5)
# Fixed predictors (Cohen 1988): predictor variables fixed by design.
ss_power_R2(population_R2 = .5, alpha_level = .05, desired_power = .85,
p = 5, random_predictors = FALSE)
# Effect size input (Cohen's f^2).
ss_power_R2(cohen_f2 = 1, alpha_level = .05, desired_power = .85, p = 5)
# Realized power at a specified N.
ss_power_R2(population_R2 = .5, specified_N = 15, alpha_level = .05,
desired_power = .85, p = 5)
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