Power Analysis for Pearson Correlation — pwrcortest • pwranova

Computes statistical power, required total sample size, \(\alpha\), or the minimal detectable correlation coefficient for a Pearson correlation test. Two computational methods are supported: exact noncentral t (method = "t") and Fisher's z-transformation with normal approximation (method = "z").

Usage

pwrcortest(
  alternative = c("two.sided", "one.sided"),
  n_total = NULL,
  alpha = NULL,
  power = NULL,
  rho = NULL,
  method = c("t", "z"),
  ncp_scale = c("n", "df"),
  bias_correction = FALSE,
  nlim = c(2, 10000)
)

Arguments

alternative: Character. Either "two.sided" or "one.sided".
n_total: Integer scalar. Total sample size (\(n\)). Must be \(\ge 3\) for method = "t" and \(\ge 4\) for method = "z". If NULL, the function solves for n_total.
alpha: Numeric in \((0,1)\). If NULL, it is solved for.
power: Numeric in \((0,1)\). If NULL, it is computed; if n_total is NULL, n_total is solved to attain this power.
rho: Numeric correlation coefficient in \((-1,1)\), nonzero. If negative, it is converted to its absolute value. If NULL, rho is solved for given the other inputs.
method: Character. Either "t" (noncentral t-distribution) or "z" (Fisher's z transformation with normal approximation).
ncp_scale: Character. Applies only to method = "t" and determines the scaling used in the noncentrality parameter. Either "n" (default; \(\sqrt{n}\), corresponding to the formulation used in GPower) or "df" (\(\sqrt{n-2}\)).
bias_correction: Logical. Applies only to method = "z". If TRUE, uses the bias-corrected Fisher z-transformation \(z_p = \operatorname{atanh}(r) + r/(2(n-1))\).
nlim: Integer vector of length 2. Search range of n_total when solving sample size.

Value

A one-row data.frame with class "cal_power", "cal_n", "cal_alpha", or "cal_es", depending on the solved quantity. Columns:

df (only for method = "t")
n_total, alpha, power
rho, t_critical or z_critical
ncp (noncentrality parameter or mean under the alternative: see Details)

Details

Exactly one of n_total, rho, alpha, or power must be NULL; that quantity is then solved.
The sign of rho is ignored; its absolute value is used, because statistical power depends on the magnitude of the effect rather than its direction.
For method = "t", computations are based on the noncentral t-distribution with noncentrality parameter \(\lambda = \tfrac{\rho}{\sqrt{1-\rho^2}} \sqrt{k}\), where the scaling factor \(k\) is determined by ncp_scale. When ncp_scale = "n" (default), \(k = n\), corresponding to the formulation used in GPower. When ncp_scale = "df", \(k = n-2\), which follows directly from the classical test statistic formula for Pearson's correlation test. These two formulations arise from different derivations of the power function: the \(\sqrt{n-2}\) form follows directly from the test statistic, whereas the \(\sqrt{n}\) form is obtained from alternative derivations used in some power-analysis implementations.
For method = "z", computations use Fisher's z transformation of the population correlation, \(z_\rho = \operatorname{atanh}(\rho)\). Let \(W = \sqrt{n-3}\, z\). Under the alternative hypothesis, \(W \sim \mathrm{Normal}(\mu,\,1)\) with \(\mu = \sqrt{n-3}\, z_\rho\). If bias_correction = TRUE, \(\rho\) is first bias-corrected before applying Fisher's transform. Critical values are taken from the central normal distribution under \(H_0:\rho=0\) (i.e., \(W \sim \mathrm{Normal}(0,1)\) under the null). The returned ncp equals \(\mu\).
Validation against GPower: Results have been confirmed to match those produced by GPower for equivalent correlation tests using the noncentral t-distribution.
Cohen (1988) suggested rough benchmarks of 0.10 (small), 0.30 (medium), and 0.50 (large) for Pearson's correlation coefficient (\(r\)). These values should be regarded as rough guidelines rather than strict criteria.
Note: Results from method = "z" will not exactly match pwr::pwr.r.test, because pwr uses a hybrid approach combining the Fisher-z approximation with a t-based critical value.

References

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum Associates.

Examples

# (1) Compute power for rho = 0.3, N = 50, two-sided test
pwrcortest(alternative = "two.sided", n_total = 50, rho = 0.3, alpha = 0.05)
#>   df n_total alpha     power rho t_critical      ncp
#> 1 48      50  0.05 0.5867505 0.3   2.010635 2.223748
#> Power (1 - beta) was calculated based on the total sample size, effect size, and alpha.

# (2) Solve required N for target power, using Fisher-z method
pwrcortest(method = "z", rho = 0.2, alpha = 0.05, power = 0.8)
#>   n_total alpha     power rho z_critical     ncp
#> 1     194  0.05 0.8000666 0.2   1.959964 2.80182
#> The required total sample size was calculated based on the effect size, alpha, and power.
#> Note: 'power' indicates the achieved power rather than the target power.

# (3) Solve minimal detectable correlation
pwrcortest(n_total = 60, alpha = 0.05, power = 0.9, rho = NULL)
#>   df n_total alpha power       rho t_critical      ncp
#> 1 58      60  0.05   0.9 0.3915971   2.001717 3.296573
#> The minimal detectable Cohen's f and partial eta squared were calculated based on the total sample size, alpha, and power.