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Power Analysis for One-/Two-Sample and Paired t Tests

Source: R/pwrttest.r
pwrttest.Rd

Computes statistical power, required total sample size, \(\alpha\), or the minimal detectable effect size for a t-test in one of three designs: one-sample, two-sample (independent), or paired/repeated measures.

Usage

pwrttest(
  paired = FALSE,
  onesample = FALSE,
  n_total = NULL,
  alpha = NULL,
  power = NULL,
  delta = NULL,
  cohensf = NULL,
  peta2 = NULL,
  alternative = c("two.sided", "one.sided"),
  nlim = c(2, 10000)
)

Arguments

paired

Logical. FALSE for two-sample (independent; default), TRUE for paired/repeated-measures. Ignored when onesample = TRUE.

onesample

Logical. TRUE for the one-sample t-test; if TRUE, paired is ignored.

n_total

Integer scalar. Total sample size. If NULL, the function solves for n_total.

alpha

Numeric in \((0,1)\). If NULL, it is solved for given the other inputs.

power

Numeric in \((0,1)\). If NULL, it is computed; if n_total is NULL, n_total is solved to attain this power.

delta

Numeric. Cohen's \(d\)-type effect size. If negative, it is converted to its absolute value. If NULL, it is derived from cohensf or peta2 when available. If all three effect-size arguments (delta, cohensf, peta2) are NULL, then the effect size is treated as the unknown quantity and is solved for given n_total, alpha, and power. The exact definition depends on the design:

  • One-sample: Cohen's \(d = (\mu - \mu_0)/\sigma\).

  • Paired: Cohen's \(d_z = (\mu_2 - \mu_1)/\sigma_D\), where \(\sigma_D\) denotes the population standard deviation of the difference scores.

  • Two-sample (equal allocation): Cohen's \(d = (\mu_2 - \mu_1)/\sigma\), where \(\sigma\) denotes the common population standard deviation.

cohensf

Numeric (non-negative). Cohen's \(f\). If NULL, it can be derived from delta; if delta is supplied, cohensf is ignored. Effect-size relations by design:

  • Two-sample (equal allocation): \(d = 2f\)

  • Paired: \(d_z = f\)

  • One-sample: \(f\) and \(\eta_p^2\) are not supported

peta2

Numeric in \((0,1)\). Partial eta squared. If NULL, it can be derived from cohensf; if delta is supplied, peta2 is ignored. Not defined for one-sample designs.

alternative

Character. Either "two.sided" or "one.sided".

nlim

Integer vector of length 2. Search range of total n 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, n_total, alpha, power, delta, cohensf, peta2, t_critical, ncp.

Details

  • The sign of delta is ignored; its absolute value is used, because statistical power depends on the magnitude of the effect rather than its direction.

  • If multiple effect-size arguments are supplied (delta, cohensf, peta2), precedence is delta \(>\) cohensf \(>\) peta2; the rest are ignored with a warning.

  • For the two-sample design, equal allocation is assumed; n_total must be even when provided, and the solved n_total will be an even number.

  • For the paired design, the effect size is interpreted as \(d_z\).

  • The noncentrality parameter is computed from the design-specific effect size and sample size: \(\lambda = \delta\sqrt{n}\) for one-sample and paired designs, and \(\lambda = \delta\sqrt{n}/2\) for the two-sample equal-allocation design, where \(n\) denotes n_total.

  • Computations use the central and noncentral t-distributions (stats::qt, stats::pt); root finding uses stats::uniroot() where needed.

  • Results have been validated to match those produced by G*Power for equivalent one-sample, paired, and two-sample t tests.

  • For the two-sample designs, Cohen (1988) suggested rough benchmarks of 0.20 (small), 0.50 (medium), and 0.80 (large) for \(d\). These values should be regarded as rough guidelines rather than strict criteria.

References

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

Examples

# (1) Two-sample (independent), compute power given N and d
pwrttest(paired = FALSE, onesample = FALSE, alternative = "two.sided",
         n_total = 128, delta = 0.50, alpha = 0.05)
#>    df n_total alpha     power delta cohensf      peta2 t_critical      ncp
#> 1 126     128  0.05 0.8014596   0.5    0.25 0.05882353   1.978971 2.828427
#> Power (1 - beta) was calculated based on the total sample size, effect size, and alpha.

# (2) Paired t-test, solve required N for target power
pwrttest(paired = TRUE, onesample = FALSE, alternative = "one.sided",
         n_total = NULL, delta = 0.40, alpha = 0.05, power = 0.90)
#>   df n_total alpha     power delta_z cohensf    peta2 t_critical      ncp
#> 1 54      55  0.05 0.9004524     0.4     0.4 0.137931   1.673565 2.966479
#> 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) One-sample t-test, solve alpha given N and power
pwrttest(onesample = TRUE, alternative = "two.sided",
         n_total = 40, delta = 0.40, alpha = NULL, power = 0.80)
#>   df n_total     alpha power delta cohensf peta2 t_critical      ncp
#> 1 39      40 0.1001708   0.8   0.4      NA    NA   1.683997 2.529822
#> Alpha was calculated based on the total sample size, effect size, and power.

# (4) Two-sample, specify effect via f or partial eta^2 (converted internally)
pwrttest(paired = FALSE, cohensf = 0.25, n_total = NULL, alpha = 0.05, power = 0.80)
#>    df n_total alpha     power delta cohensf      peta2 t_critical      ncp
#> 1 126     128  0.05 0.8014596   0.5    0.25 0.05882353   1.978971 2.828427
#> 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.