Power of the TOST or Noninferiority Test for a Linear Contrast

Computes the exact power of the Schuirmann (1987) two one-sided tests procedure, or of the one-sided noninferiority test, for a linear contrast of group means \(\psi = \sum_j c_j \mu_j\) with one pooled error term. For equivalence, the power is the probability that the \((1 - 2\alpha)\) confidence interval for \(\psi\) lies entirely inside \((-\delta_L, \delta_U)\), computed by numerical integration over the chi distribution of the estimated error standard deviation; for noninferiority, the power is a noncentral t probability in closed form. This is the contrast generalization of power_equivalence_md.

Usage

power_equivalence_c(
  c_weights,
  n,
  sigma,
  delta_lower = NULL,
  delta_upper = NULL,
  true_psi = 0,
  alpha_level = 0.05,
  side = c("equivalence", "noninferiority"),
  df_error = NULL
)

Arguments

c_weights

The contrast weights. The weights must sum to zero with the positive weights summing to 1 and the negative weights to -1, so that the bounds are on the raw scale of the response.

n

Sample sizes per group (if length 1, equal group sizes are assumed). Together with c_weights, n determines the standard error factor \(\sqrt{\sum_j c_j^2 / n_j}\).

sigma

The error standard deviation (the square root of the mean square error).

delta_lower, delta_upper

Equivalence bounds on the raw scale of the response. Both must be positive; the equivalence region is \((-\delta_L, +\delta_U)\). If only delta_upper is supplied, the bounds are symmetric. Noninferiority uses \(-\delta_L\) alone.

true_psi

The population value of the contrast at which the power is evaluated. Default 0, the most favorable point for an equivalence declaration.

alpha_level

One-sided significance level for each test. Default 0.05.

side

"equivalence" (default) for the TOST power, or "noninferiority" for the one-sided test against \(-\delta_L\).

df_error

The error degrees of freedom. Defaults to \(N - J\); supply it directly when the error term comes from a model with more groups or additional predictors than the contrast involves (for example, a five-group model supplying the pooled error for a two-group contrast).

Value

A one-row data.frame with columns term ("power") and value (the computed power, in \([0, 1]\)).

Details

Equivalence power. Conditional on the estimated error standard deviation \(S\), the \((1 - 2\alpha)\) CI fits inside the bounds on a computable event, and the unconditional power integrates that event over the scaled chi distribution of \(S\) on df_error degrees of freedom. With c_weights = c(1, -1) and equal n, the result reproduces power_equivalence_md exactly.

Noninferiority power. The one-sided test rejects when \(t = (\hat\psi + \delta_L)/\mathrm{SE}(\hat\psi)\) exceeds \(t_{1-\alpha,\nu}\), so the power is \(\Pr(T'_{\nu}(\lambda) > t_{1-\alpha,\nu})\) with noncentrality \(\lambda = (\psi + \delta_L)/(\sigma \sqrt{\sum_j c_j^2/n_j})\).

The feasibility condition. If the expected half-width of the CI is not smaller than the bounds allow, the equivalence power is zero or near zero regardless of true_psi: an imprecise design cannot declare equivalence even when the arms are truly identical. Planning should target a half-width of about half the bound; see ss_power_equivalence_c and ss_aipe_c.

References

Chattopadhyay, B., Bandyopadhyay, T., Kelley, K., & Padalunkal, J. J. (2025). A sequential approach for noninferiority or equivalence of a linear contrast under cost constraints. Psychological Methods, 30(2), 425–439. doi:10.1037/met0000570

Phillips, K. F. (1990). Power of the two one-sided tests procedure in bioequivalence. Journal of Pharmacokinetics and Biopharmaceutics, 18(2), 139–144. doi:10.1007/BF01063556

Schuirmann, D. J. (1987). A comparison of the two one-sided tests procedure and the power approach for assessing the equivalence of average bioavailability. Journal of Pharmacokinetics and Biopharmaceutics, 15(6), 657–680.

See also

power_equivalence_md, ss_power_equivalence_c, equivalence_c, ss_aipe_c

Other equivalence testing: equivalence_c(), equivalence_r(), equivalence_smd(), plot_equivalence(), power_density_equivalence_md(), power_equivalence_md(), power_equivalence_md_plot(), ss_power_equivalence_c()

Author

Ken Kelley kkelley@nd.edu

Examples

# 1. Two groups of 61 and 113 sharing a five-group pooled error term
#    (so df_error = 404 - 5 = 399), bounds of 5 raw-scale points:
#    the design's probability of declaring equivalence when the
#    groups are truly identical.
power_equivalence_c(c_weights = c(1, -1), n = c(61, 113),
                    sigma = 15.67, delta_upper = 5,
                    true_psi = 0, df_error = 399)
#>  term  value
#>  power 0.281

# 2. The same design's noninferiority power at the same point.
power_equivalence_c(c_weights = c(1, -1), n = c(61, 113),
                    sigma = 15.67, delta_upper = 5,
                    true_psi = 0, df_error = 399,
                    side = "noninferiority")
#>  term  value
#>  power 0.641

# 3. Agreement with power_equivalence_md() in the two-group case
#    (Phillips, 1990, Table 1: expected 0.8029678).
power_equivalence_c(c_weights = c(1, -1), n = 24, sigma = 0.20,
                    delta_lower = 0.2, delta_upper = 0.2,
                    true_psi = 0.05, df_error = 22)
#>  term  value
#>  power 0.803