Two-group comparison of an ordered categorical outcome (Whitehead) — calculate_two

Whitehead (1993) Eq 10 (p. 2260) with allocation ratio A:

Usage

calculate_two_ordinal(or, prop_bar, alpha, power, allocation = 1)

Arguments

or: Anticipated odds ratio (proportional odds); > 0, != 1.
prop_bar: Anticipated average category proportions; positive, summing to 1.
allocation: Allocation ratio phi = n_Test / n_Standard (Whitehead's A); number > 0, default 1 (equal groups).

Details


  n = 3 (A + 1)^2 (u_{a/2} + u_b)^2 / [ A theta^2 (1 - sum pbar_i^3) ]

and Machin et al. (2018) Eq 4.3 (p. 57) with allocation ratio phi:


  N = Gamma (1 + phi)^2 / phi * (z_{1-a/2} + z_{1-b})^2 / (log OR)^2,
  Gamma = 3 / (1 - sum pbar_i^3)                          (Eq 4.4)

The two are algebraically identical when A = phi: Whitehead's numerator 3 (A+1)^2 and Machin's Gamma (1+phi)^2/phi differ only by the allocation factor, which is 1 when the groups are equal.

Anchors: Machin Example 4.1 (p. 59): kappa = 4, phi = 1, OR = 3.65, pbar = (0.2564, 0.2792, 0.2056, 0.2619), 2-sided alpha = 0.05, power = 0.80 -> N = 60.09 -> 60 total. Whitehead Example 1 (p. 2261): OR = 0.887, pbar = (0.289, 0.486, 0.153, 0.072), alpha = 0.05, power = 0.90 -> n = 187 total (94 per group).