Table 2.

Concordance rates and equity values in alternative trial designs.

	Design
	Parallel group and fully randomised	Two-stage	Partially randomised	Zelen single consent, treatments concealed	Zelen single consent, treatments revealed	Zelen double consent, treatments concealed	Zelen double consent, treatments revealed
Concordance rate for patients who prefer treatment A	ρ	θ + (1 − θ)ρ	1	θφ	θ	1 − φ(1 − θ)	1
Concordance rate for patients who prefer treatment B	1 − ρ	θ + (1 − θ)(1 − ρ)	1	1 − θφ	1	1 − φθ	1
Concordance rate	αρ + β(1 − ρ) + γ	θ + (1 − θ) [αρ + β(1 − ρ) + γ]	1	αθφ + β(1 − θφ) + γ	αθ + β + γ	1 − φ[α(1 − θ) + βθ]	1
Scenarios to maximise concordance (C = 1)	1) γ = 1a 2) α = 1, ρ = 1a 3) β = 1, ρ = 0a	1) γ = 1a 2) θ = 1a 3) α = ρ = 1a 4) β = 1, ρ = 0a	Always	1) γ = 1a 2) α = 1, θ = φ = 1a 3) β = 1, θ = φ = 0a	1) γ = 1a 2) β = 1a 3) θ = 1a	γ = 1a φ = 0a α = θ = 0a β = θ = 1a	Always
Scenarios where concordance is conditionallyb maximised	ρ = 1/2(1)	ρ = 1/2(2)	Always	θφ = 1/2(3)	θ = 1a(4)	φ = 0a(5) θ = 1/2	Always
Conditional maximum concordance	(1 + γ)/2	θ + (1 − θ)(1 + γ)/2	1	(1 + γ)/2	1	1 (if φ = 0) 1 − 1/2 (1 − γ) (if θ = 1/2)	1
Equity (A vs. B)	2ρ − 1	(1 − θ)(2ρ − 1)	0	2θφ − 1	θ − 1	φ(2θ − 1)	0
Scenarios to optimise equity	ρ = 1/2	ρ = 1/2 θ = 1a	Always	θφ = 1/2	θ = 1a	φ = 0a θ = 1/2	Always
Conditional optimal equity	0	0	0	0	0	0	0

ρ = proportion randomised to treatment A in standard design, and in a random arm of 2-stage design; (α, β) = preference rates for treatment A or B; γ = undecided rate; θ = proportion randomised to choice arm in two-stage design, or to A-consent arms in Zelen designs; φ = proportion of patients who consent to randomised treatment in Zelen concealed designs.

^a Implies degenerate design.

^b Maximised over definitive treatment preference rates (α and β), conditional on fixed parameters (1) γ, ρ; (2) γ, ρ, θ; (3) γ, θφ; (4) γ, θ; (5) γ, φ, θ.