Clinical trials - Deriving Power and Sample size

Question 1

Between-subject design

Derive the power from the sample size

Answer 1

n = n=(2σ^2 (z_(α/2)+z_β )^2)/τ^2

power = 1--ϕ(z_(1-α/2)-τ/σλ)
β=1-power=1-(1-ϕ(z_(1-α/2)-τ/σλ))
                    = ϕ(z_(1-α/2)-τ/σλ)
ϕ^(-1) (β)=z_(1-α/2)-τ/σλ
z_β=z_(1-α/2)-τ/(σsqrt(2/n))
σz_β=σz_(1-α/2)-τ/sqrt(2/n) 
σ^2 (z_(α/2)+z_β )^2=(τ^2/2)/n
n=(2σ^2 (z_(α/2)+z_β )^2)/τ^2

Question 2

Between-subject design

Derive the sample size from the power

Answer 2

β=1-power=1-(1-ϕ(z_(1-α/2)-τ/σλ))
                    = ϕ(z_(1-α/2)-τ/σλ)
ϕ^(-1) (β)=z_(1-α/2)-τ/σλ
z_β=z_(1-α/2)-τ/(σsqrt(2/n))
σz_β=σz_(1-α/2)-τ/sqrt(2/n) 
σ^2 (z_(α/2)+z_β )^2=(τ^2/2)/n
n=(2σ^2 (z_(α/2)+z_β )^2)/τ^2

Question 3

Between-subject design with binary outcome

Derive the power from the sample size

Answer 3

n= n=(z_(α/2) sqrt(2p_bar(1-p_bar ))+z_β sqrt(p1(1-p1)+p2(1-p2)))^2/τ^2

make beta the subject, them 1-beta = power

power = 1-ϕ((z_(α/2) λ sqrt((p) ̅(1-p ̅ )-τ)/sqrt(p1(1-p1)/n1+p2(1-p2)/n2)

Question 4

Between-subject design with binary outcome

Derive the sample size from the power

Answer 4

power = 1-ϕ((z_(α/2) λ sqrt((p) ̅(1-p ̅ )-τ)/sqrt(p1(1-p1)/n1+p2(1-p2)/n2)

beta = 1- power

n= n=(z_(α/2) sqrt(2p_bar(1-p_bar ))+z_β sqrt(p1(1-p1)+p2(1-p2)))^2/τ^2

Question 5

Within-subject design

Derive the power from the sample size

Answer 5

n= n=(z_(α/2) + z_β)^2σ_w^2/τ^2

make beta the subject, them 1-beta = power

power = 1- ϕ((z_(1-α/2) - τ /(σ_w sqrt(1/n))

Question 6

Within-subject design

Derive the sample size from the power

Answer 6

power = 1- ϕ((z_(1-α/2) - τ /(σ_w sqrt(1/n))

beta = 1- power

n= n=(z_(α/2) + z_β)^2σ_w^2/τ^2

Question 7

Crossover design with binary outcome

Derive the sample size from the power

Answer 7

power =
1- ϕ((z_(1-α/2) - τ /(σ_s sqrt(1/n))

beta = 1- power

n=(z_(α/2) + z_β)^2σ_s^2/τ^2

Question 8

Crossover design with binary outcome

Derive the power from the sample size

Answer 8

n=(z_(α/2) + z_β)^2σ_s^2/τ^2

make beta the subject, them 1-beta = power

power =
1- ϕ((z_(1-α/2) - τ /(σ_s sqrt(1/n))