Chapter 4 need to know

Question 1

Type 1 error

Answer 1

alpha error or significance level

alpha = P(reject null | null hypothesis is true)

Question 2

Type 2 error

Answer 2

beta error

beta = P(do not reject null | null hypothesis is false)

Question 3

Power

Answer 3

1 - beta = P(reject null | null hypothesis is false)

trials usually should be at leat 80% powered

Question 4

Power for Between-subject design

1-ϕ(z_(1-α/2)-τ/σλ)

Answer 4

λ = sqrt(2/n)
ϕ is the cumulative distribution function of the standard normal distribution N(0,1)
τ denotes the treatment effect (difference in two group means)
σ denotes the within-group standard deviation

Question 5

Factors affecting sample size

Answer 5

Higher power needs a larger sample size
Higher significance level would need a smaller sample size
Type of outcome; continuous, binary or other
Design for the trial

Question 6

Sample size for Between-subject design

n =(2(z_α/2+z_beta)^2σ^2)/τ^2

Answer 6

n =(2(z_α/2+z_beta)^2σ^2)/τ^2

• beta is the probability of making a type 2 error
• z_α/2 and z_beta are the α/2th and the Beta th percentiles of a standard normal distribution
• z_p = ϕ^-1(p)
• z_α/2=-z_(1-α/2)
  The required total sample size for this trial is then 2n

Question 7

Between-subject with a binary outcome

Designs

Answer 7

Treatment group 1 or 2
Proportion of events in each group given by p_1 and p_2
Sample Size given by n_1 and n_2
Event counts m_1 and m_2

Question 8

Power for between-subject with a binary outcome

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

Answer 8

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

• ϕ is the cumulative distribution function of the standardised normal distribution N(0,1)
• p1 and p2 denote the proportion of events in the respective group
• λ = sqrt(1/n1+1/n2), τ=p1-p2 and p_hat = (p1+p2)/2

Question 9

Sample size for between-subject with a binary outcome

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

Answer 9

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

p1 and p2 denote the proportions of events in the respective groups
τ = p1-p2 denotes the difference in the two proportions
p_bar =(p1+p2)/2 denotes the average proportion
The required sample size for this trial is 2n

Question 10

Power for Within-subject design

1- ϕ((z_(1-α/2) - τ /(σ_w sqrt(1/n))

Answer 10

1- ϕ((z_(1-α/2) - τ /(σ_w sqrt(1/n))
where
ϕ is the cumulative distribution function of the standard normal distribution N(0,1)
τ denotes the treatment effect estimated by the mean of within subject differences
σ_w denotes the standard deviation for within-subject differences

Question 11

Sample size for Within-subject design

n=(z_(α/2) + z_β)^2σ_w^2/τ^2

Answer 11

n=(z_(α/2) + z_β)^2σ_w^2/τ^2

beta is the probability of making a type 2 error
z_p = ϕ^-1(p)
- z_α/2=-z_(1-α/2)

The required sample size is then n for the within-subject design

Question 12

Power for crossover design

1- ϕ((z_(1-α/2) - τ /(σ_s sqrt(1/n))

Answer 12

1- ϕ((z_(1-α/2) - τ /(σ_s sqrt(1/n))

ϕ is the cumulative distribution function of the standard normal distribution N(0,1)
τ denotes the treatment effect estimated by the mean of within subject differences
σ_s denotes the standard deviation for the sequence of measurements of the outcome variable in the crossover trial. Here we assume var(y11)=var(y12)=var(21)=var(y22) = σ_s ^2

Question 13

Sample size for crossover design

n=(z_(α/2) + z_β)^2σ_s^2/τ^2

Answer 13

n=(z_(α/2) + z_β)^2σ_s^2/τ^2

beta is the probability of making a type 2 error
z_p = ϕ^-1(p)
z_α/2=-z_(1-α/2)

The required sample size is then 2n
When σ_s is not available, we may assume that σ_s=σ_w/sqrt(2) where σ_w is the standard deviation for the within-subject differences