Clinical Trials Chapter 2

Question 1

What is a ‘Between Subjects’ Design?

Answer 1

Also called parallel group trial
It is a standard randomised controlled trial where different subjects are allocated at random to one of the treatments. Every subject receives only one treatment.

Question 2

What is a ‘Within Subject’ Design?

Answer 2

Every subject receives all treatments. The treatment orders may be different for each subject.

Question 3

Between-subject design

Estimate treatment effect and its variance

Answer 3

There are n1 and n2 participants in group 1 and 2, respectively.
y ̅_i -> the mean of the outcome for group i
(s_i)^2 -> the variance of the outcome for group i

Treatment effect
t_obs = y ̅_1 - y ̅_2

Variance 
var (t_obs) = var(y ̅_1 - y ̅_2) = (s_1)^2/n1 +(s_2)^2/n2

Question 4

Between Subject Design

Formulation for the variance of the treatment effect

Answer 4

Var(y ̅_1-y ̅_2 )=var(y ̅_1 )+var(y ̅_2 )
=var((∑y ̅_1j )/n_1 )+var((∑y ̅_2j )/n_2 )
=1/(n_1^2 ) var(∑y ̅_1j )+1/(n_2^2 ) var(∑y ̅_2j ) =n_1/(n_1^2 ) var(y_1 )+n_2/(n_2^2 ) var(y_2)
=(s_1^2)/n_1 +(s_2^2)/n_2
Where:
s_i^2 is the variance of the outcomes in group i
n_i is the size of group i

Question 5

Null and alternative hypotheses to asses the results of a ‘Between Subjects’ Design clinical trial for treatment effect

Answer 5

Null - treatment effect = 0

Alternative - treatment effect ≠0

Question 6

Between subject

Test Statistic to assess hypothesis

Answer 6

t`_obs/standard error(t_obs)

Question 7

Between Subject
Assess statistical significance of the test statistic
- via a critical value
- via a confidence interval

Answer 7

Via critical value
Use test statistic
If absolute value bigger than critical value reject H0

Via confidence interval
If the data are normally distributed, a (1-alpha) confidence interval for the treatment effect is given by
((y ̅_1-y ̅2 )∓z(α/2)x(sqrt(var(y ̅_1-y ̅_2 ))
for a small sample use a t-test with degrees of freedom n1+n2-2
If 0 is not in the confidence interval reject H0

Question 8

Between subject Model

Answer 8

Y_ij as the measurement on subject j of treatment i
Y_ij=μ+α_i+ϵ_ij
With ∑α_i=0 and ϵ_ij~N(0,σ^2)
Notations:
μ is the overall mean
α_i is the effect of treatment i, with i=1(A) or 2(B))
ϵ_ij is the error term, with i=1(A) or 2(B), j=1,…,n

Question 9

Within-subject design

Estimate treatment effect and its variance

Answer 9

Comparing two treatments 1 and 2
There are n participants in total
Each participant will receive both treatments
A pair of measurements for the outcome variable is taken on each participant, one after each treatment
y ̅_i -> the mean of the outcome under treatment i (1(A), 2(B))
(s_i)^2 -> the variance of the outcome for treatment I

Treatment effect is estimated by the mean of the within-subject differences:
t_obs = d ̅=y ̅_1-y ̅_2
(Difference in means = mean of differences)

Variance 
var(t_obs) = var(d ̅)= (σ_w)^2/n = var(d)/n
where (σ_w)^2 is the variance for the within-subject differences, var(Y_1-Y_2)

Question 10

Within Subject
Assess statistical significance of the test statistic
- via a critical value
- via a confidence interval

Answer 10

Via critical value
Use test statistic
If absolute value bigger than critical value reject H0

Via confidence interval
If the data are normally distributed, a (1-alpha) confidence interval for the treatment effect is given by
(d ̅)∓z_(α/2)x(sqrt(var(d bar ))
d ̅=y ̅_1-y ̅_2
var(d ̅)= (σ_w)^2/n
for a small sample use a t-test with degrees of freedom n-1
If 0 is not in the confidence interval reject H0

Question 11

Within Subject Model

Answer 11

Y_ij as the measurement on subject j of treatment i
Y_ij=μ+α_i+s_j+ϵ_ij
With ∑α_i=0, s_j~N(0,σ^2) and ϵ_ij~N(0,σ^2)
Notations:
μ is the overall mean
α_i is the effect of treatment i, with i=1(A) or 2(B))
s_j is the random effect of subject j (j=1,…n)
ϵ_ij is the error term, with i=1(A) or 2(B), j=1,…,n

Question 12

Advantages and Disadvantages of between-subject design

Answer 12

Advantages

• Simple, obvious approach
• Usually easy to analyse
• No carry over effect

Disadvantages
- More subjects needed

Question 13

Advantages and Disadvantages of within-subject design

Answer 13

Advantages
- fewer subjects are needed
- reduction in unexplained variability (same person so no difference between groups) although this may be balanced by reduced degrees of freedom
Disadvantages
- Not always suitable (e.g if the patient may be cured
- Carry over effect

Question 14

Null and alternative hypotheses to asses the results of a ‘Within Subjects’ Design clinical trial for treatment effect

Answer 14

Null - t_obs =0, no difference

Alternative - t_obs≠0, there is a difference between treatments

Question 15

Formulas for working out variance from raw data

Answer 15

Variance fo sample
s^2=sum (x_i - x_bar)^2/n-1
s^2 = [sum(x^2)- (sum(x))^2/n]/(N-1)