Bioequivalence and Bioavailability Forum

Dear Helmut.

Should be considered possibility of unbalanced design after adjusting for expected dropout-rate? See example below.
Let's plan 2x2 crossover study with following assumptions/requirements:
CV = 0.262
alpha = 0.05
power >= 0.80
ratio = 0.95
acceptance range: 0.80-1.25

It implies minimum sample size n_des = 30.
Expected dropout-rate = 15%.
n_adj=36 (n₁=18, n₂=18)

Now let's see that all assumptions are respected, but power falls under desired 80% in unbalanced cases.

In the first line of the next table "TR" stands for number of dropouts from sequence TR, "RT" likewise and "Probability" stands for probability of case when exactly 6 dropouts are expected.

TR RT Probability     Power
6  0  p_1=0.009530792 <80%
5  1  p_2=0.07917889  <80%
4  2  p_3=0.2403645   <80%
3  3  p_4=0.3418517   >80%
2  4  p_5=0.2403645   <80%
1  5  p_6=0.07917889  <80%
0  6  p_7=0.009530792 <80%

Note: Probability was calculated using combination without repetition p_1=("18 choose 6")/("36 choose 6")=18/36*17/35*16/34*15/33*14/32*13/31=0.009530792 .
# R code:
p_1=choose(18,6)/choose(36,6)
p_2=(choose(18,5)*choose(18,1))/choose(36,6)
p_3=(choose(18,4)*choose(18,2))/choose(36,6)
p_4=(choose(18,3)*choose(18,3))/choose(36,6)
2*p_1+2*p_2+2*p_3+p_4 # Check that sum of p_i is equal to 1.

In case of 3 dropouts with TR and 3 with RT, the power is 80.1%.
Otherwise the power is supposed to be lower than 80% - not much lower of course. With unbalanced case 4:2 dropouts, the power is 79.9%. Then if it's getting more unbalanced, the power is decreasing (as you know).

So in this example we have 65.8% (100*(1-p_4)) probability that we shoot ourselves in the foot .
Maybe not so big shot - planned number of subjects finished will be achieved , but sample size wasn't estimated with option of getting unbalanced in mind. Even though there is almost 2/3 probability that it will ends unbalanced (if assumptions are right, and no better GMR than expected, no lower CV estimated from residual mean square than expected, and no lower dropout-rate than expected).

Best regards,
zizou

_{"Remember, with great power comes great responsibility."}