Power for one group proportion where sample size is sufficient for lower 95%CI to be above a certain threshold

[Megan Moreton]

Hi

I am trying to replicate the sample size calculation from a 1:1 RCT. The sample size is based on the lower 95%CI for the proportion with success in each of the arms to be above a threshold - it is not powered for an formal comparison between arms.

The inputs/outputs are:

• Success proportion: 85%
• Lower 95%CI of success proportion to be above: 73%
• Power: 80%
• Alpha: 5% (one-sided)
• LTFU: 10%
• Planned enrolment: 54 per arm (not powered for formal comparison between arms)

Is there a way I can do this in Stata using a command (I am using Stata 16.1)? Or would this be something I need to build a simulation for?

Thanks!

[Ken Chui]

Does this make sense to you? It does not make sense to me.

Even with many random draws of 54 people (assuming 54 was the LTFU-adjusted number.) + a success rate of 85%, only about 40% of those draws would have a lower 95%CI excluding 0.73 (sample simulation code below). The other group is assumed to have a lower success rate (implied by "success" and one-tailed), thus it'd need a tighter confidence interval and bigger group size.

I think it may be better to post the original text of that RCT, check their official registry, or contact the researchers for more details.

capture program drop mySim
program define mySim, rclass
version 16.1
drop _all
set obs 54
gen y = runiform() > 0.15
proportion y
return scalar p1 = r(table)[5,2]
end

simulate p1 = r(p1), reps(1000): mySim
gen gt73 = (p1 > 0.73)
tab gt73

[Megan Moreton]

Thanks, Ken - that's really helpful. I've attached the power calculation text from the protocol below.

My impression is that the sample size is quite small. On rereading, I think they have estimated that 42/49 (85%) participants in each group will achieve the endpoint & have then worked out the confidence interval around that rather than run any formal sample size calculation.

. cii prop 49 42

                                                         -- Binomial Exact --
    Variable |        Obs  Proportion    Std. Err.       [95% Conf. Interval]
-------------+---------------------------------------------------------------
             |         49    .8571429    .0499896        .7275785    .9405785

[Ken Chui]

I'd agree with you, it seems they just figured the 95%CI lower bound out, and determined that 49 is a good size. And it also looks like there is no across arm comparison, so it's closer to a one-sample proportion test? This is the closest I could get:

power oneproportion 0.70(0.01)0.73 0.85, onesided

Output:

Estimated sample size for a one-sample proportion test
Score z test
Ho: p = p0  versus  Ha: p > p0

  +-------------------------------------------------+
  |   alpha   power       N   delta      p0      pa |
  |-------------------------------------------------|
  |     .05      .8      50     .15      .7     .85 |
  |     .05      .8      56     .14     .71     .85 |
  |     .05      .8      64     .13     .72     .85 |
  |     .05      .8      74     .12     .73     .85 |
  +-------------------------------------------------+