How to analyse data if sample size was increased after first results are known I am estimating a proportion p and believe it to be 0.8. I want to show that it is almost certainly greater than 0.7. I have done a power calculation by simulation, and I have figured out that I need n samples to be 80% sure that the lower confidence limit for p will be greater than 0.7 (given p is really 0.8)
I am using a 90% confidence interval, because this will have 95% of the probability mass greater than the lower limit.
So far, so good.
Suppose that I have done the experiments,  and I find that this lower confidence limit is not greater than 0.7. This could be due to a) my assumption of 0.8 is wrong or b) bad luck (or both)
Now someone says: We'll take another m samples, and see what lower limit we will have then. Suppose that now we are successful, and this lower limit is indeed greater than 0.7.
What do I know now ?
If I would have done n+m experiments to begin with, all would have been good. But now I have been unsuccessful with n samples, and I have chosen m by looking at the results of the first n experiments.
I don't have any data yet, I am just trying to get precise the implications of this strategy. I have seen this post data peeking and increasing sample size
but I still don't know the answer.
 A: Since you haven't started the experiment, I suggest a different approach to determine sample size: choose n to control the accuracy of the estimate, not the power of a statistical test. Here accuracy means the margin of error $\delta$ of the 100(1-α)% confidence interval for the population proportion $p$.
Here is how the process could work out in your case. You believe the true proportion is $p$ = 0.8 or at least $p$ > 0.7. Once you collect data you would be able to construct a 100(1-α)% for $p$:
$$
  \begin{aligned}
    \hat{p} \pm \delta = \hat{p} \pm z_{1-\alpha/2}\sqrt{\hat{p}(1-\hat{p})/n}
  \end{aligned}
  $$
You can specify the half-width $\delta$ you'd like to achieve. For example, $\delta$ = 1 is probably too wide because even if $p$ is exactly 0.8 the confidence interval might be centered at a value for $\hat{p}$ that's slightly below 0.8, so the confidence interval will include 0.7. On the other hand, $\delta$ = 0.01 might be unnecessarily precise because you want to check whether the lower limit $\hat{p} - \delta$ is above 0.7. Perhaps $\delta$ = 0.05 sounds about right? Then you compute the required sample size is a function of $\delta$ and the variance of $\hat{p}$. (I also assume the significance level is $\alpha$ = 0.05.)
$$
  \begin{aligned}
    n = z_{1-\alpha/2}^2\hat{p}(1-\hat{p})/\delta^2
  \end{aligned}
  $$
For simplicity, let's plug in 1/4 for the variance of $\hat{p}$. This is the "worst case" value because the function $p(1-p)$ is maximized at $p$ = 1/2 for $p$ between 0 and 1. You estimate, conservatively, that you'll need a sample of size $n$ = 385 (after rounding up).
See the section Sample Size Justification in Improving Your Statistical Inferences by Daniël Lakens.
A: Not only is this situation common -- some people even deliberately design their studies for two-stage sampling. Cochran mentions in Sampling Techniques (1955) the approach known as Stein's method.
First you do a regular sample and derive the statistic of interest. Then based on the information of the first, you do a second sample designed to be of the right size to get the interval you need.
The method proceeds in much the same way as the answer you have already received from dipetkov, other than the fact that you split it up into two stages.
