Continuous sample size determination based on the control group I am curious if and where the following reasoning breaks down.
Traditionally, sample size determination is done as part of the design phase. To this end, one has to have an understanding of the baseline performance, such as the mean and standard deviation of the metric in question. One might use recent historical data to obtain reasonable estimates.
Suppose we are not necessarily interested in knowing the minimal sample size ahead of time and simply launch the experiment. The control group is the baseline. Each day, we take the statistics of the control group and perform sample size determination. With each new day passed, we get a better and better understanding of when to stop.
Apart from the fact that one might launch an experiment that is doomed to fail, are there any other problems with the above logic?
Clarification 1: It should be noted that this question is not about whether it is sound to check some p-value on a daily basis and stop whenever it goes below a predefined level. The inadequacy of this procedure is well understood.
Clarification 2: Here we are not assuming any clinical or otherwise setting with serious implications and ethical concerns. Rather, one can imagine something trivial, such as an experiment with a webpage where the color of a button is subject to change.
Clarification 3: The question describes a specific stopping rule: every day, run a power analysis using the statistics of the data for the control group collected up to today and terminate the experiment as soon as the actual sample size exceeds the one given by the power analysis, potentially skipping a few first days to avoid pathological cases.

Example:  Suppose you have $n = 67$ observations from a normal population with unknown population mean $\mu$ and standard deviation $\sigma.$ You wish to test $H_0: \mu = 106$ against $H_a: \mu \ne 106.$ and you want the power of a one sample t test at the 5% level to be about 90% when the true mean is $\mu = 100.$
Then a simple simulation in R
shows that $n = 67$ observations is sufficient to give
the desired power:
set.seed(2021)
pv = replicate(10^6, t.test(rnorm(67, 100, 15), mu = 106)$p.val)
mean(pv <= 0.05)
[1] 0.896917

However, the P-values corresponding to t statistics computed after $n = 2, 3, \dots, 67$ may be smaller than 5% for some values of $n$ smaller than $67.$
Below we plots of such 'intermediate' P-values for nine experiments with $n = 67.$ About 80% samples will show P-values below $0.05$ exactly at $n = 67.$
That may be the first step at which the value is small enough to reject, but that is typically not the case.
Clearly, it would not be appropriate to make a decision
based on the first step at which a P-value is smaller than 5%.

Of course, some of these P-value traces may never dip as
low as $0.05$ within $n = 67$ observations.
R code for one trace is shown below:
pv = numeric(67)
x = rnorm(67, 100, 15)
pv = numeric(67);  pv[1]= 1  # no t stat for n=1
for(i in 2:47) 
 { pv[i] = t.test(x[1:i], mu=106)$p.val }
plot(pv, type="l", ylim=0:1)
 abline(h=.05, col="blue")
 m = min(which(pv < .05));m
 abline(v=m, col="red") 

