Consider the the situation you want to know whether a new medicine will help to reduce the blood pressure.
Assume now you do a random draw from your target population, say you have 20 people in the lab, then you do randomized assignment, let's say it leads to 10 people in treatment group and 10 people in control group. Let's say there is no placebo effect. So you given the 10 people in the treatment group the new medicine, then after some time, you measure the blood pressure of two group of people, then you conduct independent unpaired t-test.
Alternatively, you could do the following: you first measure the blood pressure of all these 20 people, then you give all of them the new medicine, then after some time you take their blood pressure again, then you run a paired t-test.
I am questioning the 2nd method in the following way:
Under Rubin's potential outcome framework, the population will have 2 potential outcome distributions, one is the distribution of current blood pressure denoted by $F_c$, the other is the distribution that forces all people in the population take the new medicine and then take their blood pressure,denoted by $F_t$. The logic of Method 1 is theoretically correct, i.e., the 10 people from control group is a representative sample of the distribution of $F_c$ and the 10 people from the treatment group, is a representative sample of the distribution of $F_t$, then using the sample mean, we can make inference about the true mean difference of $F_c$ and $F_t$. Because the treatment group sample and control group sample are independent, in particular, they are all different people.
Now let's look at Method 2, I agree that these 20 people is a representative sample of $F_c$, now here comes the problem, after you give all these 20 people the new medicine, these 20 data points get mapped into the data points that form $F_t$. However, there is no guarantee that "these 20 data" is a representative of the $F_t$! Hence, we cannot draw any conclusion about the true mean difference between $F_c$ and $F_t$ using the sample in Method 2. Did I miss something?
I understand that for Method 2, you perform paired-sample t test, which will in general has less variability as the unpaired t-test, which sounds like pretty attractive. In practice, I don't know whether Method 1 or Method 2, which is more popular. What is the disadvantage of Method 2 comparing to Method 1? Why not just abandon Method 1 and all use Method 2?