A random sample is drawn from a population by using a probability device. We might put everyone's name on a slip of paper, mix thoroughly, and select the number of names we need, or we might have a computer generate random numbers and use them to select our sample. If you don't trust the computer to generate your random numbers, there are always http://random.org, http://www.fourmilab.ch/hotbits/, and even http://www.lavarnd.org/. The use of a probability device to select the subjects allows us to make valid generalizations from the sample to the population.
In an intervention trial, randomization refers to the use of a probability device to assign subjects to treatment. This allows us to use statistical methods to make valid statements about the difference between treatments for this set of subjects. The subjects who are randomized may or may not be a random sample from some larger population. Typically, when human subjects are involved, they are volunteers. If they are a random sample, then statistical theory lets us generalize from this trial to the population from which the sample was drawn. If they are not a random sample from some larger population, then generalizing beyond the trial is a matter of nonstatistical judgement.
Intervention trials are typically analyzed by using the same statistical methods for the analyzing random samples. Almost all intervention trials involve volunteers, usually recruited locally. If convenience samples are inappropriate for surveys, how can they be appropriate for intervention trials?
There are two distinct issues to address--validity and generalizability.
The reason volunteers can be used to make valid comparisons comes from the use of randomization in the assignment of treatments. It is beyond the scope of these notes to give mathematical proofs, but the common statistical methods that are appropriate to compare simple random samples are also valid for deciding whether the observed difference between the two treatments is greater than would be expected when subjects are assigned to treatments at random and the treatments are equivalent. The probability models for random sampling and the probability models for randomization lead to the same statistical methods.
Within broad limits, the results of intervention trials can be genralized because all human beings are made out of the same stuff. While this justification cannot be applied blindly, it may be comforting to know that many of the surgical advances of the mid 20-th century were developed in VA hospitals on middle-age white males. However, the ability to generalize results does not immediately follow from the use of particular numerical methods. Rather, it comes from the subject matter specialist's knowledge of those who were studied and the group to whom the generalization will be made.
It is worth noting here that the quality of statistical evidence differs according to whether subjects can be randomized. For example, consider an intervention trial that compares the effects of two diets in smoking and nonsmoking pregnant women. The use of statistical methods to compare diets can be justified by the random assignment of subjects to treatment. However, the comparison between smokers and nonsmokers is subjective. It depends on an enrollment procedure that does not introduce artificial differences between smokers and nonsmokers, what epidemiologists call selection bias.