How can I handle the non-response in this simple random sampling? I am planning to conduct a simple random sampling for a population of $N$ patients with index $1,...N$. I have enough money to interview $n$ patients (sample size=$n$). In order to determine the sample patients, I first generate $n$ random integers(with replacement) and ask the patients with such index whether they would like to join the interview. However, due to some reasons, several patients (non-responder) were not able to join.
My question is: what shall I do if in this case I have patients who can not join, but I also want to maximize the use of $n$ samples? 
I can think of:
If 3 patients can not join, re-generate 3 random integers and assign three new patients. Repeat this process until all patients are willing to join. However, I will still record information from the 3 non-responders, and the total sample size is $n+3$. The non-responder's information will be analyzed later to see if there is any bias. 
Is this method reasonable? 
 A: Non-response is what researchers in the survey world have long lived with, tried to understand, describe, and account for. The wisdom has been accumulated in a 2001 edited volume; I wouldn't be surprised if a new edition is being prepared.
If you have the luxury of being able to add samples until you run out of money (assuming that sampling itself is very cheap as compared to collecting the data that probably requires more expensive instrumentation), then what you propose is a very reasonable strategy. 
Comparing respondents with non-respondents is a part of non-response bias analysis, which is often a good thing to do when you have non-response (and sometimes is required when the magnitude of non-response may cast doubts on validity of the study). You need to identify the variables that are correlated with non-response, the outcome(s), and, better, with both, to see how the respondents and non-respondent samples compare to one another.
A: This is a very broad question. To give a proper answer there are several additional questions to be asked:


*

*Do you have any idea about the possible response rate of your survey?

*Is there a reason to choose simple random sampling?

*Is there a reason to choose sampling with replacement?

*You have a budget to do $n$ interviews. What about non-response cases? Usually there are some cost as well to handle non-response cases (travel, phone call, documentation of non-response cases, data entry of non-response cases, etc.). Costs for non-response cases have to be planed as well.

*Can you give the values $N$ and $n$?


Update
This is an update of my answer after receiving your answers to my questions.
Firstly I have to say it will be very hard to get any sensible estimates if sample size is only 10. Here the population size does not play a big role. To have a reasonable sample size is important to achieve low sampling errors.
Secondly, your strategy could be good. However I advise to keep the response rate as high as possible. Here high response rate (not number of respondents) is important to have low response bias. It means try to put as much efforts as you can (are allowed) to convince the sampled person to take part in the survey. Draw an additional sample only when you are sure that it is not possible to get responses from non-respondents.
