Sample size determination for a pilot study I am struggling to figure out how should I approach to decide the sample size for a pilot study. The pilot study will be funded so a proposal is required justifying the sample size. The study participants are patients from a city hospital which will be extended to multiple cities in the main study. Patients will be evaluated based on the present of an attribute (this is not a randomized control trial, however, based on a character, we will have two groups of patients). Most of the measurements obtained will be numeric. From a previous study with a broader goal has a response rate of 40%. 
I read some literature and liked the idea of deciding the sample size based on feasibility a good literature. I do have a confusion, though. When I calculate the sample size based on the proportion of nonresponse for a particular level of confidence, what does the margin of error mean here? Is is the margin of error for the estimates from my study or, the margin of error for the nonresponse rate? 
Any help is greatly appreciated! Thank you.
 A: 
The ultimate goal is to see, out of all these patients those who have a treadmill, for example, in the house are more physically active. We can have the average physical activity of these patients per week. We can also find the correlation of physical activity with well-being and test if patients with a treadmill have a higher correlation between physical activity and personal well-being.

I think in your case you would want to start with a simple cohort study as opposed to a randomized control trial (RCT). An RCT would be ideal (but more costly) for your eventual pivotal study. The goal of this cohort study would be to ask: does exposure to x (having a treadmill in the house) associate with outcome Y (increased physical activity)? 
Such a study would recruit a group of people with a treadmill in the house and a group of people without a treadmill (the unexposed group) and follow them for a set period of time and note differences in the incidence of physical activity (PA) between the groups at the end of this time.
The groups should be matched in terms of other variables such as economic status and other health status so that the variable being assessed, the independent variable (in this case, owning a treadmill) can be isolated as the cause of the dependent variable (in this case, PA).
your statistical method would be based on your measure(s) of PA.  One thing to consider is how many PA measures can you obtain from each cohort, and how often will you take these measures (daily, weekly, monthly).
From my perspective, focusing on a large group of PA measures and identifying their variance within the two groups and between the two groups would be critically important for this pilot study.  You may find some measures vary a lot and would be terrible as metrics to design a future pivotal study, whereas others are less variable and would require a much smaller sample size in a future pivotal study to obtain an appropriate Power.
If you do perform any statistical significance testing using this pilot study data, take it with a grain of salt.  The family-wise error rate will come into play.  If multiple comparisons are done or multiple hypotheses are tested, the chance of a rare event increases, and therefore, the likelihood of incorrectly rejecting a null hypothesis (i.e., making a Type I error) increases.
Here there is an explanation and advice given on sample size estimations for cohort studies.  Remember to consider any sample size will need to be adjusted to account for the expected drop-outs from your study.
Lastly, you may be able to perform a less costly Case-Control Study to determine if people that have higher levels of PA tend to own treadmills.  It may be difficult to calculate your measures of PA on people based on a questionnaire or survey, but the study would be simpler to perform, because you would not need to find and follow people over time.  However, it would be less valuable to help you plan for a pivotal study.
Hope this helps.
