I am not sure what you mean by sampling frame here? In simple, sampling frame is the list of people in your target population. Sampling frame is usually required for probability sampling techniques. The presence or absence of sampling frame doesn't affect the sample size. If you don't have a sampling frame you can use any non-probability sampling and recruit people for the survey from your target population.
** if probability sampling like multistage sampling in the survey that will be conducted >>>> so you have to calculate ( or know ) : design effect
The design effect is a correction factor that is used to adjust required sample size for cluster sampling. The required sample size is estimated assuming a random sample, and then multiplied by the design effect. This accounts for the loss of information inherent in the clustered design.
Check epi info program software for free from CDC site
in population survey
https://www.cdc.gov/epiinfo/pc.html
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What is the difference between cluster sampling and multistage sampling?
from larger clusters that were randomly selected previously. What is the difference between cluster and multistage sampling? A. Cluster sampling uses several levels of clusters. B. Cluster sampling uses clusters whereas multistage sampling uses stages.
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In Wikipedia , it shows types of sampling frame nearly to data collection as I understood
Types of sampling frames
The most straightforward type of frame is a list of elements of the population (preferably the entire population) with appropriate contact information.
For example, in an opinion poll, possible sampling frames include an electoral register or a telephone directory.
Other sampling frames can include employment records, school class lists, patient files in a hospital, organizations listed in a thematic database, and so on. On a more practical levels, sampling frames have the form of computer files.
Not all frames explicitly list population elements; some list only 'clusters'. For example, a street map can be used as a frame for a door-to-door survey; although it doesn't show individual houses, we can select streets from the map and then select houses on those streets. This offers some advantages: such a frame would include people who have recently moved and are not yet on the list frames discussed above, and it may be easier to use because it doesn't require storing data for every unit in the population, only for a smaller number of clusters.
Sampling frame has nothing to do with number of units in the sample (sample size). Population standard deviation along with Z and error value, can be used to determine the sample size.
I assume that you mean that you cannot list the members of the population. In that case, it sounds like you want to have a probability-of-selection-based (design based) approach, not a model-based approach, as that would require further identification of regressor data for the population.
The problem is then how to select your sample. If you assumed a relatively homogeneous (and infinite??) population, you might assume simple random sampling (and you will not need a finite population correction (fpc) factor, if infinite - but you did not say that). But simple random sampling could be a very bad choice if you should have stratified.
You must know something about your population, even if what you know is that you don't know about certain parts of it.
If a reasonable approach is to stratify, and you don't know the relative sizes of those parts of the population (N_h), then even if you estimate standard deviation for each of those parts of the population, you can't look at contributions to the overall population. In that case, perhaps you could specify what characteristics categorize each stratum, which then really become subpopulations which you sample separately. You might then report results by each subpopulation, assuming that you know each is then fairly homogeneous - no "hidden" parts of the population.
If you can assume that a population does not have an odd shaped distribution, so that simple random sampling makes sense, you still have a problem: How do you randomly select members of your sample if you cannot identify them to give them equal chances of selection?
If you can say your selection is random, how do you support that statement? If you can't say that for simple random sampling, then how can you ever have a more complicated design?
My guess is that you want to say that your unknown population is homogeneous, and that your sample selection gives each member of the population an equal chance of selection, and you can then estimate standard deviation and see what sample size with that standard deviation would give you an acceptable standard error, say perhaps for an estimated mean. But how do you know? What makes your sampling mechanism random, etc?
In my understanding of this case, to simplify the situation, I would like to suggest the use of the formula for sample size estimation which does not include sampling frame. Your sampling technique will depend on your study design which you have not mentioned. Best of luck.