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In fitting a logistic regression, as in applying any statistic method, the required sample size depends on the level of dispersion in the population and the quality of the statistics that you are prepared to accept.

Usually there will be some information available somewhere (in the literature, say, or from colleagues) suggesting what level of variability to expect in data that is collected. This can be used to simulate some data sets and the logistic regression results that would arise from them. By varying the sizes of the data sets you can make a judgement. Once you have collected your first sample and fit the actual logistic regression to it you will have a much better idea how much data is actually required for useful estimates.

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Q: What is the minimum required sample size for running logistic regression?
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