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Under the assumption of the distribution of error and also the form that X takes to affect Y, we get this probability function: P = exp(β'x)/ (1 + exp(β'x))

Therefore, does it mean that we are looking for such a set of betas that will make the error to satisfy our logistic distribution form of error?

Since we use the maximum likelihood method to fit the model, does the maximum likelihood also serve the need for statistical inference of the parameters? If not, what should be?

Please feel free to rectify me, thanks in advance!

Yuan
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  • There is no "error" in logistic distribution. See https://stats.stackexchange.com/questions/29325 for an explanation. – whuber Jan 13 '20 at 23:24
  • Should I say we assume the variable left out follows the logistic distribution instead? – Yuan Jan 14 '20 at 00:38
  • I don't understand what you might mean by "the variable left out." Could you clarify what you are asking? It's unclear whether you are referring to ordinary regression with logistically-distributed errors or to logistic regression (which is entirely different). – whuber Jan 14 '20 at 13:27
  • We define there is a deterministic model between X and Y(which takes the value of either 0 or 1). However, due to some unobservable variables were left out, our equation becomes Y=X+ ϵ. Furthur, P(Y=1|X=x) is equal to the probability that ϵ takes the value that makes Y=1. Suppose we assume the density function of ϵ is logistic(along with other assumption about the way X affect Y), and therefore we can deduce that P(Y=1|X=x) = exp(β'x)/ (1 + exp(β'x)). reference: https://pdfs.semanticscholar.org/6994/03fc68a4ec773a8e19438a448dd075a5fb6e.pdf – Yuan Jan 14 '20 at 20:14
  • Thank you: properly interpreted, that looks like one formulation of logistic regression. (You need to incorporate the parameters $\beta$ and a threshold to compare $x\beta+\epsilon$ to.) However, the values of $\epsilon$ are not observable, so it's impossible to address your main question about making it follow the logistic regression. – whuber Jan 14 '20 at 20:43
  • You are right. So my doubt is, is there a set of betas we are searching for in population? And is the goal of Maximum likelihood to estimate that set of betas? If not, what is tested by the hypothesis testing in the case of logistic regression? – Yuan Jan 14 '20 at 21:39

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