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Logistic regression aims at transforming the linear regression formula and fitting the s curve or logistic function to a particular dataset in order to calculate the probability of a categorical outcome variable given a number of predictor variables.

Also a formula aims at demonstrating what needs to happen to yield the desired results (s curve). What is the standard formula that makes regression model logistic? I am a bit confused because they seem to be written differently (although could be meaning the same thing in different sources).After some research the below formula appears to be what I am looking for (docs.tibco.com/pub/sfire-dsc/6.5.0/doc/html/TIB_sfire-dsc_user-guide/GUID-C4D05ED0-3392-4407-B62A-7D29B26DC566.html). First the logistic function below is applied to transform linear to logistic;

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And then the below final formula is obtained which is what I intend to use as the standard Logistic regression formula. Is this correct?

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Which is interpreted as P = Probability of Event, and and X1,X2,… are the independent variable values.

The idea is to provide a holistic base of the model, from theory to practice in my methodology section.

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    $\begingroup$ What is your question? Eg, I don't see a "?" anywhere. $\endgroup$ Apr 21 at 12:13
  • $\begingroup$ I have revised the question. Thank you. I am looking for guidance to the standard logistic regression formula and how to interpret it, to provide theoretical background in my data analysis section. $\endgroup$ Apr 21 at 12:24
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    $\begingroup$ Now you only have a question as to whether the formula is correct. It is. Whether you want to call it the "standard logistic formula" is up to you. There are other equivalent formulations. But I don't know what a "holistic base of the model" means and a methodology section could probably just say "I used logistic regression". $\endgroup$
    – Peter Flom
    Apr 21 at 13:16
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    $\begingroup$ What you say seems broadly correct & it's still not clear to me exactly what your question is, but a missing piece (in your understanding) might be that $\text{log}(p/(1-p))=\mu \Leftrightarrow p=1/(1+\text{e}^{-\mu})$, i.e. the second function -- sometimes called 'expit' -- is the inverse of the logit. The model fit is performed in the logistic (link) scale, and you can obtain probabilities via back-transformation. $\endgroup$
    – PBulls
    Apr 21 at 13:19
  • $\begingroup$ Thank you all for your patience, my question was perhaps rather to confirm whether that is the correct formula for logistic regression because I am not sure, and whether it is complete. In my data analysis section, I want to give a theoretical background on what logistic regression is, including how it is done by giving the formula, interpreting it before giving the practical results from r. I am struggling to understand because I have no math or statistic background. $\endgroup$ Apr 22 at 17:45

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It sounds like you're new to logistic regression as a concept. If your purpose is to properly understand it, and then provide a rigorous justification of its use for an academic manuscript, I suggest you turn to more reputable and comprehensive sources than us random commenters on StackExchange. Jurafsky & Martin have a nice chapter on it.

Your question specifically seems to be about what the "logistic regression formula" is. This term is a bit vague, as you could be asking about several different things.

  • If you have a trained logistic regression model with coefficients $[\beta_i]$ and you want to classify a data point given by a vector $[X_i]$, the formula to get the probability of a success is the "standard logistic function" or "inverse logit" $$p=\frac{1}{1+e^{-(\beta_0+\beta_1X_1+...)}}$$This is the equation for the "S" or "sigmoid" curve that you've fitted.

  • Then there is the "logit" itself, which is the inverse function of the above and what you've written in your question. $$\ln\frac{p}{1-p}=\beta_0+\beta_1X_1+...$$ This is essentially the statement that you're modelling the log-odds as a linear regression of your data, which is the central idea of the logistic regression.

Does this help? If you still have confusions over different sources writing things differently, please provide examples of any discrepancies.

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