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I do diagnostic research and am interested in determining how the probability of a patient having cancer is related to the concentration of different molecules in the blood. I have received conflicting information on what specifically counts as logistic regression. Some people say that it is regression when the outcome is categorical (eg. cancer vs not) while someone I work with says that it is when the relationship between the probability of the outcome and the measured variables is not linear and includes exponents.

ie they are saying that if the probability of cancer is a function of:

β0 + (β1* molecule x) + (β2* (molecule y)^2) +(β3* (molecule z)^0.5)

then this is logistic regression but if the probability is based off of:

β0 + (β1* molecule x) + (β2* molecule y) +(β3* molecule z)

Then I should use linear regression. (beta reefers to different constants). I am very confused.

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    $\begingroup$ Both equations can be logistic or can be something else. You have logistic if your response is binary and you fit your model with the appropriate likelihood function. How you specify the two equations is common for all GLM models. $\endgroup$ – SmallChess Mar 9 '17 at 3:53
  • $\begingroup$ Is there any chance you could say "based on", which is standard English, rather than "based off of", which appears to be a usage from children's cartoons? $\endgroup$ – Michael Hardy Mar 9 '17 at 5:37
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You are getting advice from profoundly ignorant people.

In logistic regression the outcome is categorical and the predictor is a quantity or a vector of quanities, and the regression function is a logistic, or sigmoid function, so that $$ \operatorname{logit} \Pr(\text{something}) = \beta_0 + \beta_1 x + \beta_2 y + \beta_3 z + \text{etc.} $$ where $$ \operatorname{logit} p = \log \frac p {1-p}. $$ Each data point has values of $x,y,z,\ldots$ and tells you whether "something" occurred in that case or not. The parameters $\beta_0, \beta_1, \beta_3, \beta_3, \ldots$ are estimated by maximum likelihood.

This sort of occurrence of the logit function is what makes it logistic regression.

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