What is the (Multivariate) Logistic Regression Equation?

In logistic regression, the log odds is modeled as a linear functions of the X-variables. Thus, the logistic regression equation is:

    \[\log\left( p \over 1-p \right) = \alpha + \beta_1 X_1 + \beta_2 X_2 + \cdots + \beta_p X_p .\]

Once the logistic regression has been “run,” the software will calculate estimates for \alpha, \beta_1, \beta_2, … , \beta_p which are usually denoted by \hat\alpha, \hat\beta_1, \hat\beta_2, … , \hat\beta_p and read “alpha hat,” “beta hat sub 1,” and so one. Plugging in these estimates yields the fitted logistic regression equation:

    \[\log\left( p \over 1-p \right) = \hat\alpha + \hat\beta_1 X_1 + \hat\beta_2 X_2 + \cdots + \hat\beta_p X_p .\]

Notes:

  • “log” refers to the natural logarithm (log to the base e) as is discussed further here.
  • “odds” are discussed further here.
  • For a basic introduction to logistic regression analysis, go to the home page of this web site (click here).
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