Definition
An explanation of logistic regression begins with an explanation of the logistic function, which, like probabilities, always takes on values between zero and one:
and
and
A graph of the function is shown in figure 1. The input is and the output is . The logistic function is useful because it can take as an input any value from negative infinity to positive infinity, whereas the output is confined to values between 0 and 1. In the above equations, g(X) refers to the logit function of some given predictor X, ln denotes the natural logarithm, is the probability of being a case, is the intercept from the linear regression equation (the value of the criterion when the predictor is equal to zero), is the regression coefficient multiplied by some value of the predictor, base e denotes the exponential function. The first formula illustrates that the probability of being a case is equal to the odds of the exponential function of the linear regression equation. This is important in that it shows that the input of the logistic regression equation (the linear regression equation) can vary from negative to positive infinity and yet, after exponentiating the odds of the equation, the output will vary between zero and one. The second equation illustrates that the logit (i.e., log-odds or natural logarithm of the odds) is equivalent to the linear regression equation. Likewise, the third equation illustrates that the odds of being a case is equivalent to the exponential function of the linear regression equation. This illustrates how the logit serves as a link function between the odds and the linear regression equation. Given that the logit varies from it provides an adequate criterion upon which to conduct linear regression and the logit is easily converted back into the odds.
Read more about this topic: Logistic Regression
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