What is the difference between linear regression and logistic regression?

Question

When we have to predict the value of a categorical (or discrete) outcome we use logistic regression. I believe we use linear regression to also predict the value of an outcome given the input values.

Then, what is the difference between the two methodologies?

Answer 1

• Linear regression output as probabilities

  It's tempting to use the linear regression output as probabilities but it's a mistake because the output can be negative, and greater than 1 whereas probability can not. As regression might actually produce probabilities that could be less than 0, or even bigger than 1, logistic regression was introduced.

  Source: http://gerardnico.com/wiki/data_mining/simple_logistic_regression

  

• Outcome

  In linear regression, the outcome (dependent variable) is continuous. It can have any one of an infinite number of possible values.

  In logistic regression, the outcome (dependent variable) has only a limited number of possible values.

• The dependent variable

  Logistic regression is used when the response variable is categorical in nature. For instance, yes/no, true/false, red/green/blue, 1st/2nd/3rd/4th, etc.

  Linear regression is used when your response variable is continuous. For instance, weight, height, number of hours, etc.

• Equation

  Linear regression gives an equation which is of the form Y = mX + C, means equation with degree 1.

  However, logistic regression gives an equation which is of the form Y = e^X + e^-X

• Coefficient interpretation

  In linear regression, the coefficient interpretation of independent variables are quite straightforward (i.e. holding all other variables constant, with a unit increase in this variable, the dependent variable is expected to increase/decrease by xxx).

  However, in logistic regression, depends on the family (binomial, Poisson, etc.) and link (log, logit, inverse-log, etc.) you use, the interpretation is different.

• Error minimization technique

  Linear regression uses ordinary least squares method to minimise the errors and arrive at a best possible fit, while logistic regression uses maximum likelihood method to arrive at the solution.

  Linear regression is usually solved by minimizing the least squares error of the model to the data, therefore large errors are penalized quadratically.

  Logistic regression is just the opposite. Using the logistic loss function causes large errors to be penalized to an asymptotically constant.

  Consider linear regression on categorical {0, 1} outcomes to see why this is a problem. If your model predicts the outcome is 38, when the truth is 1, you've lost nothing. Linear regression would try to reduce that 38, logistic wouldn't (as much)².