Logistic Regression | How to derive Logistic Regression | Deriving Logistic Regression Equation - P5

Step 1.
We know the equation of simple linear regression is Y = m*X + c
X is a continuous / dummy variable (categorical) whose range can be from
–inf to +inf
The output variable Y is a continuous variable whose range is also from -inf to +inf
So, the ranges of variable on both the sides of the equation matched.

Step 2
As the outcome variable is binary in case of logistic regression, the Y variable can take only discrete values, 0 or 1.
However, predicting 0 or 1 using an equation similar to linear regression is not possible.
So, what if we try to predict the probabilities associated with the two events rather than the binary outcomes? Predicting the probabilities will be feasible as their range spans from 0 to 1.

Step 3
Earlier, we calculated the conditional probability of a customer purchasing a particular product, given he is male or female. These are the probabilities we are thinking of predicting.
In case there is one predictor variable, so it was very easy. However, as the number of predictor variables increase, these conditional probabilities will become more and more difficult to calculate. However, anyways, predicting probability is a better choice than predicting 0 or 1. Hence, for logistic regression, something like the below equation will be better:
P = m* X + c

Step 4
Even with this new equation, the problem of non-matching ranges on both the sides of the equation persists.
The P ranges from 0 to 1, while X ranges from –inf to +inf.
What if we replace the P with odds, that is, P/1-P. We have seen earlier that the odds can range from 0 to +infinity. So, the proposed equation becomes:
P / ( 1 – P) = m*X + c

Step 5
Still the LHS of the equation ranges from 0 to +infinity, while the RHS ranges from –infinity to +infinity. How to get rid of this? We need to transform one side of the equation so that the ranges on both the sides match.
What if we take a natural logarithm of the odds (LHS of the equation)? Then, the range on the LHS also becomes –infinity to +infinity.
So, the final equation becomes as follows:
Log (P / 1 – P) = m* X + c
The term Loge(Odds) is called Logit.


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