In logistic regression of binary data, the natural logarithm of the odds is equal to the linear predictor.
\(\ln \left[\frac{P(Y_{i}=1|X_{i}=x_{i})}{P(Y_{i}=0|X_{i}=x_{i})}\right] = \ln \left[\frac{P(Y_{i}=1|X_{i}=x_{i})}{1-P(Y_{i}=1|X_{i}=x_{i})}\right]=\beta_{0}+\beta_{1} x_{i} \ \ \ \ \ \ \ (1)\)
This may be a complicated equation to absorb for a newcomer, so let’s break this down.
In this context, the odds is the ratio of the probability of success to the probability of failure. (The subscript “1” denotes success, and the subscript “0” denotes failure.)
\(\frac{P(Y_{i}=1|X_{i}=x_{i})}{P(Y_{i}=0|X_{i}=x_{i})} \ \ \ \ \ \ \ (2)\)
Since Yᵢ is a binary variable, it can hold only one of two values: 1 or 0. By the Kolmogorov laws of probability,
\(P(Y_{i}=0|X_{i}=x_{i}) = 1-P(Y_{i}=1|X_{i}=x_{i}) \ \ \ \ \ \ \ (3)\)
This is how I re-wrote the denominator in Equation (1).
Based on the modelling assumption of logistic regression, the natural logarithm of the odds is set to equal the linear predictor 𝛽₀ + 𝛽₁xᵢ. This is the statement as expressed mathematically in Equation (1).