Logistic Function and Odds
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Calculate
Example: a model estimates the chance of passing an exam from hours of study, with β₀ = −4 and β₁ = 1.5. What is the probability of passing for someone who studied 3 hours? Change the values to work out your own case.
Step by step
What each term means
- Probability (p)
- Always between 0 and 1, whatever z is. It is what logistic regression predicts: the chance of the event happening for a given x.
- One in the Numerator
- Makes the result run from 0 (when e⁻ᶻ is huge, z very negative) to 1 (when e⁻ᶻ vanishes, z very positive).
- One Plus e⁻ᶻ
- The denominator that flattens the curve at the ends. Since e⁻ᶻ = (1 − p)/p, it holds the odds: the ratio of the probability of happening to that of not happening.
- Linear Predictor (z)
- A straight line in x, as in linear regression. z is the log of the odds, ln(p/(1 − p)): that is why each unit of x multiplies the odds by e^β₁.