ETC1000 Chap.10 Logistic Regression for Binary Outcomes
Logistic Regression for Binary Outcomes
Why the obvious approach fails
Running an ordinary regression on a zero one outcome half works.
The fitted values rise with the predictor and sit near one half in the middle, and then far enough along they return 1.4 or minus 0.3. A probability cannot take those values, so the failure is not cosmetic and cannot be patched by rounding.
What the curve guarantees
Logistic regression keeps the linear part and passes it through a function that squashes any number into the open interval between zero and one.
The result is an S shape: nearly flat where the outcome is almost certain either way, steepest in the middle where the prediction is genuinely uncertain, and asymptotic at both ends so the bounds can never be crossed.
Read the coefficient on the odds scale
The model says the log odds of the outcome are a straight line in the predictors, which restores everything you know about reading coefficients on condition that you read them there.
A coefficient adds a constant to the log odds, which multiplies the odds by a constant factor. It does not change the probability by a constant amount, because how much probability a unit buys depends on where you started.
How the model is judged
Not by residuals and not by a share of variation, neither of which is defined for a binary outcome.
A logistic model is judged the way week 9 judges any classifier: pick a threshold, build the four counts on data the model has not seen, and report sensitivity and precision beside accuracy with the base rate stated.
What this chapter covers
- 01
Why a straight line predicts probabilities outside zero and one
- 02
The logistic curve, its steep middle and its flat ends
- 03
Probability, odds and log odds, and what a coefficient does to each
- 04
Turning a coefficient into a defensible sentence
- 05
The tipping point where the predicted probability is one half
- 06
Judging the model with the week 9 classifier rates
Interpret an authored model and find its tipping point
- 1Interpret each coefficient on the odds scale.
- 1Predict the probability for an established customer on 30 days.
- 1Find the tipping point and say what to do with it.
Key terms
- Binary outcome
- An outcome taking one of two values, usually coded zero and one.
- Logistic curve
- The S shaped function mapping any real number into the interval between zero and one.
- Odds
- A probability divided by one minus that probability, ranging from zero upwards.
- Log odds
- The natural logarithm of the odds, which the model treats as a straight line in the predictors.
- Odds ratio
- The constant factor by which a one unit change in a predictor multiplies the odds.
- Tipping point
- The predictor value at which the estimated probability is exactly one half.
Logistic Regression for Binary Outcomes FAQ
Why not just run an ordinary regression on a zero one outcome?
Because a straight line is unbounded. Far enough along the predictor it returns values above one or below zero, and no reading of those numbers is a probability. Near the middle of the range the two models agree closely, which is why the ordinary version looks acceptable until it is used at the ends, where decisions about the clearest cases actually live.
What does a coefficient of 0.8 mean in this model?
That a one unit increase in the predictor adds 0.8 to the log odds, which multiplies the odds by about 2.2, holding the other predictors fixed. So each unit roughly doubles the odds of the outcome. It does not mean the probability rises by a fixed amount, because the same coefficient is worth a great deal in the steep middle of the curve and almost nothing at either end.
Is doubling the odds the same as doubling the chance?
No, and the gap grows towards the top. Doubling the odds from one to two moves the probability from 0.50 to about 0.67. Doubling them from eight to sixteen moves it from about 0.89 to 0.94. Writing doubles the chance for a doubling of the odds overstates the effect near the top of the range and understates it in the middle.
How is a logistic model's fit reported?
Through the classifier rates from the previous chapter, not through residuals or a share of variation, neither of which is defined here. Choose a threshold, build the four counts on data the model has not seen, and report sensitivity and precision alongside accuracy with the base rate stated so a reader can see what the accuracy figure had to beat.
Exam move
Rehearse the scale discipline until it is automatic: sign, then odds multiplier with the holding fixed clause, then a probability only after naming where on the curve you are standing. Then practise finding the tipping point, because it is the one number from this model a non technical reader can use directly.
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