BIOL2001 Chap.14 Interpreting Multiple Logistic Regression
Interpreting Multiple Logistic Regression
Three scales, one coefficient
A logistic coefficient adds to the log-odds and multiplies the odds by the same amount wherever x starts, but its effect on probability depends on where on the S-curve you are. This chapter makes those three readings precise, and derives the divide-by-4 rule: the steepest slope of the curve, at a probability of 0.5, is the coefficient divided by four.
A computed figure draws the tangent at the midpoint and at a point further out for an invented nest-predation model.
Interpreting with several predictors
A worked example interprets that model fully: the intercept as a probability, the odds multiplier, the midpoint slope and a predictive probability.
The chapter covers inference with two standard errors, multiple predictors held constant, interactions read two ways, centring predictors so main effects refer to average cases, deviance as a guide to keeping predictors, and the separation problem that sends a coefficient to infinity.
What this chapter covers
- 01
Adding to log-odds, multiplying odds
- 02
Uneven changes in probability
- 03
The divide-by-4 rule
- 04
Evaluating probabilities at chosen values
- 05
Inference with two standard errors
- 06
Several predictors held constant
- 07
Centring and interactions
- 08
Deviance and perfect separation
Worked example · free
The midpoint slope of a nest-predation curve
- 1P = 0.5 where the log-odds are 0: 1.2 − 0.6 × distance = 0 gives a distance of 2, that is 200 m.
- 1At the midpoint the slope is β / 4 = −0.6 / 4 = −0.15 per 100 m.
- 1At 400 m the log-odds are 1.2 − 2.4 = −1.2, so P = 0.231.
- 1The slope there is βP(1 − P) = −0.6 × 0.231 × 0.769 = −0.107 per 100 m.
Key terms
- Divide-By-4 Rule
- The rule of thumb that a logistic coefficient divided by four bounds the change in probability per unit.
- Predictive Probability
- The fitted probability that the outcome is 1 for given predictor values.
- Centring
- Subtracting a predictor's mean so that zero corresponds to an average case.
- Perfect Separation
- A predictor that splits the outcomes completely, which drives its estimated coefficient towards infinity.
- Inverse Logit
- The function that converts a log-odds value back into a probability.
Interpreting Multiple Logistic Regression FAQ
What is the divide-by-4 rule in logistic regression?
The slope of the logistic curve is beta times P times one minus P, which is largest at P equal to 0.5, where it equals beta divided by four. Dividing a coefficient by four therefore gives an upper bound on the change in probability per unit of the predictor, accurate near the midpoint.
How do I interpret an interaction in a logistic model?
An interaction coefficient can be read two ways: it is added to the coefficient of one predictor for each extra unit of the other, and vice versa. Writing the model equation and substituting values such as the means shows the resulting slopes directly.
Why centre predictors before fitting an interaction?
Without centring, each main effect is the effect when the other predictor is zero, which may be an impossible value. After centring, the main effects refer to average values of the other predictor. The model, its fitted probabilities and its interaction coefficient do not change.
Assessment move
Take one fitted logistic model and write four sentences about each coefficient: on the log-odds scale, as an odds multiplier, as a probability slope at the midpoint and as a probability at a stated value. Practise the divide-by-4 rule against the exact slope at two other points. Refit a model with centred predictors and confirm that only the main effects and intercept change.
Work through the wells example from the lecture in R yourself, refitting it with centred predictors and checking that the probability at the means and the divided-by-four slopes match the printed values.
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