12.2 - Multiple Logistic Regression

Multiple covariates (predictor variables) can be included in the logistic model. The variables may be binary, ordinal, nominal or continuous.

\(In\left ( \frac{P}{1-P} \right )=\alpha +\beta_{1}x_{1}+ \beta_{2}x_{2}+...+ \beta_{i}x_{i}\)

For multiple logistic regression, the \(\beta_i\)

  • are interpreted as the increase in log-odds for a one unit increase in \(x_i\) with all the other \(x_i\) constant
  • measure the association between \(x_i\) and log-odds adjusted for all other \(x_i\)

Example

P : Probability for cardiac arrest
Exc: 1 = lack of exercise, 0 = exercise
Smk: 1 = smokers, 0 = non-smokers

\(\begin{align}In\left ( \frac{P}{1-P} \right )&=\alpha +\beta_{1}Exc+ \beta_{2}Smk\\ &= 0.7102 + 1.0047 Exc + 0.7005 Smk\end{align}\)

(SE 0.2614) (SE 0.2664)

OR for lack of exercise = \(e^{1.0047} = 2.73\) (adjusted for smoking)

95% CI for \(\Theta = e^{(1.0047 \pm 1.96 \times 0.2614)} = (1.64 , 4.56) \)

Lack of exercise is associated with cardiac arrest, after adjusting for smoking.