Understanding output from logistic regression
Generalized Linear Models in R
Richard Erickson
Instructor
Communicating results from logistic regression?
Linear regression is straight forward:
- Add intercepts
- Multiply slopes
Poisson regression:
- Requires exponential transformation
- Similar to linear regression post-transformation
- Logistic regression???
Odds-ratios
- Not as straightforward as Poisson exponential
- Used to compare relative odds of two events occurring
Example odds-ratios
- Unfair coin:
- Compare heads to tails
- Heads 3 times for every 1 tails
- 3-to-1 odds
- Odds-ratios 3.0
- Often used in sports/gambling
- Medical studies
Logistic derivation of odds-ratio
Log-odds ("logit"):
$\phi(x) = \text{ln}(\frac{p(x)}{1 - p(x)}) = \beta_0 + \beta_1 x$
Odds, take exponential ($e^{x}$):
$\frac{p(x)}{1 - p(x)} = e^{\beta_0 + \beta_1x}$
Odd-ratio for continuous variable
Odds-ratio (OR) for continuous variable:
$\text{OR} = \frac{e^{\beta_0 + \beta_1(x +1)}}{e^{\beta_0 + \beta_1x}} = e^{\beta_1}$
Interpretation
OR Values:
- OR = 1: Coefficient has no effect
- OR < 1: Coefficient decreases odds
- OR > 1: Coefficient increases odds
Cancer example
Non-smoking vs smoking males (Pesch et al. 2012)
- OR: 103.5 (95% CI 74.8-143.2)
- $>$ 100-to-1 odds of getting cancer for smoking men!
- Medical literature often reports 95% confidence intervals rather than p-values
- Broader trend away from p-values
Extract from GLM
glm_out <- glm(y ~ x, family = 'binomial')
coef(glm_out)
exp(coef(glm_out))
confint(glm_out)
exp(confint(glm_out))
Tidyverse
library(broom)
glm_out <- glm(y ~ x, family = 'binomial')
tidy(glm_out, exponentiate = TRUE, conf.int= TRUE)
Let's practice!
Generalized Linear Models in R
