Poisson regression coefficients

Generalized Linear Models in R

Richard Erickson

Instructor

Chapter overview

• Describing Poisson regressions
• Plotting Poisson GLMs with ggplot2
• Describing logistic regression with odds-ratios
• Plotting binomial GLMs with ggplot2

Fire injury data

Linear model coefficients overview

• Estimate expected daily injury per month
• Estimate reference intercept
• Estimate intercept for other months

Linear model equation

• $ y \sim \beta_0 + \beta_m x_m + \ldots + \epsilon$
• $\beta_0$: Reference intercept
• $\beta_m$: Month $m$ effect
• $y$ injuries per day (e.g., 1, 0, 4)
• $x$ dummy variable to code for month (0 or 1)
• $m$ corresponds to month intercept, dummy variable

Linear model results

• $\beta_0$ is expected (or average) in reference month
• $\beta_m$ is effect of month $m$ (or difference from reference)
• e.g., $\beta_0$ + $\beta_m$ = ave. daily injuries for month $m$
• More complicated models covered in chapter 4
• Linear models are additive

Poisson model

• $y \sim \text{Poisson}(\lambda)$
• Link: $\lambda = e^{(\beta_0 + \beta_m x_m + \epsilon)}$
• Multiplicative
• Example results:
  • $\beta_0 \times \beta_1 = \text{ln}(\text{mean daily injuries for month}\ m)$
  • Take exponential to convert to raw units

Difference between Poisson and linear models

• Poisson model: $e^{\beta_0 \times \beta_m} = \text{expected daily injuries for month}\ m$
• Linear model: $\beta_0$ + $\beta_m = \text{expected daily injuries for month}\ m$

Extract in R

poisson_out <- glm(y ~ x, family = 'poisson')
coef(poisson_out)
exp(coef(poisson_out))

Tidy solution

library(broom)
poisson_out <- glm(y ~ x, family = 'poisson')
tidy(poisson_out, exponentiate = TRUE)

Statistical inferences

• Similar as linear model on link-scale
• Do coefficients differ from zero?
• On data-scale, different
• Do coefficients differ from 1?
• Exponential-scale rather than raw-scale

Let's practice!

Generalized Linear Models in R