SOCI 620: Quantitative methods 2

Agenda

Expanding on the Poisson regression

  1. Administrative
  2. Interpreting coefficients from
    Poisson regressions
  3. “Over-dispersed” Poisson
    regressions
  4. Zero-inflated Poisson
    regressions
  5. Hands on:
    Poisson models in R

Slides are licensed under CC BY-NC-SA 4.0

Interpreting Poisson coefficients

Movie still of a man (Nicolas Cage) in glasses studiously reading an old book

Interpreting coefficients

Mean exp(Mean)

α

 0.27 1.32

βM

 1.11 3.05

βG

 -0.14 0.87

α (baseline)

  • A student who is not a boy (Mi = 0) and is in grade 10 (Gi = 0) is predicted to play about 1.32 hours of games per week.

βM (gender)

  • Boys (Mi = 1) are expected to spend about 3.05 times more time than non-boys (Mi = 0) playing games.

βG (grade)

  • A one-year increase in grade is associated with playing 0.87 times as much. This is a decrease of 13% each year.

Over-dispersion

Image of a doctor standing behind a woman making a slightly disgusted look as he stretches the skin from her face out grotesquely

Over-dispersion

To illustrate, consider the simpler model with only gender as a predictor:

Mean exp(Mean)

α

 0.38 1.46

β

 1.10 3.00

Over-dispersion

Actual distribution

Posterior predicted distribution
(Poisson)

Potential problem:

Our model under-predicts the number of large and small values in the outcome (not enough variation)

Over-dispersion

Gamma-Poisson regression:

AKA negative binomial
AKA over-dispersed Poisson

Extra “dispersion” from gamma

Two students who look identical based on covariates can have different Poisson rates λi.

One more prior

Over-dispersion

Gamma-Poisson regression:

Data
story:

Over-dispersion

Gamma-Poisson and negative-binomial regressions are the same:

“Negative binomial regression” is the typical terminology

Over-dispersion

Mean 95% CI exp(Mean)

α

 0.38 (0.32, 0.45) 1.47

β

 1.09 (1.01, 1.18) 2.99

θ

 0.35 (0.33, 0.37) —

Over-dispersion

Actual distribution

Posterior predicted distribution
(Gamma-Poisson)

Zero
inflation

Packaging advertizing a inflatable Halloween decoration of Zero (the ghost dog from Nightmare Before Christmas)

Zero inflation

Actual distribution

Posterior predicted distribution
(Poisson)

Potential problem:

Our model under-predicts the number of zero-valued outcomes

Zero inflation

Outcome variable is the result of one of two processes:

Either the student is structurally constrained to play zero hours per week–e.g. they do not own a game console ()

Or the student is able to play games and does so at a rate ()

Zero inflation

Each student’s probability of not owning a console is modeled with

Zero inflation

The probability is modeled with a linear function of family income

Zero inflation

The rate is modeled with a linear function of gender

Zero inflation

Data story:

Model comparisons

Data distribution
Poisson regression

WAIC: 39,628.2
Negative binomial
(gamma-Poisson)
WAIC: 19,411.4
Zero-inflated Poisson

WAIC: 31,848.9

Image credit

Figures by Peter McMahan (source code)

Movie still of a man (Nicolas Cage) in glasses studiously reading an old book

Still from National Treasure (2004)

Image of a doctor standing behind a woman making a slightly disgusted look as he stretches the skin from her face out grotesquely

Still from Brazil (1985)

Packaging advertizing a inflatable Halloween decoration of Zero (the ghost dog from Nightmare Before Christmas)

Merchandise from The Nightmare Before Christmas (1993)