BIOL2001 Chap.15 Poisson Regression, Offsets and Overdispersion
Poisson Regression, Offsets and Overdispersion
Counts as responses
Species richness and abundance are counts, and a straight-line model for counts can predict negative values. Poisson regression models the log of the expected count as a linear function of the covariates, so the fitted mean, an exponential, is always positive.
The chapter compares the linear and Poisson models, sets out the assumptions of independence and equal mean and variance, and reads coefficients as multipliers of the expected count; a computed figure shows tadpole numbers rising multiplicatively with vegetation cover, with a band showing how the spread grows with the mean.
Rates and extra variation
A worked example uses an offset, the log of quadrat area, to model seedlings per square metre when sampling effort differs.
The chapter closes with overdispersion: the ratio of summed squared standardised residuals to n minus k, why even 2 is large, and the quasi-Poisson correction that inflates the standard errors, with the negative binomial as an alternative.
What this chapter covers
- 01
Counts and the log link
- 02
Why fitted means stay positive
- 03
Assumptions: independence and equidispersion
- 04
Exponentiated coefficients as percentage changes
- 05
Exposure and the offset
- 06
Overdispersion ratio
- 07
Quasi-Poisson and negative binomial models
Worked example · free
Expected tadpole counts from a log-link model
- 1With no cover, the expected count is e1.2 = 3.32 tadpoles.
- 1At 50% cover, cover = 5, so the expected count is e1.2 + 0.75 = e1.95 = 7.03.
- 1Each extra 10% of cover multiplies the expected count by e0.15 = 1.162.
- 1e0.15 − 1 = 0.162, an increase of 16.2% per extra 10% of cover.
Key terms
- Poisson Regression
- A generalised linear model for counts that uses the log link and a Poisson distribution.
- Offset
- The log of an exposure measure entered with its coefficient fixed at one, so the model describes rates.
- Equidispersion
- The Poisson property that the variance of a count equals its mean.
- Quasi-Poisson Model
- A Poisson-type model whose standard errors are inflated to allow for overdispersion.
- Standardised Residual
- An observed count minus its fitted mean, divided by the square root of the fitted mean.
- Exposure
- The size of the sampling unit, such as area or time, over which events are counted.
Poisson Regression, Offsets and Overdispersion FAQ
Why use Poisson regression instead of linear regression for counts?
A linear model can predict negative counts and assumes a constant variance, while counts cannot be negative and their spread grows with their mean. Poisson regression models the log of the mean, so fitted values stay positive, and assumes a variance equal to the mean.
How do I interpret a Poisson regression coefficient?
Exponentiate it: e to the power of the coefficient multiplies the expected count for each unit of the covariate, and subtracting one gives the percentage change. In the lecture's beach example, a coefficient of minus 0.56 means 43% fewer species for each metre above sea level.
What is an offset in a Poisson model?
An offset is the log of an exposure measure, such as quadrat area or the number of vehicles, added to the model with its coefficient fixed at one. It turns the model for counts into a model for rates, so units sampled with different effort can be compared fairly.
How do I detect and handle overdispersion?
Sum the squared standardised residuals and divide by the number of observations minus the number of parameters. A value well above 1, and the lecture treats even 2 as large, means overdispersion. Refit with a quasi-Poisson family to inflate the standard errors, or use a negative binomial model.
Assessment move
Write the equation for every Poisson output, then compute expected counts at two covariate values and the percentage change between them. Practise one model with an offset by predicting counts for units of different sizes. Compute an overdispersion ratio from given numbers and say what you would change in the model.
End Part III by putting a linear, a logistic and a Poisson output side by side and writing one interpretation sentence for each slope on its natural scale: units, odds and expected counts.
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