CVEN9407 Chap.9 Poisson and Negative Binomial Count Models
Poisson and Negative Binomial Count Models
Crash frequency is a count, and this chapter replaces the linear model with models built for counts. It sets out Poisson regression with its log link and its equal mean and variance restriction, shows how an exposure offset such as segment length enters, and explains why overdispersed crash data call for the negative binomial.
A worked Poisson model is turned into predicted counts and incidence rate ratios, and the chapter ends by comparing models with log-likelihood and BIC, as the course reading on speed enforcement does.
This step forms the count half of the second deliverable.
The brief asks for a count specification of crash frequency, Poisson or negative binomial, set against the linear regression from the first report, and discuss overdispersion and model fit. Rate ratios are the most useful way to report the results to a road safety reader.
What this chapter covers
- 01
Why crash counts are not continuous outcomes
- 02
Poisson regression and the log link
- 03
Exposure offsets for segment length or traffic
- 04
Overdispersion and its consequences
- 05
The negative binomial model
- 06
Incidence rate ratios
- 07
Comparing models with BIC
Worked example · free
Predicted crashes and a zero-crash probability
- 1Linear predictor: -0.40 + 0.50(0) + 0.02(60) = 0.80.
- 1Expected count: lambda = e to the power 0.80 = 2.23 crashes a year.
- 1Crash-free year: P(0) = e to the power -2.23 = 0.108, about one segment-year in nine.
Key terms
- Log link
- A log link connects the expected count to the covariates through its logarithm, so predicted counts are always positive.
- Exposure offset
- An exposure offset is the log of an exposure measure, such as segment length, entered with its coefficient fixed at one.
- Negative binomial model
- The negative binomial model is a count regression that adds an overdispersion parameter so the variance can exceed the mean.
- Incidence rate ratio
- An incidence rate ratio is the exponentiated count-model coefficient: the factor by which expected crashes change for a one-unit change.
- Bayesian information criterion
- The Bayesian information criterion is minus twice the log-likelihood plus a penalty for parameters; the lower value is preferred.
- Equidispersion
- Equidispersion is the Poisson model's assumption that the variance of a count equals its mean.
Poisson and Negative Binomial Count Models FAQ
How do I know if my crash counts are overdispersed?
Compare the variance of the counts with their mean, or the residual deviance with its degrees of freedom after fitting a Poisson model. Values well above 1 point to overdispersion, and a negative binomial fit with a clearly positive dispersion parameter confirms it.
Why can't I compare a count model with the linear model by R squared?
Count models are estimated by maximum likelihood on a log scale, so R squared is not defined in the same way. Compare log-likelihoods or BIC across count models, and compare predictions and signs against the linear model in words.
What does an offset do?
It puts exposure into the model without estimating a coefficient for it. With log length as an offset, the other coefficients describe crashes per kilometre, so long and short segments are compared fairly.
What is a random parameters negative binomial model?
It is an extension, used in the course reading on speed enforcement, in which some coefficients vary across road segments to capture unobserved differences between sites. It goes beyond what the project requires, but it shows where count modelling of crashes leads in research.
Why do so many segments record zero crashes?
Many sites carry little traffic or are simply short, so in a given year the most likely count is zero. A Poisson model with a small expected count predicts many zeros; if there are even more zeros than a negative binomial predicts, a zero-inflated model may be considered.
Assessment move
Fit a Poisson model to the counts you used for the linear regression, check the variance-to-mean ratio, and refit as negative binomial if it is well above 1. Convert every coefficient into a rate ratio before writing about it. In Report 2, set the linear and count results side by side and explain each disagreement. Note the dispersion parameter and its test in your model log, since it decides which standard errors you report.
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