Properties of a poisson distribution:
- Discrete (0, 1, 2, etc.)
- Strictly positive
- No natural upper limit (unlike binomial)
What is the sole parameter of the poisson distribution?
Lambda
- The mean rate of occurrence of the event
What’s an offset and what the use?
- An offset allows to model counts per unit time rather than simply counts
- It’s a variable that is forced to have a regression coefficient of 1
What effect measure can we model with poisson?
Incidence rates
Btw, what effect measure can we model with logistic?
Cumulative incidence in cohort studies with equal follow-up of study participants
3 commonalities between poisson and other regression models
- Outcomes are independent
- A linear model is fitted
- Coefficients are obtained and interpreted in usual manner
4 differences between poisson and other regression models
- It models expected counts (rate)
- The underlying mathematics and underlying probability distribution theory are different
- E(mean) = E(variance)
- No danger with negative predicted values (antilog of negative regression values is between 0 and 1)
What is overdispersion?
When there is a larger variance than what is assumed in a model
Also: observed variance is greater than the mean
4 sources of overdispersion
- Unobserved heterogeneity
- Clustering
- Contagion or diffusion
- Classical measurement error
What is the effect of overdispersion?
The point estimates are accurate but less precise than they say
When would underdispersion occur?
Negative correlations induced by contagion and clustering (rare)
What are two alternatives in the case of overdispersion?
- quasi poisson: assumes var(Y)= theta*u
- negative binomial: assumes Var(Y)=u(1+ ku)
theta = quasi-poisson overdispersion parameter k(1/theta) = shape parameter of negative binomial distribution
What is zero-inflation?
When we have a bunch of 0s in a variable - it addresses both excess zeros and implicitly over-dispersion
What’s up with the maximum likelihood estimator in the case of over-dispersion?
If the mean does not equal the variance,
the mle is consistent
but gives the wrong standard errors
Two types of residuals for poisson?
- Pearson residuals
- Deviance residuals
What’s the equivalent of the F tests for multiple linear regressions?
- The deviance test
- Null: any subset of the betas is equal to 0
- Has chi2 distribution with p-r degrees of freedom
How does a poisson distribution approach a binomial distribution?
When the probability of a success grows very small while the number of trials grows very large in such a way that the number of successes stays finite
Interpretation of intercept?
Baseline rate of outcome for 0 covariate pattern
Key assumptions of poisson? (3)
- Count outcome
- Mean = Variance (of outcome)
- Independent residuals
Quasi poisson
- Quasi likelihood
- Fits scale parameter to allow flexible variance/mean relationship
Negative binomial
- Fits scale parameter to allow flexible variance/mean relationship
Zero-inflated poisson
- Slightly complicated but accounts for extra zeros and overdispersion
