POISSON REGRESSION

Question 1

Properties of a poisson distribution:

Answer 1

• Discrete (0, 1, 2, etc.)
• Strictly positive
• No natural upper limit (unlike binomial)

Question 2

What is the sole parameter of the poisson distribution?

Answer 2

Lambda

- The mean rate of occurrence of the event

Question 3

What’s an offset and what the use?

Answer 3

• 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

Question 4

What effect measure can we model with poisson?

Answer 4

Incidence rates

Question 5

Btw, what effect measure can we model with logistic?

Answer 5

Cumulative incidence in cohort studies with equal follow-up of study participants

Question 6

3 commonalities between poisson and other regression models

Answer 6

1. Outcomes are independent
2. A linear model is fitted
3. Coefficients are obtained and interpreted in usual manner

Question 7

4 differences between poisson and other regression models

Answer 7

1. It models expected counts (rate)
2. The underlying mathematics and underlying probability distribution theory are different
3. E(mean) = E(variance)
4. No danger with negative predicted values (antilog of negative regression values is between 0 and 1)

Question 8

What is overdispersion?

Answer 8

When there is a larger variance than what is assumed in a model

Also: observed variance is greater than the mean

Question 9

4 sources of overdispersion

Answer 9

1. Unobserved heterogeneity
2. Clustering
3. Contagion or diffusion
4. Classical measurement error

Question 10

What is the effect of overdispersion?

Answer 10

The point estimates are accurate but less precise than they say

Question 11

When would underdispersion occur?

Answer 11

Negative correlations induced by contagion and clustering (rare)

Question 12

What are two alternatives in the case of overdispersion?

Answer 12

• 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

Question 13

What is zero-inflation?

Answer 13

When we have a bunch of 0s in a variable - it addresses both excess zeros and implicitly over-dispersion

Question 14

What’s up with the maximum likelihood estimator in the case of over-dispersion?

Answer 14

If the mean does not equal the variance,
the mle is consistent
but gives the wrong standard errors

Question 15

Two types of residuals for poisson?

Answer 15

• Pearson residuals

- Deviance residuals

Question 16

What’s the equivalent of the F tests for multiple linear regressions?

Answer 16

• The deviance test
• Null: any subset of the betas is equal to 0
• Has chi2 distribution with p-r degrees of freedom

Question 17

How does a poisson distribution approach a binomial distribution?

Answer 17

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

Question 18

Interpretation of intercept?

Answer 18

Baseline rate of outcome for 0 covariate pattern

Question 19

Key assumptions of poisson? (3)

Answer 19

1. Count outcome
2. Mean = Variance (of outcome)
3. Independent residuals

Question 20

Quasi poisson

Answer 20

• Quasi likelihood

- Fits scale parameter to allow flexible variance/mean relationship

Question 21

Negative binomial

Answer 21

• Fits scale parameter to allow flexible variance/mean relationship

Question 22

Zero-inflated poisson

Answer 22

• Slightly complicated but accounts for extra zeros and overdispersion