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Predicting Count Data

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Predicting Count Data

Poisson Regression

General Linear Model (GLM)

-Anything that can be written like this:

-Solved using ordinary least squares

-Assumptions revolve around the Normal Dist.

Generalized Linear Model

-Anything that can be written like this:

-Solved using maximum likelihood

-Assumptions use many different distributions

- Linear Regression: Assuming normal errors around the predicted score
- When we violate this assumptions, our estimates of the distributions of the B’s are incorrect
- Also…in some case our estimates of the effect size are inaccurate (usually too small)

- Linear regression is really a predictive model before anything else. (The statistical aspect is extra).

B1

B0

- (Criminal Justice) Number of offenses per year
- (Domestic Violence) Number of DV events per person
- (Epidemiology) Number of seizures per week

- This type of data can only have discrete values that are greater than or equal to zero.
- In situations, this data follows the Poisson Distribution

- The Poisson random variable is defined by one parameter: the mean (μ)
- It has the strong assumption that the mean is equal to the variance
μ=σ

- In this model, instead of predicting mean of a normal distribution, you are predicting the mean of a Poisson distribution (given some predictors)

- In linear regression:
- In Poisson regression:

- In your outcome variable (Y), the mean equals the variance. (There is a test for this)
- For violations you can use Negative Binomial…which is just a Poisson where the variance is separate from the mean.

- Observations are independent (as with most analyses)
- And, basically, that the predictive model makes sense ( )

- Like logistic, we have to interpret the EXP(B)
- (This is the notation for )

- Instead of an odds ratio, this is a relative risk ratio: it is the additional rate given a one unit increase in X
- 1 is the null hypothesis
- 1.2 would be an increase of .2 in the relative rate for a one unit increase

- Turns out that not using Poisson isn’t the worst thing ever.
- Actually get alpha deflation

- BUT- Many journals that are used to this kind of data will reject articles that do not use the proper technique