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Poisson Regression Notes

Poisson Regression Notes - Count data Refers to number of...

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Count data Refers to number of times and events. Frequency data. ”Number of tornados per month”, ”hurricanes per year”. Often modeled with a Poisson distribution of parameter λ . f ( y ; λ ) = e - λ λ y y ! ; y = 0 , 1 , 2 , . . . , λ is the average number of ocurrances. Poisson regression: Y 1 , Y 2 , . . . , Y n are n counts. where Y i denotes the number of events for ”exposures” η i . UNM
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So E ( Y i ) = η i θ i and observation i has a specific covariance pattern. θ i is explained through covariates, θ i = exp ( β 0 + β 1 X i 1 + β 2 X i 2 + . . . + β p X ip ) = exp ( x t i β ) . The model is Y i Poisson ( λ i ); λ i = η i θ i = η i exp ( x t i β ) , i = 1 , 2 , . . . , n In log-scale, log ( λ i ) = log ( η i ) + β 1 X i 1 + β 2 X i 2 + . . . + β p X ip = offset + linear predictor UNM
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For a covariate X j , factor is absent X j = 0 and factor present if X j = 1. The Rate ratio (RR) RR = E ( Y i | present ) E ( Y i | absent ) = η i exp ( β 0 + β 1 ) η i exp ( β 0 ) = exp ( β 1 ) If a covariate is increased by one unit, exp ( β 1 ) is the effect due to the increase.
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