For three distinct times, the columns are linearly independent by the Vandermonde determinant. Thus a quadratic Poisson regression for the log mean can reproduce any three positive time-group means. A grouped goodness-of-fit test of that mean form has zero residual degrees of freedom. Replicated observations still allow checks of the Poisson distribution and overdispersion within each time group; saturation of the mean does not establish the response distribution.
A categorical covariate can be grouped in an interaction term while retaining its original main effect. This shares selected slopes but keeps separate intercepts. Nested Poisson deviance differences test slope pooling and intercept pooling separately.
In a Poisson regression with , the treatment logarithm of the mean ratio at baseline is . The main treatment coefficient tests the contrast at . An overall no-treatment test requires both and , with their joint covariance matrix for a Wald test.
For independent Poisson observations with fitted means , the deviance from the saturated model iswith .
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Poisson regression is a type of statistical modeling used primarily for count data. It is particularly useful when the response variable represents counts of events that occur within a fixed period of time or space. The key characteristics of Poisson regression are: 1. **Count Data**: The dependent variable is a count (e.g., number of events, occurrences, etc.), typically non-negative integers (0, 1, 2, ...).