Ideally, the observed data should have the sample mean equal to the sample variance (or close to one another except for randomness) in order for the Poisson Regression model to be appropriate for that data. If the observed count data for a sample has a variance that is larger than the sample mean, it is said to be over dispersed.

Question

Ideally, the observed data should have the sample mean equal to the sample variance (or close to one another except for randomness) in order for the Poisson Regression model to be appropriate for that data.  If the observed count data for a sample has a variance that is larger than the sample mean, it is said to be over dispersed.

BlackTom AI · Accepted Answer

The question discusses a property of Poisson regression and when it is appropriate to use this model. 
Option 1: True. In a Poisson distribution, the mean and the variance are equal (both equal to the rate parameter λ). Therefore, for Poisson regression to be appropriate, the observed data should exhibit roughly equal sample mean and sample variance, allowing for randomness. When the observed variance is larger than the mean, this is termed overdispersion, which suggests that the Poisson assumption (mean equals variance) is violated and alternative models (like negative binomial) may be more suitable. 
Option 2: False. If one assumes Poisson regression and observes overdispersion (variance greater than the mean), the statement that Poisson is appropriate is not supported. The presence of overdispersion indicates a mismatch between the data and the Poisson assumptions, making False an incorrect stance in this context. 
To summarize the core idea: the Poisson model implies mean equals variance, so equal or near-equal sample mean and variance is desirable; overdispersion (variance > mean) signals potential model misspecification for Poisson rather than suitability. Hence the true description is that observed overdispersion contradicts Poisson appropriateness, reinforcing why the True option aligns with the underlying theory.

2025WI_MSDS_410-DL_SEC56 Quiz #5: Poisson and ZIP Regression

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