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In statistics, Poisson regression is a generalized linear model form of regression analysis used to model count data and contingency tables. [1] Poisson regression assumes the response variable Y has a Poisson distribution, and assumes the logarithm of its expected value can be modeled by a linear combination of unknown parameters.
Hilbe [3] notes that "Poisson regression is traditionally conceived of as the basic count model upon which a variety of other count models are based." In a Poisson model, "… the random variable y {\displaystyle y} is the count response and parameter λ {\displaystyle \lambda } (lambda) is the mean.
The Poisson model is a popular model for recurrent event data, which models the number of recurrences that have occurred. Poisson regression assumes that the number of recurrences has a Poisson distribution with a fixed rate of recurrence over time.
In probability theory and statistics, the Poisson distribution (/ ˈ p w ɑː s ɒ n /; French pronunciation:) is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known constant mean rate and independently of the time since the last event. [1]
Some authors use the term Cox proportional hazards model even when specifying the underlying hazard function, [14] to acknowledge the debt of the entire field to David Cox. The term Cox regression model (omitting proportional hazards) is sometimes used to describe the extension of the Cox model to include time-dependent factors. However, this ...
A classical GLM formulation for a CMP regression has been developed which generalizes Poisson regression and logistic regression. [13] This takes advantage of the exponential family properties of the CMP distribution to obtain elegant model estimation (via maximum likelihood ), inference, diagnostics, and interpretation.
The Poisson distribution has one free parameter and does not allow for the variance to be adjusted independently of the mean. The choice of a distribution from the Poisson family is often dictated by the nature of the empirical data. For example, Poisson regression analysis is commonly used to model count data. If overdispersion is a feature ...
First, regression analysis is widely used for prediction and forecasting, where its use has substantial overlap with the field of machine learning. Second, in some situations regression analysis can be used to show causal relationships between the independent and dependent variables. Importantly, regressions by themselves only reveal ...