Poisson regression model

We have a sequence of Poisson observations t/i. where each observation j/j has its own mean value /Xj. We have also observed the values (xn. Xip) that each of the p predictor variables has for the observation. Let pi come from a Poisson(pi) distribution. The likelihood is 

The likelihood of the random sample Pi, - , Pn will be the product of the individual likelihoods and is given by 

The linear predictor. We want to relate the Poisson observations to the set of predictor variables xi. Xp. We let. x be the value of predictor j for the 

This is called the log link function and it relates the linear predictor to the parameter. Other link functions could be used, but the log link is the most commonly used for Poisson observations. We use the link function to rewrite the likelihood function as a function of the parameters Po.Pp. The likelihood becomes 

The maximum likelihood estimates can be found by taking derivatives with respect to the parameters, setting them equal to zero, and finding the simultaneous solutions of the resulting equations. The likelihood and its logarithm will have maximums at the same values since 

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Regardless of the choice of analytic framework, the assumptions underlying any modeling need to be understood and examined. For example, models of rates, for example, Poisson regression models, require the assumption... 

Maximum likelihood estimation in the Poisson regression model. Since the Poisson regression model is a generalized linear model, the maximum likelihood... 

COMPUTATIONAL APPROACH TO POISSON REGRESSION MODEL 207 2. Then update the parameter vector to step n by... 

The observations of the censoring variable Wi come from the Poisson distribution, a member of the exponential family. The logarithm of the parameter /r, is linked to the linear predictor rj. The observations are independent. Clearly the proportional hazards model is a generalized linear model and can be analyzed the same way as the Poisson regression model. 

The Poisson regression model allows each observation to come from the Poisson distribution having its own parameter pi, which is linearly regressed on the known values of the predictors for that observation. 

The link function relates the linear predictor to a function of the parameter. The log link function is commonly used for the Poisson regression model. 

The Poisson regression model is an example of the generalized linear model. The maximum likelihood estimates of the coefficients of the predictors can be found by iteratively reweighted least squares. This also finds the covariance matrix of the normal distribution that matches the curvature of the likelihood... 

The likelihood turns out to be the same as if the censoring variables are a sample of Poisson random variables where the means are linear functions of the predictors. So this is like the Poisson regression model. 

The data from Table 9.1 are given in Minitab worksheet ExerciseQ. 1. mtw. We wish to find out how y, the number of fractures that occur in an upper seam of a coal mine depends on the predictor variables xi, the inner burden thickness, X2, the percent extraction of the lower previously mined seam, X3 the lower seam height, and X4 the length of time mine has been opened. We will use the Poisson regression model. Since we don t have any particular prior knowledge we will use "flat" priors for the parameters. 

Table A, 14 Minitab commands for three parallel runs of the Metropolis-Hastings algorithm for the Poisson regression model with five parameters (four predictors) coupled with the past. The chains have different starting points, but all have the same random inputs. Note we are using flat prior in this example.

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