Methods and formulas for the estimated equation in Fit Poisson Model

Select the method or formula of your choice.

Coefficients

Two methods to find the maximum likelihood estimates of the coefficients exist. One method is to maximize directly the likelihood function with respect to the coefficients. These expressions are nonlinear in the coefficients. The alternative method is to use an iterative reweighted least squares approach, which is the method that Minitab uses to obtain the estimates of the coefficients. McCullagh and Nelder1 show that the two methods are equivalent. However, the iterative reweighted least squares method is easier to implement. For details, see 1.

[1] P. McCullagh and J. A. Nelder (1989). Generalized Linear Models, 2nd Ed., Chapman & Hall/CRC, London.

Standard error of coefficients

The standard error of the ith coefficient is the positive square root of the ith diagonal element of the variance-covariance matrix. The variance-covariance matrix has the following form:

W is a diagonal matrix where the diagonal elements are given by the following formula:

where

This variance-covariance matrix is based on the observed Hessian matrix as opposed to the Fisher's information matrix. Minitab uses the observed Hessian matrix because the model that results is more robust against any conditional mean misspecification.

If the canonical link is used then the observed Hessian matrix and the Fisher's information matrix are identical.

Notation

TermDescription
yithe response value for the ith row
the estimated mean response for the ith row
V(·)the variance function given in the table below
g(·)the link function
V '(·)the first derivative of the variance function
g'(·)the first derivative of the link function
g''(·)the second derivative of the link function

The variance function depends on the model:

Model Variance function
Binomial
Poisson

See [1] and [2] for more information.

[1] A. Agresti (1990). Categorical Data Analysis. John Wiley & Sons, Inc.

[2] P. McCullagh and J.A. Nelder (1992). Generalized Linear Model. Chapman & Hall.

Z

The Z-statistic used to determine whether the predictor is significantly related to the response. Larger absolute values of Z indicate a significant relationship. The formula is:

Notation

TermDescription
Zi The test statistic for a standard normal distribution
The estimated coefficient
The standard error of the estimated coefficient

For small samples, the likelihood-ratio test may be a more reliable test of significance. The likelihood ratio p-values are in the deviance table. When the sample size is large enough, the p-values for the Z statistics approximate the p-values for the likelihood ratio statistics.

p-value (P)

Used in hypothesis tests to help you decide whether to reject or fail to reject a null hypothesis. The p-value is the probability of obtaining a test statistic that is at least as extreme as the actual calculated value, if the null hypothesis is true. A commonly used cut-off value for the p-value is 0.05. For example, if the calculated p-value of a test statistic is less than 0.05, you reject the null hypothesis.

Confidence interval

The large sample confidence interval for an estimated coefficient is:

For binary logistic regression, Minitab provides confidence intervals for the odds ratios. To obtain the confidence interval of the odds ratio, exponentiate the lower and upper limits of the confidence interval. The interval provides the range in which the odds may fall for every unit change in the predictor.

Notation

TermDescription
the ith coefficient
the inverse cumulative probability of the standard normal distribution at
the significance level
the standard error of the estimated coefficient

Variance-covariance matrix

A d x d matrix, where d is the number of predictors plus one. The variance of each coefficient is in the diagonal cell and the covariance of each pair of coefficients is in the appropriate off-diagonal cell. The variance is the standard error of the coefficient squared.

The variance-covariance matrix is from the final iteration of the inverse of the information matrix. The variance-covariance matrix has the following form:

W is a diagonal matrix where the diagonal elements are given by the following formula:

where

This variance-covariance matrix is based on the observed Hessian matrix as opposed to the Fisher's information matrix. Minitab uses the observed Hessian matrix because the model that results is more robust against any conditional mean misspecification.

If the canonical link is used then the observed Hessian matrix and the Fisher's information matrix are identical.

Notation

TermDescription
yi the response value for the ith row
the estimated mean response for the ith row
V(·)the variance function given in the table below
g(·)the link function
V '(·)the first derivative of the variance function
g'(·)the first derivative of the link function
g''(·)the second derivative of the link function

The variance function depends on the model:

Model Variance function
Binomial
Poisson

See [1] and [2] for more information.

[1] A. Agresti (1990). Categorical Data Analysis. John Wiley & Sons, Inc.

[2] P. McCullagh and J.A. Nelder (1992). Generalized Linear Model. Chapman & Hall.

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