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Methods and formulas for the diagnostic measures in Fit Poisson Model

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Pearson residuals

Elements of the Pearson chi-square that can be used to detect ill-fitted factor/covariate patterns. Minitab stores the Pearson residual for the ith factor/covariate pattern. The formula is:

Notation

TermDescription
yithe response value for the ith factor/covariate pattern
the fitted value for the ith factor/covariate pattern
Vthe variance function for the model at

The variance function depends on the model:

Model Variance function
Binomial
Poisson

Standardized and deleted Pearson residuals

Used to detect ill-fitted factor/covariate patterns. Minitab stores the standardized Pearson residual for the ith factor/covariate pattern. Deleted Pearson residuals are also called likelihood ratio Pearson residuals. For the deleted Pearson residual, Minitab calculates the one-step approximation described in Pregibon.1 This approximation is equal to the standardized Pearson residual. The formula is:

Notation

TermDescription
the Pearson residual for the ith factor/covariate pattern
1, for the binomial and Poisson models
the leverage for the ith factor/covariate pattern

Standardized Pearson residuals with validation

For validation data, the denominator of the formula for the standardized Pearson residual adds the leverage instead of subtracting the leverage.

Formula

Notation

TermDescription
the Pearson residual for the ith validation row
1, for the binomial and Poisson models
the leverage for the ith validation row

Deviance residuals

Deviance residuals are based on the model deviance and are useful in identifying ill-fitted factor/covariate patterns. The model deviance is a goodness-of-fit statistic based on the log-likelihood function. The deviance residual defined for the ith factor/covariate pattern is:

Notation

TermDescription
yi the response value for the ith factor/covariate pattern
the fitted value for the ith factor/covariate pattern
the deviance for the ith factor/covariate pattern

Standardized deviance residual

The standardized deviance residual is helpful in the identification of outliers. The formula is:

Notation

TermDescription
rD,iThe deviance residual for the ith factor/covariate pattern
hiThe leverage for the ith factor/covariate pattern

Standardized deviance residual with validation

For validation data, the denominator of the formula for the standardized deviance residual adds the leverage instead of subtracting the leverage.

Formula

Notation

TermDescription
rD,iThe deviance residual for the ith validation row
hiThe leverage for the ith validation row

Deleted deviance residual

The deleted deviance residual measures the change in the deviance due to the omission of the ith case from the data. Deleted deviance residuals are also called likelihood ratio deviance residuals. For the deleted deviance residual, Minitab calculates a one-step approximation based on the Pregibon one-step approximation method1. The formula is as follows:

Notation

TermDescription
yithe response value at the ith factor/covariate pattern
the fitted value for the ith factor covariate pattern
hithe leverage for the ith factor/covariate pattern
r'D,ithe standardized deviance residual for the ith factor/covariate pattern
r'P,ithe standardized Pearson residual for the ith factor/covariate pattern

1. Pregibon, D. (1981). "Logistic Regression Diagnostics." The Annals of Statistics, Vol. 9, No. 4 pp. 705–724.

Delta chi-square

Minitab calculates the change in the Pearson chi-square due to deleting all the observations with the jth factor/covariate pattern. Minitab stores one delta chi-square value for each distinct factor/covariate pattern in the data. You can use delta chi-square to detect ill-fitted factor/covariate patterns. The formula for the delta chi-square is:

Formula

Notation

TermDescription
hj leverage
rj Pearson residuals

Delta deviance

Minitab calculates the change in the deviance statistic by deleting all the observations with the jth factor/covariate pattern. Minitab stores one value for each distinct factor/covariate pattern in the data. You can use delta deviance to detect ill-fitted factor/covariate patterns. The change in the deviance statistic is:

Notation

TermDescription
hjleverage
rjPearson residuals
djdeviance residuals

Delta beta (standardized)

Minitab calculates the change by deleting all observations with the jth factor/covariate pattern. One value is stored for each distinct factor/covariate pattern in the data. You can use standardized delta β to detect factor/covariate patterns that have a strong influence on the estimates of the coefficients. This value is based on the standardized Pearson residual.

Formula

Notation

TermDescription
hj leverage
rs j standardized Pearson residuals

Delta beta

Minitab calculates the change by deleting all observations with the jth factor/covariate pattern. One value is stored for each distinct factor/covariate pattern in the data. You can use delta β to detect factor/covariate patterns that have a strong influence on the estimates of the coefficients. This value is based on the Pearson residual.

Formula

Notation

TermDescription
hj leverage
rj Pearson residuals

Leverages

The leverages are the diagonal elements of the generalized hat matrix. The leverages are useful in detecting factor/covariate patterns that may have a significant influence on the results.

Formula

Notation

TermDescription
wj the jth diagonal element of the weight matrix from fitting the coefficients
xjthe jth row of the design matrix
Xthe design matrix
X'the transpose of X
Wthe weight matrix from the estimation of the coefficients

Leverages with validation

Notation

TermDescription
wi the internal weight for the ith validation row
xithe row of the design matrix for the predictors in the ith validation row
Xthe design matrix for the training data set
X'the transpose of X
Wthe diagonal matrix of internal weights for the training data set

Cook's distance

Minitab calculates an approximate Cook's distance.

Formula

Notation

TermDescription
hithe leverage for the ith factor/covariate pattern
the standardized Pearson residual for the ith factor/covariate pattern
pthe regression degrees of freedom

DFITS

A measure of the influence of a single deletion on the fitted values. Observations with large DFITS values may be outliers. Minitab calculates an approximate value for DFITS.

Formula

Notation

TermDescription
hiThe leverage for the data point
The deleted Pearson residual for the data point

Variance inflation factor (VIF)

To calculate a VIF, perform a weighted regression on the predictor with the remaining predictors. The weight matrix is that given in McCullagh and Nelder1 for the estimation of the coefficients. In this case, the VIF formula is equivalent to the formula for a linear regression. For example, for predictor xj the formula for the VIF is:

Notation

TermDescription
coefficient of determination with xj as the response variable and the other terms in the model as the predictors

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

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