glmtoolbox (version 0.1.12)

dfbeta.gnm: Dfbeta statistic for Generalized Nonlinear Models

Description

Calculates an approximation of the parameter estimates that would be produced by deleting each case in turn, which is known as the one-step approximation. Additionally, the function can produce an index plot of the Dfbeta statistic for some parameter specified by the argument coefs.

Usage

# S3 method for gnm
dfbeta(model, coefs, identify, ...)

Value

A matrix with as many rows as cases in the sample and as many columns as parameters in the linear predictor. The \(i\)-th row in that matrix corresponds to the difference between the parameters estimates obtained using all cases and the one-step approximation of those estimates when excluding the \(i\)-th case from the dataset.

Arguments

model

an object of class gnm.

coefs

an (optional) character string which (partially) match with the names of some model parameters.

identify

an (optional) integer indicating the number of individuals to identify on the plot of the Dfbeta statistic. This is only appropriate if coefs is specified.

...

further arguments passed to or from other methods. If plot.it=TRUE then ... may be used to include graphical parameters to customize the plot. For example, col, pch, cex, main, sub, xlab, ylab.

Details

The one-step approximation of the parameters estimates when the \(i\)-th case is excluded from the dataset consists of the vector obtained as a result of the first iteration of the Fisher Scoring algorithm when it is performed using: (1) a dataset in which the \(i\)-th case is excluded; and (2) a starting value that is the estimate of the same model but based on the dataset including all cases.

References

Pregibon D. (1981). Logistic regression diagnostics. The Annals of Statistics, 9, 705-724.

Wei B.C. (1998). Exponential Family Nonlinear Models. Springer, Singapore.

Examples

###### Example 1: The effects of fertilizers on coastal Bermuda grass
data(Grass)
fit1 <- gnm(Yield ~ b0 + b1/(Nitrogen + a1) + b2/(Phosphorus + a2) + b3/(Potassium + a3),
            family=gaussian(inverse), start=c(b0=0.1,b1=13,b2=1,b3=1,a1=45,a2=15,a3=30), data=Grass)

fit1a <- update(fit1, subset=-c(1), start=coef(fit1), maxit=1)
coef(fit1) - coef(fit1a)

dfbetas <- dfbeta(fit1)
round(dfbetas[1,],5)

###### Example 2: Assay of an Insecticide with a Synergist
data(Melanopus)
fit2 <- gnm(Killed/Exposed ~ b0 + b1*log(Insecticide-a1) + b2*Synergist/(a2 + Synergist),
            family=binomial(logit), weights=Exposed, start=c(b0=-3,b1=1.2,a1=1.7,b2=1.7,a2=2),
		   data=Melanopus)

fit2a <- update(fit2, subset=-c(2), start=coef(fit2), maxit=1)
coef(fit2) - coef(fit2a)

dfbetas <- dfbeta(fit2)
round(dfbetas[2,],5)

###### Example 3: Developmental rate of Drosophila melanogaster
data(Drosophila)
fit3 <- gnm(Duration ~ b0 + b1*Temp + b2/(Temp-a), family=Gamma(log),
            start=c(b0=3,b1=-0.25,b2=-210,a=55), weights=Size, data=Drosophila)

fit3a <- update(fit3, subset=-c(3), start=coef(fit3), maxit=1)
coef(fit3) - coef(fit3a)

dfbetas <- dfbeta(fit3)
round(dfbetas[3,],5)

###### Example 4: Radioimmunological Assay of Cortisol
data(Cortisol)
fit4 <- gnm(Y ~ b0 + (b1-b0)/(1 + exp(b2+ b3*lDose))^b4, family=Gamma(identity),
            start=c(b0=130,b1=2800,b2=3,b3=3,b4=0.5), data=Cortisol)

fit4a <- update(fit4, subset=-c(4), start=coef(fit4), maxit=1)
coef(fit4) - coef(fit4a)

dfbetas <- dfbeta(fit4)
round(dfbetas[4,],5)