Description Usage Arguments Details Value Side Effects Acknowledgments Note References See Also Examples
Calculates different types of residuals, influence measures and leverages for a nonlinear heteroscedastic model.
1 
fitted 
a 
hoa 
logical value indicating whether higher order asymptotics should
be used for calculating the regression diagnostics. Default is

infl 
logical value indicating whether influence measures should be
calculated on the basis of a leaveoneout analysis. Default is

trace 
logical value. If 
The regression diagnostics implemented in the nlreg.diag
routine follow two approaches. The first exploits, where possible,
the analogy with linear models, that is, it applies the classical
definitions of residuals, leverages and Cook's distance after having
linearized the nonlinear model through Taylor series expansion
(Carroll and Ruppert, 1988, Section 2.8). The second
approach uses the mean shift outlier model (Cook and Weisberg,
1982, Section 2.2.2), where a dummy variable is included for each
observation at a time, the model refitted and the significance of
the corresponding coefficient assessed.
The leverages are defined in analogy to the linear case (Brazzale, 2000, Appendix A.2.2). Two versions are available. In the first case the subblock of the inverse of the expected information matrix corresponding to the regression coefficients is used in the definition. In the second case, this matrix is replaced by the inverse of M'WM, where M is the n x p matrix whose ith row is the gradient of the mean function evaluated at the ith data point and W is a diagonal matrix whose elements are the inverses of the variance function evaluated at each data point.
If the model is correctly specified, all residuals follow the standard normal distribution. The second kind of leverages described above are used to calculate the approximate studentized residuals, whereas the generalized Pearson residuals use the first kind. The ith generalized Pearson residual can also be obtained as the score statistic for testing the significance of the dummy coefficient in the mean shift outlier model for observation i. Accordingly, the ith deletion and r*type residuals are defined as respectively the likelihood root and modified likelihood root statistics (r and r*) for the same situation (Bellio, 2000, Section 2.6.1).
Different influence measures were implemented in
nlreg.diag
. If infl = TRUE
, the global measure
(Cook and Weisberg, 1982, Section 5.2) and two partial ones
(Bellio, 2000, Section 2.6.2), the first measuring the
influence of each observation on the regression coefficients and the
second on the variance parameters, are returned. They are calculated
through a leaveoneout analysis, where one observation at a time is
deleted and the model refitted. In order to avoid a further model
fit, the constrained maximum likelihood estimates that would be
needed are approximated by means of a Taylor series expansion
centered at the MLEs. If infl = FALSE
, only an
approximation to Cook's distance, obtained from a first order Taylor
series expansion of the partial influence measure for the regression
coefficients, is returned.
A detailed account of regression diagnostics can be found in Davison and Snell (1991) and Davison and Tsai (1992). The details and in particular the definitions of the above residuals and diagnostics are given in Brazzale (2000, Section 6.3.1 and Appendix A.2.2).
Returns an object of class nlreg.diag
with the following
components:
fitted 
the fitted values, that is, the mean function evaluated at each data point. 
resid 
the response (or standardized) residuals from the fit. 
rp 
the generalized Pearson residuals from the fit. 
rs 
the approximate studentized residuals from the fit. 
rj 
the deletion residuals from the fit; only if 
rsj 
the r*type residuals from the fit; only if

h 
the leverages of the observations. 
ha 
the approximate leverages of the observations. 
cook 
an approximation to Cook's distance for the regression coefficients. 
ld 
the global influence of each observation; only for heteroscedastic
errors and if 
ld.rc 
the partial influence of each observation on the estimates of the
regression coefficients; only for heteroscedastic errors and if

ld.vp 
the partial influence of each observation on the estimates of the
variance parameters; only for heteroscedastic errors and if

npar 
the number of regression coefficients. 
If trace = TRUE
, the number of the observation currently
considered in the mean shift outlier model or omitted in the
leaveoneout analysis (see Details section above) is printed;
only if hoa = TRUE
or infl = TRUE
.
This function is based on A. J. Canty's function
glm.diag
contained in library boot
.
The calculation of the deletion and r*type residuals and
of the influence measures can be timeconsuming. In the first case,
the mean shift outlier model has to be refitted as many times as the
total number of observations. In the second case, the original model
is refitted the same amount of times, where one observation at a time
is deleted. Furthermore, the definition of the r*type
residuals
requires differentiation of the mean function of the mean shift
outlier model. These calculations can be avoided by changing the
default setting of the arguments hoa
and infl
to
FALSE
.
To obtain some of the regression diagnostics (typically those based
on higher order statistics), the model is repeatedly refitted for
different values of the mean shift outlier model parameter. Although
rarely, convergence problems may occur as the starting values are
chosen in an automatic way. A try
construct is
used to prevent the nlreg.diag
method from breaking down.
Hence, the values of the diagnostics are not available where a
convergence problem was encountered. A warning is issued whenever
this occurs.
Bellio, R. (2000) Likelihood Asymptotics: Applications in Biostatistics. Ph.D. Thesis, Department of Statistics, University of Padova.
Brazzale, A. R. (2000) Practical SmallSample Parametric Inference. Ph.D. Thesis N. 2230, Department of Mathematics, Swiss Federal Institute of Technology Lausanne.
Carroll, R. J. and Ruppert, D. (1988) Transformation and Weighting in Regression. London: Chapman & Hall.
Cook, R. D. and Weisberg, S. (1982) Residuals and Influence in Regression. New York: Chapman & Hall.
Davison, A. C. and Snell, E. J. (1991) Residuals and diagnostics. In Statistical Theory and Modelling: In Honour of Sir David Cox (eds. D. V. Hinkley, N. Reid, and E. J. Snell), 83–106. London: Chapman & Hall.
Davison, A. C. and Tsai, C.L. (1992) Regression model diagnostics. Int. Stat. Rev., 60, 337–353.
nlreg.diag.plots
,
nlreg.object
1 2 3 4 5 6 7 8 9 10 11 12  library(boot)
data(calcium)
calcium.nl < nlreg( cal ~ b0*(1exp(b1*time)), weights = ~ ( 1+time^g )^2,
data=calcium, start = c(b0 = 4, b1 = 0.1, g = 1),
hoa = TRUE )
##
calcium.diag < nlreg.diag( calcium.nl )
plot( calcium.diag, which = 9 )
##
calcium.diag < nlreg.diag( calcium.nl, hoa = FALSE, infl = FALSE)
plot(calcium.diag, which = 9)
## Not available

Loading required package: statmod
Loading required package: survival
Package "nlreg" 1.22.2 (20190130)
Copyright (C) 20002019 R. Bellio & A. R. Brazzale
This is free software, and you are welcome to redistribute
it and/or modify it under the terms of the GNU General
Public License published by the Free Software Foundation.
Package "nlreg" comes with ABSOLUTELY NO WARRANTY.
type `help(package="nlreg")' for summary information
Attaching package: ‘boot’
The following object is masked from ‘package:survival’:
aml
differentiating mean and variance function  may take a while
differentiating mean function  may take a while
differentiating variance function  may take a whileWarning messages:
1: In par(old.par) : graphical parameter "cin" cannot be set
2: In par(old.par) : graphical parameter "cra" cannot be set
3: In par(old.par) : graphical parameter "csi" cannot be set
4: In par(old.par) : graphical parameter "cxy" cannot be set
5: In par(old.par) : graphical parameter "din" cannot be set
6: In par(old.par) : graphical parameter "page" cannot be set
Warning messages:
1: In par(old.par) : graphical parameter "cin" cannot be set
2: In par(old.par) : graphical parameter "cra" cannot be set
3: In par(old.par) : graphical parameter "csi" cannot be set
4: In par(old.par) : graphical parameter "cxy" cannot be set
5: In par(old.par) : graphical parameter "din" cannot be set
6: In par(old.par) : graphical parameter "page" cannot be set
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