Higher Order Inference for Nonlinear Heteroscedastic Models

Description

Implements likelihood inference based on higher order approximations for nonlinear models with possibly non constant variance.

Details

The DESCRIPTION file:

Index of help topics:

C1                      Six Herbicide Data Sets
Dmean                   Differentiate the Mean Function of a Nonlinear
                        Model
Dvar                    Differentiate the Variance Function of a
                        Nonlinear Model
chlorsulfuron           Chlorsulfuron Data
contour.all.nlreg.profiles
                        Contour Method for 'nlreg' Objects
daphnia                 'Daphnia Magna' Data
expInfo                 Returns the Expected Information Matrix -
                        Generic Function
expInfo.nlreg           Expected Information Matrix for 'nlreg' Objects
helicopter              Helicopter Data
logLik.nlreg            Compute the Log Likelihood for Nonlinear
                        Heteroscedastic Models
metsulfuron             Metsulfuron Methyl Data
mpl                     Maximum Adjusted Profile Likelihood Estimation
                        - Generic Function
mpl.nlreg               Maximum Adjusted Profile Likelihood Estimates
                        for a 'nlreg' Object
mpl.object              Maximum Adjusted Profile Likelihood Object
nlreg                   Fit a Nonlinear Heteroscedastic Model via
                        Maximum Likelihood
nlreg-package           Higher Order Inference for Nonlinear
                        Heteroscedastic Models
nlreg.diag              Nonlinear Heteroscedastic Model Diagnostics
nlreg.object            Nonlinear Heteroscedastic Model Object
obsInfo                 Returns the Observed Information Matrix -
                        Generic Function
obsInfo.nlreg           Observed Information Matrix for 'nlreg' Objects
param                   Extract All Parameters from a Model - Generic
                        Function
plot.nlreg.contours     Use plot() on a 'nlreg.contours' object
plot.nlreg.diag         Diagnostic Plots for Nonlinear Heteroscedastic
                        Models
plot.nlreg.profile      Use plot() on a 'profile.nlreg' and
                        'all.profiles.nlreg' object
profile.nlreg           Profile Method for 'nlreg' Objects
ria                     Radioimmunoassay Data
summary.all.nlreg.profiles
                        Summary Method for Objects of Class
                        'all.nlreg.profiles'
summary.mpl             Summary Method for 'mpl' Objects
summary.nlreg           Summary Method for Nonlinear Heteroscedastic
                        Models
summary.nlreg.profile   Summary Method for Objects of Class
                        'nlreg.profile'
var2cor                 Convert Covariance Matrix to Correlation Matrix
                        - Generic Function

Likelihood inference based on higher order approximations for nonlinear models with possibly non constant variance

Index:

Functions:
=========
Dmean                   Differentiate the Mean Function of a Nonlinear
                        Model
Dvar                    Differentiate the Variance Function of a
                        Nonlinear Model
contour.all.nlreg.profiles
                        Contour Method for 'nlreg' Objects
expInfo                 Returns the Expected Information Matrix --
                        Generic Function
expInfo.nlreg           Expected Information Matrix for 'nlreg'
                        Objects
logLik.nlreg            Compute the Log Likelihood for Nonlinear
                        Heteroscedastic Models
mpl                     Maximum Adjusted Profile Likelihood Estimation
                        -- Generic Function
mpl.nlreg               Maximum Adjusted Profile Likelihood Estimates
                        for a 'nlreg' Object
mpl.object              Maximum Adjusted Profile Likelihood Object
nlreg                   Fit a Nonlinear Heteroscedastic Model via
                        Maximum Likelihood
nlreg.diag              Nonlinear Heteroscedastic Model Diagnostics
nlreg.object            Nonlinear Heteroscedastic Model Object
obsInfo                 Returns the Observed Information Matrix --
                        Generic Function
obsInfo.nlreg           Observed Information Matrix for 'nlreg'
                        Objects
param                   Extract All Parameters from a Model -- Generic
                        Function
plot.nlreg.contours     Use plot() on a 'nlreg.contours' object
plot.nlreg.diag         Diagnostic Plots for Nonlinear Heteroscedastic
                        Models
plot.nlreg.profile      Use plot() on a 'profile.nlreg' and
                        'all.profiles.nlreg' object
profile.nlreg           Profile Method for 'nlreg' Objects
summary.all.nlreg.profiles
                        Summary Method for Objects of Class
                        'all.nlreg.profiles'
summary.mpl             Summary Method for 'mpl' Objects
summary.nlreg           Summary Method for Nonlinear Heteroscedastic
                        Models
summary.nlreg.profile   Summary Method for Objects of Class
                        'nlreg.profile'
var2cor                 Convert Covariance Matrix to Correlation
                        Matrix -- Generic Function


Datasets:
========
C1                      Herbicide Data (Chlorsulfuron)
C2                      Herbicide Data (Chlorsulfuron)
C3                      Herbicide Data (Chlorsulfuron)
C4                      Herbicide Data (Chlorsulfuron)
M2                      Herbicide Data (Metsulfuron Methyl)
M4                      Herbicide Data (Metsulfuron Methyl)
chlorsulfuron           Chlorsulfuron Data
daphnia                 'Daphnia Magna' Data
helicopter              Paper Helicopter Data
metsulfuron             Metsulfuron Methyl Data
ria                     Radioimmunoassay Data

Further information is available in the following vignettes:

Author(s)

Alessandra R. Brazzale [aut, cre] (author of original code for S, author of R port following earlier work by Douglas Bates)

Maintainer: Alessandra R. Brazzale <alessandra.brazzale@unipd.it>

References

Brazzale, A.R. (2005). hoa: An R package bundle for higher order likelihood inference. The R Journal, 5/1, May 2005, 20-27. ISSN 609-3631. URL: https://journal.r-project.org/articles/RN-2005-004/RN-2005-004.pdf

Examples of applications, and generally of the use of likelihood asymptotics, are given in: Brazzale, A.R., Davison, A.C. and Reid, N. (2007). Applied Asymptotics: Case Studies in Small-Sample Statistics. Cambridge University Press, Cambridge.

See Also

cond marg csampling

Six Herbicide Data Sets

Description

The C1–C4, M2 and M4 data frames have 40 to 72 rows and three columns.

Six bioassay on the action of the herbicides chlorsulfuron and metsulfuron methyl on the callus area of colonies of Brassica napus L. The experiments consist of measurements for different dose levels and can be balanced (C4, M2) or unbalanced (C1, C2, C3, M4).

Usage

  data(C1)
  data(C2)
  data(C3)
  data(C4)
  data(M2)
  data(M4)

Format

These data frame contain the following columns:

group

indicator variable for each tested dose;

dose

the tested dose (nmol/l);

area

the callus area (mm^2).

Note

Data sets C3 and chlorsulfuron are the same. Data sets M2 and metsulfuron are the same.

Source

The data were obtained from

Seiden, P., Kappel, D. and Streibig, J. C. (1998) Response of Brassica napus L. tissue culture to metsulfuron methyl and chlorsulfuron. Weed Research, 38, 221–228.

References

Bellio, R., Jensen, J.E. and Seiden, P. (2000). Applications of likelihood asymptotics for nonlinear regression in herbicide bioassays. Biometrics, 56, 1204–1212.

Brazzale, A. R. (2000) Practical Small-Sample Parametric Inference. Ph.D. Thesis N. 2230, Department of Mathematics, Swiss Federal Institute of Technology Lausanne. Section 5.3, Example 8.

See Also

chlorsulfuron, metsulfuron

Examples

data(C3)
attach(C3)
plot(dose, area, xlab = "tested dose (nmol/l)", 
     ylab = "log callus area (mm^2)", log = "y")
detach()

Differentiate the Mean Function of a Nonlinear Model

Description

Calculates the gradient and Hessian of the mean function of a nonlinear heteroscedastic model.

Usage

Dmean(nlregObj, hessian = TRUE)

Arguments

Details

The mean function is differentiated with respect to the regression coefficients as specified in the coef component of the nlreg object. The returned function definition, however, includes all parameters — regression coefficients and variance parameters — as arguments. When evaluated, it implicitly refers to the data to whom the model was fitted and which must be on the search list. The gradient and Hessian are calculated for each data point: the gradient attribute is a n\times p matrix and the hessian attribute is a n\times p\times p array, where n and p are respectively the number of data points and the number of regression coefficients.

Value

a function whose arguments are named according to the parameters of the nonlinear model nlregObj. When evaluated, it returns the value of the mean function along with attributes called gradient and hessian, the latter if requested. These are the gradient and Hessian of the mean function with respect to the regression coefficients.

Note

Dmean and Dvar are the two workhorse functions of the nlreg library. The details are given in Brazzale (2000, Section 6.1.2).

The symbolic differentiation algorithm is based upon the D function. As this algorithm is highly recursive, the hessian = TRUE argument should only be used if the Hessian matrix is needed. Whenever possible, derivatives should be stored so as to be re-used in further calculations. This is, for instance, achieved by the nonlinear heteroscedastic model fitting routine nlreg through the argument hoa = TRUE.

References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language: A Programming Environment for Data Analysis and Graphics. London: Chapman & Hall. Section 9.6.

Brazzale, A. R. (2000) Practical Small-Sample Parametric Inference. Ph.D. Thesis N. 2230, Department of Mathematics, Swiss Federal Institute of Technology Lausanne.

See Also

Dvar, nlreg.object, deriv3, D

Examples

library(boot)
data(calcium)
calcium.nl <- nlreg( cal ~ b0*(1-exp(-b1*time)), 
                     start = c(b0 = 4, b1 = 0.1),  data = calcium )
Dmean( calcium.nl )
##function (b0, b1, logs)
##{
##    .expr3 <- exp(-b1 * time)
##    .expr4 <- 1 - .expr3
##    .expr6 <- .expr3 * time
##    .value <- b0 * .expr4
##    .grad <- array(0, c(length(.value), 2), list(NULL, c("b0",
##        "b1")))
##    .hessian <- array(0, c(length(.value), 2, 2), list(NULL,
##        c("b0", "b1"), c("b0", "b1")))
##    .grad[, "b0"] <- .expr4
##    .hessian[, "b0", "b0"] <- 0
##    .hessian[, "b0", "b1"] <- .hessian[, "b1", "b0"] <- .expr6
##    .grad[, "b1"] <- b0 * .expr6
##    .hessian[, "b1", "b1"] <- -(b0 * (.expr6 * time))
##    attr(.value, "gradient") <- .grad
##    attr(.value, "hessian") <- .hessian
##    .value
##}
##
param( calcium.nl )
##        b0         b1       logs
## 4.3093653  0.2084780 -1.2856765
##
attach( calcium )
calcium.md <- Dmean( calcium.nl )
attr( calcium.md( 4.31, 0.208, -1.29 ), "gradient" )
##              b0       b1
## [1,] 0.08935305 1.766200
## [2,] 0.08935305 1.766200
## [3,] 0.08935305 1.766200
## [4,] 0.23692580 4.275505
## \dots
detach()

Differentiate the Variance Function of a Nonlinear Model

Description

Calculates the gradient and Hessian of the variance function of a nonlinear heteroscedastic model.

Usage

Dvar(nlregObj, hessian = TRUE)

Arguments

Details

The variance function is differentiated with respect to the variance parameters specified in the varPar component of the nlregObj object and, if the variance function depends on them, with respect to the regression coefficients specified in the coef component. The returned function definition includes all parameters. When evaluated, it implicitly refers to the data to whom the nlreg object was fitted and which must be on the search list. The gradient and Hessian are calculated for each data point: the gradient attribute is a n\times p matrix, and the hessian attribute is a n\times p\times p array, where n and p are respectively the number of data points and the number of regression coefficients.

Value

a function whose arguments are named according to the parameters of the nonlinear model nlregObj. When evaluated, it returns the value of the variance function along with attributes called gradient and hessian, the latter if requested. These are the gradient and Hessian of the variance function with respect to the model parameters.

Note

Dmean and Dvar are the two workhorse functions of the nlreg library. The details are given in Brazzale (2000, Section 6.1.2).

The symbolic differentiation algorithm is based upon the D function. As this algorithm is highly recursive, the hessian = TRUE argument should only be used if the Hessian matrix is needed. Whenever possible, derivatives should be stored so as to be re-used in further calculations. This is, for instance, achieved for the nonlinear heteroscedastic model fitting routine nlreg through the argument hoa = TRUE.

References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language: A Programming Environment for Data Analysis and Graphics. London: Chapman & Hall. Section 9.6.

Brazzale, A. R. (2000) Practical Small-Sample Parametric Inference. Ph.D. Thesis N. 2230, Department of Mathematics, Swiss Federal Institute of Technology Lausanne.

See Also

Dmean, nlreg.object, deriv3, D

Examples

library(boot)
data(calcium)
calcium.nl <- nlreg( cal ~ b0*(1-exp(-b1*time)), 
                     start = c(b0 = 4, b1 = 0.1), data = calcium )
Dvar( calcium.nl )
##function (b0, b1, logs)
##{
##    .expr1 <- exp(logs)
##    .value <- .expr1
##    .grad <- array(0, c(length(.value), 1), list(NULL, c("logs")))
##    .hessian <- array(0, c(length(.value), 1, 1), list(NULL,
##        c("logs"), c("logs")))
##    .grad[, "logs"] <- .expr1
##    .hessian[, "logs", "logs"] <- .expr1
##    attr(.value, "gradient") <- .grad
##    attr(.value, "hessian") <- .hessian
##    .value
##}
##
attach( calcium )
calcium.vd <- Dvar( calcium.nl )
param( calcium.nl )
##        b0         b1       logs
## 4.3093653  0.2084780 -1.2856765
##
attr( calcium.vd( 4.31, 0.208, -1.29 ), "gradient" )
##          logs
##[1,] 0.2752708
##
calcium.nl <- update( calcium.nl, weights = ~ ( 1+time^g )^2, 
                      start = c(b0 = 4, b1 = 0.1, g = 1))
Dvar( calcium.nl )
##function (b0, b1, g, logs) 
##{
##    .expr1 <- time^g
##    .expr2 <- 1 + .expr1
##    .expr4 <- exp(logs)
##    .expr5 <- .expr2^2 * .expr4
##    .expr6 <- log(time)
##    .expr7 <- .expr1 * .expr6
##    .expr10 <- 2 * (.expr7 * .expr2) * .expr4
##    .value <- .expr5
##    .grad <- array(0, c(length(.value), 2), list(NULL, c("g",
##        "logs")))
##    .hessian <- array(0, c(length(.value), 2, 2), list(NULL,
##        c("g", "logs"), c("g", "logs")))
##    .grad[, "g"] <- .expr10
##    .hessian[, "g", "g"] <- 2 * (.expr7 * .expr6 * .expr2 + .expr7 *
##        .expr7) * .expr4
##    .hessian[, "g", "logs"] <- .hessian[, "logs", "g"] <- .expr10
##    .grad[, "logs"] <- .expr5
##    .hessian[, "logs", "logs"] <- .expr5
##    attr(.value, "gradient") <- .grad
##    attr(.value, "hessian") <- .hessian
##    .value
##}
##
calcium.vd <- Dvar( calcium.nl )
param( calcium.nl )
##        b0         b1          g       logs 
## 4.3160408  0.2075937  0.3300134 -3.3447585
##
attr( calcium.vd(4.32, 0.208, 0.600, -2.66 ), "gradient" )
##                g      logs
## [1,] -0.11203422 0.1834220
## [2,] -0.11203422 0.1834220
## [3,] -0.11203422 0.1834220
## [4,]  0.09324687 0.3295266
## \dots
##
detach()

Support for function ‘profile.nlreg’ — Generic Function

Description

This is support for the function profile.nlreg. It is not intended to be called directly by users.

Usage

allProfiles(fitted, ...)

Arguments

Details

The allProfiles.nlreg function is called internally by the profile.nlreg routine. It is not intended to be called directly by users.

Value

a list of elements of class all.nlreg.profiles for profiling all parameters of a nonlinear heteroscedastic model.

See Also

profile.nlreg, nlreg.profile.objects, nlreg.object

Support for function ‘profile.nlreg’

Description

This is support for the function profile.nlreg. It is not intended to be called directly by users.

Usage

## S3 method for class 'nlreg'
allProfiles(fitted, hoa = TRUE, precision = 6, signif = 30, 
                   n = 50, omit = 0.5, trace = FALSE, call, ...)

Arguments

Details

The allProfiles.nlreg function is called internally by the profile.nlreg routine. It is not intended to be called directly by users.

Value

a list of elements of class all.nlreg.profiles for profiling all parameters of a nonlinear heteroscedastic model.

See Also

profile.nlreg, nlreg.profile.objects, nlreg.object

Chlorsulfuron Data

Description

The chlorsulfuron data frame has 51 rows and 3 columns.

Bioassay on the action of the herbicide chlorsulfuron on the callus area of colonies of Brassica napus L. The experiment consists of 51 measurements for 10 different dose levels. The design is unbalanced: the number of replicates per dose varies from a minimum of 5 to a maximum of 8.

Usage

data(chlorsulfuron)

Format

This data frame contains the following columns:

group

indicator variable for each tested dose;

dose

the tested dose (nmol/l);

area

the callus area (mm^2).

Source

The data were obtained from

Seiden, P., Kappel, D. and Streibig, J. C. (1998) Response of Brassica napus L. tissue culture to metsulfuron methyl and chlorsulfuron. Weed Research, 38, 221–228. Dataset C3.

References

Bellio, R., Jensen, J.E. and Seiden, P. (2000). Applications of likelihood asymptotics for nonlinear regression in herbicide bioassays. Biometrics, 56, 1204–1212.

Brazzale, A. R. (2000) Practical Small-Sample Parametric Inference. Ph.D. Thesis N. 2230, Department of Mathematics, Swiss Federal Institute of Technology Lausanne. Section 5.3, Example 8.

Examples

data(chlorsulfuron)
attach(chlorsulfuron)
plot(dose, area, xlab = "tested dose (nmol/l)", 
     ylab = "log callus area (mm^2)", log = "y")
detach()

Use coef() on a ‘nlreg’ object

Description

This is a method for the function coef() for objects inheriting from class nlreg. See coef for the general behavior of this function and for the interpretation of object.

Usage

## S3 method for class 'nlreg'
coef(object, ...)

See Also

nlreg.object, coef

Contour Method for ‘nlreg’ Objects

Description

Draws the approximate bivariate contour plots for two or all parameters of a nonlinear heteroscedastic model and, on request, returns the list of elements used.

Usage

## S3 method for class 'all.nlreg.profiles'
contour(x, offset1, offset2, alpha = c(0.1, 0.05),
        stats = c("sk", "fr"), ret = FALSE, plotit = TRUE, 
        drawlabels = FALSE, lwd1 = 1, lwd2 = 1, lty1 = "solid", 
        lty2 = "solid", cl1 = "blue", cl2 = "red", col = "black", 
        pch1 = 1, pch2 = 16, cex = 0.5, ...)

Arguments

Details

The function contour.all.nlreg.profiles calculates all elements needed to draw the profile and approximate bivariate contour plots for respectively two parameters of interest and all parameters in the model, depending on whether the offset1 and offset2 arguments are used.

Contour plots represent the bivariate extension of profile plots. Given two parameters of interst, they plot the corresponding joint confidence regions of levels 1-\alpha obtained from the likelihood ratio statistic and the Wald statistic (Bates and Watts, 1988, Section 6.1.2). The closer the two curves are, the more the likelihood surface is quadratic. Usually profile traces are added, that is, the curves showing the constrained maximum likelihood estimates of one parameter as a function of the other, as they provide useful information on how the estimates affect each other. If the asymptotic correlation is zero, the angle between the traces is close to \pi/2. The calculation of exact contour plots is computationally very intesive, as the model has to be refitted several times to obtain the constrained estimates. Bates and Watts (1988, Appendix A.6) present an approximate solution, which only requires the computation of the parameter profiles and which gives rise to the so-called profile pair sketches.

The function contour.all.nlreg.profiles extends the classical profile plots and profile pair sketches by including the higher order solutions r^* (Barndorff-Nielsen, 1991) and w^* (Skovgaard, 2001). The idea is to provide insight into the behaviour of first order methods such as detecting possible bias of the estimates or the influence of the model curvature. More precisely, the sample space derivatives in Barndorff-Nielsens' (1991) r^* statistic are replaced by respectively the approximations proposed in Skovgaard (1996) and Fraser, Reid and Wu (1999) depending on the value of the stats argument. The r^* statistic is used to calculate an approximation to Skovgaard's (2001) w^* statistic adopting the method by Bates and Watts (1988, Appendix A.6). This method can break down, if the two parameter estimates are strongly correlated. The approximate contours of w^* are then missing in the corresponding panels; four bullets indicate where they intersect the profile traces.

All necessary quantities are retrieved from the all.nlreg.profiles object passed through the x argument. The offset1 and offset2 arguments can be used to specifiy two parameters of interest, in which case only the profile pair sketches for these two parameters are returned, one on the original scale and one on the normal scale. On the normal scale, the units do not express the parameter values themselves, but the associated likelihood root statistics. (See Bates and Watts, 1988, Section 6.1.2, for explanation.) If the offset1 and offset2 arguments are missing, profile plots and approximate contour plots are drawn for all model parameters. The plots are organized in form of a matrix. The main diagonal contains the profile plots. The approximate bivariate contour plots in the lower triangle are plotted on the original scale, whereas the ones in the upper triangle are on the r scale.

The theory and statistics used are summarized in Brazzale (2000, Chapters 2 and 3). More details of the implementation are given in Brazzale (2000, Section 6.3.2).

Value

If ret = TRUE, a list of class nlreg.contours is returned which contains the elements needed to draw the profiles and approximate bivariate contours for two or all parameters in a nonlinear heteroscedastic model. Otherwise, no value is returned.

Side Effects

If plotit = TRUE, a plot is produced on the current graphics device.

Note

contour.all.nlreg.profiles is a method for the generic function contour for class all.nlreg.profiles. It can be invoked by calling contour for an object of the appropriate class, or directly by calling contour.all.nlreg.profiles.

References

Barndorff-Nielsen, O. E. (1991) Modified signed log likelihood ratio. Biometrika, 78, 557–564.

Bates, D. M. and Watts, D. G. (1988) Nonlinear Regression Analysis and Its Applications. New York: Wiley.

Brazzale, A. R. (2000) Practical Small-Sample Parametric Inference. Ph.D. Thesis N. 2230, Department of Mathematics, Swiss Federal Institute of Technology Lausanne.

Fraser, D.A.S., Reid, N. and Wu, J. (1999). A simple general formula for tail probabilities for frequentist and Bayesian inference. Biometrika, 86, 249–264.

Skovgaard, I. M (1996) An explicit large-deviation approximation to one-parameter tests. Bernoulli, 2, 145–165.

Skovgaard, I. M. (2001) Likelihood asymptotics. Scandinavian Journal of Statistics, 28, 3–32.

See Also

nlreg.profile.objects, plot.nlreg.contours, contour

Examples

## Not run: 
data(metsulfuron)
metsulfuron.nl <- 
    nlreg( formula = log(area) ~ log( b1+(b2-b1) / (1+(dose/b4)^b3) ), 
           weights = ~ ( 1+dose^exp(g) )^2, data = metsulfuron, 
           start = c(b1 = 138, b2 = 2470, b3 = 2, b4 = 0.07, g = log(0.3)),
           hoa = TRUE )
##
metsulfuron.prof <- profile( metsulfuron.nl, trace = TRUE )
par( mai = rep(0.2, 4) )
contour( metsulfuron.prof )
## End(Not run)

‘Daphnia Magna’ Data

Description

The daphnia data frame has 136 rows and 2 columns.

Ecotoxicity study to assess the impact of the herbicide dinoseb on the survival of Daphnia magna Strauss, 1820, a micro-crustacean widely used as test organism in aquatic ecotoxicological assays. The design of the experiment includes 35 irregularly spaced concentrations ranging from 0.006 to 11.3 mg/l and a control group. The upper endpoint of 11.3 mg/l is the highest concentration at which the test substance is soluble in the test medium. The number of replicates per concentration varies from 1 to 11 experimental units. The survival time is measured in days.

Usage

data(daphnia)

Format

This data frame contains the following columns:

conc

the tested concentration (mg/l);

time

the survival time in days.

Source

The data were obtained from

Ch\'evre, N. (2000) Etude et mod\'elisation des effets \'ecotoxiques d'un micropolluant organique sur Daphnia magna et Pseudokirchneriella subcapitata (in French). Ph.D. Thesis N. 2117, Department of Rural Engineering, Swiss Federal Institute of Technology Lausanne.

References

Brazzale, A.R. (2000) Practical Small-Sample Parametric Inference. Ph.D. Thesis N. 2230, Department of Mathematics, Swiss Federal Institute of Technology Lausanne. Section 5.3, Example 5.

Ch\'evre, N., Becker-van Slooten, K., Tarradellas, J., Brazzale, A. R., Behra, R. and Guettinger, H. (2001) Effects of dinoseb on the entire life-cycle of Daphnia magna. Part II: Modelling of survival and proposal of an alternative to No-Observed-Effect-Concentration (NOEC). Environmental Toxicology and Chemistry, 21, 828–833.

Examples

data(daphnia)
attach(daphnia)
plot(conc, time, xlab = "test concentration (mg/l)", 
     ylab = "survival time (d)", log = "y")
detach()

Returns the Expected Information Matrix — Generic Function

Description

Returns the expected information matrix from a fitted model object.

Usage

expInfo(object, ...)

Arguments

Details

This function is generic (see methods); method functions can be written to handle specific classes of data. Classes which already have methods for this function include: nlreg.

Value

the expected information matrix for a fitted regression model.

See Also

expInfo.nlreg, nlreg.object, obsInfo

Examples

data(metsulfuron)
metsulfuron.nl <- 
    nlreg( log(area) ~ log( b1+(b2-b1) / (1+(dose/b4)^b3) ),
           weights = ~ ( 1+dose^exp(g) )^2, data = metsulfuron, 
           start = c(b1 = 138, b2 = 2470, b3 = 2, b4 = 0.07, g = log(0.3)), 
           hoa = TRUE)
expInfo( metsulfuron.nl )

Expected Information Matrix for ‘nlreg’ Objects

Description

Returns the expected information matrix for a fitted nlreg model.

Usage

## S3 method for class 'nlreg'
expInfo(object, par, mu, v, m1 = NULL, v1 = NULL, ...)

Arguments

Details

This function is a method for the generic function expInfo for objects inheriting from class nlreg.

Value

the expected information matrix of the fitted nonlinear model passed through the object argument.

Note

This function is mostly intended for internal use. It is called by functions such as nlreg.diag, summary.nlreg and profile.nlreg. To extract the expected information matrix from a fitted nlreg object, the generic method expInfo should be used.

See Also

expInfo, nlreg.object, obsInfo

Use fitted() on a ‘nlreg’ object

Description

This is a method for the function fitted() for objects inheriting from class nlreg. See fitted for the general behavior of this function and for the interpretation of object.

Usage

## S3 method for class 'nlreg'
fitted(object, ...)

See Also

nlreg.object, fitted

Helicopter Data

Description

The helicopter data frame has 9 rows and 6 columns.

Experimental design for studying the influences of the factors wing length and wing width on a paper helicopter's flight time. The goal is to find the factor setting that maximizes flight time when the paper helicopter is dropped from a fixed height of 15.5 feet

Usage

data(helicopter)

Format

A data frame with 9 observations on the following 6 variables:

L

wing length in inches;

W

wing width in inches;

B

base length (always set to 3in);

H

base height (always set to 2in);

Order

run order;

Time

flight time in seconds.

Source

The data were obtained from

Annis, D. H. (2006) Rethinking the paper helicopter: Combining statistical and engineering knowledge. The American Statistician, 59, 320–326.

References

Box, G. E. P. (1992) Teaching engineers experimental design with a paper helicopter. Quality Engineering, 4, 453–459.

Examples

data(helicopter)
##
## fit model (5) of Annis (2005)
## -----------------------------
heli <- helicopter
##
heli$LW <- heli$L * heli$W
heli$S <- heli$B * heli$H + ( 2 * heli$L + 1 ) * heli$W
heli$logTime <- log( heli$Time )
heli$Y <- heli$logTime + log( heli$S ) / 2
#
heli.nlreg <- nlreg( Y ~ b0 + b1 * log( b2^2 / LW + LW ), data = heli, 
                    start = c( b0 = 6, b1 = -1, b2 = 20 ) )

Compute the Log Likelihood for Nonlinear Heteroscedastic Models

Description

Computes the log likelihood for a nonlinear model with possibly non constant variance.

Usage

## S3 method for class 'nlreg'
logLik(object, ...)

Arguments

Details

This is a method for the function logLik() for objects inheriting from class nlreg.

Value

Returns an object class logLik which is a number with attributes nobs, npar and df giving respectively the number of observations, the number of parameters (regression coefficients plus variance parameters) and the degrees of freedom in the model.

Note

The default print method for logLik objects is used.

See Also

rsm.object, logLik

Examples

library(boot)
data(calcium)
calcium.nl <- nlreg( cal ~ b0*(1-exp(-b1*time)), 
                     start = c(b0 = 4, b1 = 0.1), data = calcium )
logLik( calcium.nl )
##
data(metsulfuron)
metsulfuron.nl <- 
    nlreg( log(area) ~ log( b1+(b2-b1) / (1+(dose/b4)^b3) ), 
           weights = ~ ( 1+dose^exp(g) )^2, data = metsulfuron, 
           start = c(b1 = 138, b2 = 2470, b3 = 2, b4 = 0.07, g = log(0.3)),
           hoa = TRUE )
logLik( metsulfuron.nl )

Metsulfuron Methyl Data

Description

The metsulfuron data frame has 40 rows and 3 columns.

Bioassay on the action of metsulfuron methyl, a sulfunylurea herbicide, on a tissue culture of Brassica napus L. The experiment consists of 8 doses and 5 replications at each level.

Usage

data(metsulfuron)

Format

This data frame contains the following columns:

group

indicator variable for each tested dose;

dose

the tested dose (nmol/l);

area

the callus area (mm^2).

Source

The data were obtained from

Seiden, P., Kappel, D. and Streibig, J. C. (1998) Response of Brassica napus L. tissue culture to metsulfuron methyl and chlorsulfuron. Weed Research, 38, 221–228. Dataset M2.

References

Bellio, R., Jensen, J.E. and Seiden, P. (2000). Applications of likelihood asymptotics for nonlinear regression in herbicide bioassays. Biometrics, 56, 1204–1212.

Brazzale, A. R. (2000) Practical Small-Sample Parametric Inference. Ph.D. Thesis N. 2230, Department of Mathematics, Swiss Federal Institute of Technology Lausanne. Section 5.3, Example 7.

Examples

data(metsulfuron)
attach(metsulfuron)
plot(dose, area, xlab = "tested dose (nmol/l)", 
     ylab = "log callus area (mm^2)", log = "y")
detach()

Maximum Adjusted Profile Likelihood Estimation — Generic Function

Description

Calculates the maximum adjusted profile likelihood estimates.

Usage

mpl(fitted, ...)

Arguments

Details

This function is generic (see methods); method functions can be written to handle specific classes of data. Classes which already have methods for this function include: nlreg.

Value

the maximum adjusted profile likelihood estimates for all parameters of a regression model or for a subset of them.

See Also

mpl.nlreg, nlreg.object, methods

Examples

data(metsulfuron)
metsulfuron.nl <- 
    nlreg( formula = log(area) ~ log( b1+(b2-b1) / (1+(dose/b4)^b3) ), 
           weights = ~ ( 1+dose^exp(g) )^2, data = metsulfuron, hoa = TRUE, 
           start = c(b1 = 138, b2 = 2470, b3 = 2, b4 = 0.07, g = log(0.3)) )
mpl( metsulfuron.nl, trace = TRUE )
##
options( object.size = 10000000 )
data(chlorsulfuron)
chlorsulfuron.nl <- 
    nlreg( log(area) ~ log( b1+(b2-b1) / (1+(dose/b4)^b3) ), 
        weights = ~ ( 1+k*dose^g*(b2-b1)^2/(1+(dose/b4)^b3)^4*b3^2*dose^(2*b3-2)/
                    b4^(2*b3)/(b1+(b2-b1)/(1+(dose/b4)^b3))^2 ),
        start = c(b1 = 2.2, b2 = 1700, b3 = 2.8, b4 = 0.28, g = 2.7, k = 1), 
        data = chlorsulfuron, hoa = TRUE, trace = TRUE,  
        control = list(x.tol = 10^-12, rel.tol = 10^-12, step.min = 10^-12) )
mpl( chlorsulfuron.nl, trace = TRUE )

Maximum Adjusted Profile Likelihood Estimates for a ‘nlreg’ Object

Description

Calculates the maximum adjusted profile likelihood estimates of the variance parameters for a nonlinear heteroscedastic model.

Usage

## S3 method for class 'nlreg'
mpl(fitted, offset = NULL, stats = c("sk", "fr"), 
    control = list(x.tol = 1e-6, rel.tol = 1e-6, step.min = 1/2048,
    maxit = 100), trace = FALSE, ... )

Arguments

Details

The mpl.nlreg routine returns nearly unbiased estimates of the variance parameters of a nonlinear heteroscedastic regression model by maximizing the corresponding adjusted profile likelihood (Barndorff-Nielsen, 1983). More precisely, it implements two approximations derived from the theories developed respectively by Skovgaard (1996) and Fraser, Reid and Wu (1999). The core algorithm alternates minimization of minus the adjusted profile log likelihood with respect to the variance parameters, and minimization of minus the profile log likelihood with respect to the regression coefficients. The first step is omitted if the offset argument is used in which case mpl.nlreg returns the constrained maximum likelihood estimates of the regression coefficients. The quasi-Newton optimizer optim is used in both steps. Starting values are retrieved from the nlreg object passed through the fitted argument.

The algorithm iterates until convergence or until the maximum number of iterations is reached. The stopping rule considers the relative difference between successive estimates of the variance parameters and the relative increment of the adjusted profile log likelihood. These are governed by the parameters x.tol and rel.tol/step.min, respectively. If the offset argument is used, the relative difference between successive estimates of the regression coefficients and the relative increment of the profile log likelihood are considered instead. If convergence has been reached, the results are saved in an object of class mpl. The output can be examined by print and summary. Components can be extracted using coef and param.

The theory is outlined in Brazzale (2000, Sections 3.1 and 3.2.3). Details of the implementation are given in Brazzale (2000, Section 6.3.1).

Value

an object of class mpl which inherits from nlreg. See mpl.object for details.

Side Effects

If trace = TRUE and offset = NULL, the iteration number and the corresponding adjusted profile log likelihood are printed.

Note

The argument control which controls the convergence criteria plays an important role. Fine-tuning of this argument helps surrounding a well-known problem in nonlinear regression, that is, convergence failure in cases where the likelihood and/or the adjusted profile likelihood are very flat.

References

Barndorff-Nielsen, O. E. (1983) On a formula for the distribution of the maximum likelihood estimator. Biometrika, 70, 343–365.

Brazzale, A. R. (2000) Practical Small-Sample Parametric Inference. Ph.D. Thesis N. 2230, Department of Mathematics, Swiss Federal Institute of Technology Lausanne.

Fraser, D.A.S., Reid, N. and Wu, J. (1999). A simple general formula for tail probabilities for frequentist and Bayesian inference. Biometrika, 86, 249–264.

Skovgaard, I. (1996) An explicit large-deviation approximation to one-parameter tests. Bernoulli, 2, 145–165.

See Also

mpl, mpl.object, nlreg.object, optim

Examples

data(metsulfuron)
metsulfuron.nl <- 
    nlreg( formula = log(area) ~ log( b1+(b2-b1) / (1+(dose/b4)^b3) ), 
           weights = ~ ( 1+dose^exp(g) )^2, data = metsulfuron, hoa = TRUE, 
          start = c(b1 = 138, b2 = 2470, b3 = 2, b4 = 0.07, g = log(0.3)) )
##
## MMPLE of the variance parameters
##
metsulfuron.mpl <- mpl( metsulfuron.nl, trace = TRUE )
summary( metsulfuron.mpl, corr = FALSE )
##
## constrained MLEs of the regression coefficients  
##
metsulfuron.mpl <- mpl( metsulfuron.nl, offset = metsulfuron.nl$varPar, 
                        trace = TRUE )
summary( metsulfuron.mpl, corr = FALSE )

Maximum Adjusted Profile Likelihood Object

Description

Class of objects returned when calculating the maximum adjusted profile likelihood estimates of the variance parameters of a nonlinear heteroscedastic model.

Arguments

The following components must be included in a mpl object:

Generation

This class of objects is returned by the mpl.nlreg function. Class mpl inherits from class nlreg.

Methods

Objects of this class have methods for the functions print, summary, coef and param.

Note

The coefficients and variance parameters should be extracted by the generic functions of the same name, rather than by the $ operator.

The data and ws components are not intended to be directly used by users, but rather contain information used by functions such as summary.

See Also

mpl.nlreg, mpl, nlreg.object

Fit a Nonlinear Heteroscedastic Model via Maximum Likelihood

Description

Returns an object of class nlreg which represents a nonlinear heteroscedastic model fit of the data obtained by maximizing the corresponding likelihood function.

Usage

nlreg(formula, weights = NULL, data = sys.frame(sys.parent()), start, 
      offset = NULL, subset = NULL, 
      control = list(x.tol = 1e-06, rel.tol = 1e-06, 
                     step.min = 1/2048, maxit = 100), trace = FALSE, 
      hoa = FALSE)

Arguments

Details

A nonlinear heteroscedastic model representing the relationship between two scalar quantities is fitted. The response is specified on the left-hand side of the formula argument. The predictor appears in the right-hand side of the formula and, if specified, weights arguments. Only one predictor variable can be included. Missing values in the data are not allowed.

The fitting criterion is maximum likelihood. The core algorithm implemented in nlreg alternates minimization of minus the log likelihood with respect to the regression coefficients and the variance parameters. The quasi-Newton optimizer optim is used in both steps. The constant term \sigma^2 in Var(error) = s^2 V(predictor) is included by default. In order to work with a real value, \sigma^2 is estimated on the logarithmic scale, that is, the model is reparametrized into \log(\sigma^2) which gives rise to the parameter name logs. If the errors are homoscedastic, the second step is omitted and the algorithm switches automatically to nls. If the weights argument is omitted, homoscedasticity of the errors is assumed.

Starting values for all parameters have to be passed through the start argument except for \sigma^2 for which the maximum likelihood estimate is available in closed form. Starting values should be chosen carefully in order to avoid convergence to a local maximum.

The algorithm iterates until convergence or until the maximum number of iterations defined by maxit is reached. The stopping rule considers the relative difference between successive estimates and the relative increment of the log likelihood. These are governed by the parameters x.tol and rel.tol/step.min, respectively.

If convergence has been reached, the results are saved in an object of class nlreg. The output can be examined by print and summary. Components can be extracted using coef, param, fitted and residuals. The model fit can be updated using update. Profile plots and profile pair sketches are provided by profile, and contour. Diagnostic plots are obtained from nlreg.diag.plots or simply plot.

The details are given in Brazzale (2000, Section 6.3.1).

Value

An object of class nlreg is returned which inherits from nls. See nlreg.object for details.

Side Effects

If trace = TRUE, the iteration number and the corresponding log likelihood are printed.

Note

The arguments hoa and control play an important role. The first forces the algorithm to save the derivatives of the mean and variance functions in the fitted model object. This is imperative if one wants to save execution time, especially for complex models. Fine-tuning of the control argument which controls the convergence criteria helps surrounding a well-known problem in nonlinear regression, that is, convergence failure in cases where the likelihood is very flat.

If the errors are homoscedastic, the nlreg routine switches automatically to nls which, although rarely, dumps because of convergence problems. To avoid this, either reparametrize the model (see Bates and Watts, 1988) or choose starting values that are more realistic. This advice also holds in case of convergence problems for models with non constant variance function. Use the trace = TRUE argument to gain insight into what goes on at the different iteration steps.

The weights argument has a different meaning than in other model fitting routines such as lm and glm. It defines the variance function of the nonlinear model and not a vector of observation weights that are multiplied into the squared residuals.

References

Bates, D. M. and Watts, D. G. (1988) Nonlinear Regression Analysis and Its Applications. New York: Wiley.

Brazzale, A. R. (2000) Practical Small-Sample Parametric Inference. Ph.D. Thesis N. 2230, Department of Mathematics, Swiss Federal Institute of Technology Lausanne.

Seber, G. A. F. and Wild, C. J. (1989) Nonlinear Regression. New York: Wiley.

See Also

nlreg.object, nls

Examples

library(boot)
data(calcium)
##
## Homoscedastic model fit
calcium.nl <- nlreg( cal ~ b0*(1-exp(-b1*time)), start = c(b0 = 4, b1 = 0.1),
                     data = calcium )
##
## Heteroscedastic model fit 
calcium.nl <- nlreg( cal ~ b0*(1-exp(-b1*time)), weights = ~ ( 1+time^g )^2,
                     start = c(b0 = 4, b1 = 0.1, g = 1), data = calcium, 
                     hoa = TRUE)
## or
calcium.nl <- update(calcium.nl, weights = ~ (1+time^g)^2, 
                     start = c(b0 = 4, b1 = 0.1, g = 1), hoa = TRUE )
##
##
## Power-of-X (POX) variance function
##
data(metsulfuron)
metsulfuron.nl <- 
    nlreg( log(area) ~ log( b1+(b2-b1) / (1+(dose/b4)^b3) ), 
           weights = ~ ( 1+dose^exp(g) )^2, data = metsulfuron, 
           start = c(b1 = 138, b2 = 2470, b3 = 2, b4 = 0.07, g = log(0.3)),
           hoa = TRUE )
##
##
## Power-of-mean (POM) variance function
##
data(ria)
ria.nl <- nlreg( count ~ b1+(b2-b1) / (1+(conc/b4)^b3), 
                 weights = ~ ( b1+(b2-b1) / (1+(conc/b4)^b3) )^g, data = ria, 
                 start = c(b1 = 1.6, b2 = 20, b3 = 2, b4 = 300, g = 2),
                 hoa = TRUE, trace = TRUE )
##
##
## Error-in-variables (EIV) variance function
##
data(chlorsulfuron)
options( object.size = 10000000 )
chlorsulfuron.nl <- 
    nlreg( log(area) ~ log( b1+(b2-b1) / (1+(dose/b4)^b3) ), 
        weights = ~ ( 1+k*dose^g*(b2-b1)^2/(1+(dose/b4)^b3)^4*b3^2*dose^(2*b3-2)/
                    b4^(2*b3)/(b1+(b2-b1)/(1+(dose/b4)^b3))^2 ),
        start = c(b1 = 2.2, b2 = 1700, b3 = 2.8, b4 = 0.28, g = 2.7, k = 1), 
        data = chlorsulfuron, hoa = TRUE, trace = TRUE,  
        control = list(x.tol = 10^-12, rel.tol = 10^-12, step.min = 10^-12) )

Contour Object for Nonlinear Heteroscedastic Models

Description

Class of objects returned when creating the contour plots of a nonlinear heteroscedatic model.

Generation

This class of objects is returned by the contour.all.nlreg.profiles function. It contains all elements needed to trace the univariate and bivariate profile plots for a fitted nonlinear heteroscedastic model.

Methods

Objects of this class must be plotted with the corresponding plot methods. No print or summary method is available.

Note

The components of this object are not intended to be accessed directly by users. They contain information used by the corresponding plot method.

See Also

contour.all.nlreg.profiles, plot.nlreg.contours, nlreg.object

Nonlinear Heteroscedastic Model Diagnostics

Description

Calculates different types of residuals, influence measures and leverages for a nonlinear heteroscedastic model.

Usage

  nlreg.diag(fitted, hoa = TRUE, infl = TRUE, trace = FALSE)

Arguments

Details

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 sub-block 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\times 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 leave-one-out 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).

Value

Returns an object of class nlreg.diag with the following components:

Side Effects

If trace = TRUE, the number of the observation currently considered in the mean shift outlier model or omitted in the leave-one-out analysis (see Details section above) is printed; only if hoa = TRUE or infl = TRUE.

Acknowledgments

This function is based on A. J. Canty's function glm.diag contained in library boot.

Note

The calculation of the deletion and r^*-type residuals and of the influence measures can be time-consuming. 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.

References

Bellio, R. (2000) Likelihood Asymptotics: Applications in Biostatistics. Ph.D. Thesis, Department of Statistics, University of Padova.

Brazzale, A. R. (2000) Practical Small-Sample 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.

See Also

nlreg.diag.plots, nlreg.object

Examples

library(boot)
data(calcium)
calcium.nl <- nlreg( cal ~ b0*(1-exp(-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

Nonlinear Heteroscedastic Model Object

Description

Class of objects returned when fitting a nonlinear heteroscedastic model.

Arguments

The following components must be included in a nlreg object:

Generation

This class of objects is returned by the nlreg function to represent a fitted nonlinear heteroscedastic model. Class nlreg inherits from class nls, which represents a homoscedastic nonlinear model fit.

Methods

Objects of this class have methods for the functions print, summary, fitted among others.

Note

The residuals, fitted values and coefficients should be extracted by the generic functions of the same name, rather than by the $ operator.

The data and ws components are not intended to be directly accessed by users, but rather contain information invoked by functions such as profile and nlreg.diag.

See Also

nlreg, nls

Profile Objects for Nonlinear Heteroscedastic Models

Description

Class of objects returned when profiling a nonlinear heteroscedatic model.

Generation

These classes of objects are returned by the profile.nlreg function. They contain all elements needed to trace the profile plots and profile pair sketches for a fitted nonlinear heteroscedastic model.

Methods

Objects of these classes must be inspected or plotted with the corresponding summary or plot methods. No print method is available.

Note

The components of these objects are not intended to be accessed directly by users. They contain information used by the corresponding summary and plot methods.

See Also

profile.nlreg, summary.nlreg.profile, summary.all.nlreg.profiles, plot.nlreg.profile, plot.all.nlreg.profiles, contour.all.nlreg.profiles, nlreg.object

Returns the Observed Information Matrix — Generic Function

Description

Returns the observed information matrix from a fitted model object.

Usage

obsInfo(object, ...)

Arguments

Details

This function is generic (see methods); method functions can be written to handle specific classes of data. Classes which already have methods for this function include: nlreg.

Value

the observed information matrix for a fitted regression model.

See Also

obsInfo.nlreg, nlreg.object, expInfo

Examples

data(metsulfuron)
metsulfuron.nl <- 
    nlreg( log(area) ~ log( b1+(b2-b1) / (1+(dose/b4)^b3) ),
           weights = ~ ( 1+dose^exp(g) )^2, data = metsulfuron, 
           start = c(b1 = 138, b2 = 2470, b3 = 2, b4 = 0.07, g = log(0.3)), 
           hoa = TRUE)
obsInfo( metsulfuron.nl )

Observed Information Matrix for ‘nlreg’ Objects

Description

Returns the observed information matrix from a fitted nlreg model.

Usage

## S3 method for class 'nlreg'
obsInfo(object, par, mu, v, m1 = NULL, m2 = NULL, v1 = NULL, 
        v2 = NULL, ...)

Arguments

Details

This function is a method for the generic function obsInfo for objects inheriting from class nlreg.

Value

the observed information matrix of the fitted nonlinear model passed through the object argument.

Note

This function is mostly intended for internal use. It is called by functions such as summary.nlreg and profile.nlreg. To extract the observed information matrix from a fitted nlreg object, the generic method obsInfo should be used.

See Also

obsInfo, nlreg.object, expInfo

Extract All Parameters from a Model — Generic Function

Description

This function extracts all parameters (regression coefficients, variance parameters etc.) from a fitted model.

Usage

param(object, ...)

Arguments

Details

This function is generic (see methods); method functions can be written to handle specific classes of data. Classes which already have methods for this function include: nlreg.

Value

all parameters (regression coefficients, variance parameters etc.) of a fitted model.

See Also

param.nlreg, methods

Examples

data(metsulfuron)
metsulfuron.nl <- 
    nlreg( log(area) ~ log( b1+(b2-b1) / (1+(dose/b4)^b3) ), 
           weights = ~( 1+dose^exp(g) )^2, data = metsulfuron, 
           start = c(b1 = 138, b2 = 2470, b3 = 2, b4 = 0.07, g = log(0.3)),
           hoa = TRUE )
param( metsulfuron.nl )
##          b1           b2           b3           b4            g         logs
## 139.0395322 2471.5097481    1.7091297    0.0772535   -1.2582597   -3.8198406

Use param() on a ‘nlreg’ object

Description

This is a method for the function param() for objects inheriting from class nlreg. See param for the general behavior of this function and for the interpretation of object.

Usage

## S3 method for class 'nlreg'
param(object, ...)

See Also

nlreg.object, param

Use plot() on a ‘nlreg.contours’ object

Description

This is a method for the function plot for objects inheriting from class nlreg.contours.

Usage

## S3 method for class 'nlreg.contours'
plot(x, alpha = c(0.1, 0.05), drawlabels = FALSE, lwd1 = 1, lwd2 = 1, 
     lty1 = "solid", lty2 = "solid", cl1 = "blue", cl2 = "red", 
     col = "black", pch1 = 1, pch2 = 16, cex = 0.5, ...)

Arguments

Value

No value is returned.

Side Effects

A plot is produced on the current graphics device.

See Also

nlreg.contours.object, contour.all.nlreg.profiles

Examples

## Not run: 
data(metsulfuron)
metsulfuron.nl <- 
    nlreg( formula = log(area) ~ log( b1+(b2-b1) / (1+(dose/b4)^b3) ), 
           weights = ~ ( 1+dose^exp(g) )^2, data = metsulfuron, 
           start = c(b1 = 138, b2 = 2470, b3 = 2, b4 = 0.07, g = log(0.3)),
           hoa = TRUE )
##
metsulfuron.prof <- profile( metsulfuron.nl, trace = TRUE )
metsulfuron.cont <- contour( metsulfuron.prof, ret = TRUE, plotit = FALSE )
par( mai = rep(0.2, 4) )
plot( metsulfuron.cont )
## End(Not run)

Diagnostic Plots for Nonlinear Heteroscedastic Models

Description

The nlreg.diag.plots routine generates diagnostic plots for a nonlinear heteroscedastic model using different types of residuals, influence measures and leverages. This is equivalent to using the plot.nlreg.diag method for function plot for objects inheriting from class nlreg.diag.

Usage

nlreg.diag.plots(fitted, which = "all", subset = NULL, iden = FALSE, 
                 labels = NULL, hoa = TRUE, infl = TRUE, 
                 trace = FALSE, ret = FALSE, ...)
## S3 method for class 'nlreg.diag'
plot(x, which = "all", subset = NULL, iden = FALSE, labels = NULL, 
     ...)

Arguments

Details

The diagnostics required for the plots are calculated by nlreg.diag, either by passing a nlreg.diag object or by applying nlreg.diag internally to the nlreg object specified through fitted. These are then used to produce the plots on the current graphics device. A menu lists all possible choices. They may be one or all of the following.

 Make a plot selection (or 0 to exit)

1:plot: Summary
2:plot: Studentized residuals against fitted values
3:plot: r* residuals against fitted values
4:plot: Normal QQ-plot of studentized residuals
5:plot: Normal QQ-plot of r* residuals
6:plot: Cook statistic against h/(1-h)
7:plot: Global influence against h/(1-h)
8:plot: Cook statistic against observation number
9:plot: Influence measures against observation number

Selection:

In the normal scores plots, the dotted line represents the expected line if the residuals are normally distributed, that is, it is the line with intercept 0 and slope 1.

In general, when plotting Cook's distance or the global influence measure against the standardized leverages, there will be two dotted lines on the plot. The horizontal line is at 8/(n-2p), where n is the number of observations and p is the number of regression coefficients estimated. Points above this line may be points with high influence on the model. The vertical line is at 2p/(n-2p) and points to the right of this line have high leverage compared to the variance of the raw residual at that point. If all points are below the horizontal line or to the left of the vertical line then the line is not shown.

Use of iden = TRUE is encouraged for proper exploration of these plots as a guide to how well the model fits the data and whether certain observations have an unduly large effect on parameter estimates.

Value

If ret = TRUE, the nlreg.diag object is returned. Otherwise, there is no returned value.

Side Effects

The current device is cleared. If iden = TRUE, interactive identification of points is enabled. All screens are closed, but not cleared, on termination of the function.

Acknowledgments

This function is based on A. J. Canty's function glm.diag.plots contained in library boot.

Note

Choices 3 and 5 are not available if hoa = FALSE in the call to nlreg.diag that generated the x argument. Choices 7 and 9 are not available if infl = FALSE in the same call. Plot number 9 is furthermore not available if the variance function is constant.

References

Brazzale, A. R. (2000) Practical Small-Sample Parametric Inference. Ph.D. Thesis N. 2230, Department of Mathematics, Swiss Federal Institute of Technology Lausanne. Section 6.3.1 and Appendix A.2.2.

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.

See Also

nlreg.diag, nlreg.object, identify

Examples

library(boot)
data(calcium)
calcium.nl <- nlreg( cal ~ b0*(1-exp(-b1*time)), weights = ~ ( 1+time^g )^2, 
                     start = c(b0 = 4, b1 = 0.1, g = 1), data = calcium, 
                     hoa = TRUE )
##
calcium.diag <- nlreg.diag( calcium.nl, trace = TRUE )
##
## menu-driven
## Not run: 
plot( calcium.diag )
##
##  Make a plot selection (or 0 to exit)
##
## 1:plot: Summary
## 2:plot: Studentized residuals against fitted values
## 3:plot: r* residuals against fitted values
## 4:plot: Normal QQ-plot of studentized residuals
## 5:plot: Normal QQ-plot of r* residuals
## 6:plot: Cook statistic against h/(1-h)
## 7:plot: Global influence against h/(1-h)
## 8:plot: Cook statistic against observation number
## 9:plot: Influence measures against observation number
##
## Selection:
## End(Not run)
##
## plot 5: Normal QQ-plot of r* residuals
plot( calcium.diag, which = 5, las = 1 )
##
nlreg.diag.plots( calcium.nl, which = 5, las = 1 )

Use plot() on a ‘profile.nlreg’ and ‘all.profiles.nlreg’ object

Description

These are methods for the function plot for objects inheriting from class "profile.nlreg" or "all.profiles.nlreg" .

Usage

## S3 method for class 'nlreg.profile'
plot(x, alpha = 0.05, add.leg = FALSE, stats = c("sk", "fr"), 
     cex = 0.7, cex.lab = 1, cex.axis = 1, cex.main = 1, lwd1 = 1, 
     lwd2 = 2, lty1 = "solid", lty2 = "solid", cl1 = "blue", 
     cl2 = "red", col = "black", ylim = c(-3,3), ...)
## S3 method for class 'all.nlreg.profiles'
plot(x, nframe, alpha = 0.05, stats = c("sk", "fr"), cex = 0.7, 
     cex.lab = 1, cex.axis = 1, cex.main = 1, lwd1 = 1, lwd2 = 2, 
     lty1 = "solid", lty2 = "solid", cl1 = "blue", cl2 = "red", 
     col = "black", ylim = c(-3,3), ...)

Arguments

Details

The function defaults to:

plot.nlreg.profile(x = stop("nothing to plot"), alpha = 0.05, add.leg = FALSE, 
                   stats = c("sk", "fr"), cex = 0.7, cex.lab = 1, cex.axis = 1, 
                   cex.main = 1, lwd1 = 1, lwd2 = 2, lty1 = "solid", lty2 = "solid", 
                   cl1 = "blue", cl2 = "red", col = "black", ylim = c(-3,3), \dots)

plot.all.nlreg.profiles(x = stop("nothing to plot"), nframe, alpha = 0.05, 
                        stats = c("sk", "fr"), cex = 0.7, cex.lab = 1, cex.axis = 1, 
                        cex.main = 1, lwd1 = 1, lwd2 = 2, lty1 = "solid", lty2 = "solid", 
                        cl1 = "blue", cl2 = "red", col = "black", ylim = c(-3,3), \dots)

Value

No value is returned.

Side Effects

A plot is produced on the current graphics device.

References

Fraser, D.A.S., Reid, N. and Wu, J. (1999). A simple general formula for tail probabilities for frequentist and Bayesian inference. Biometrika, 86, 249–264.

Skovgaard, I. (1996) An explicit large-deviation approximation to one-parameter tests. Bernoulli, 2, 145–165.

See Also

profile.nlreg, nlreg.profile.objects, plot

Examples

## Not run: 
data(metsulfuron)
metsulfuron.nl <- 
    nlreg( formula = log(area) ~ log( b1+(b2-b1) / (1+(dose/b4)^b3) ), 
           weights = ~ ( 1+dose^exp(g) )^2, data = metsulfuron, 
           start = c(b1 = 138, b2 = 2470, b3 = 2, b4 = 0.07, g = log(0.3)),
           hoa = TRUE)
##
metsulfuron.prof <- profile( metsulfuron.nl, offset = g, trace = TRUE )
plot( metsulfuron.prof, lwd2 = 2 )
##
metsulfuron.prof <- profile( metsulfuron.nl, trace = TRUE )
plot( metsulfuron.prof, lwd2 = 2, nframe = c(2,3) )
## End(Not run)

Use print() on a ‘mpl’ object

Description

This is a method for the function print() for objects inheriting from class mpl. See print and print.default for the general behaviour of this function and for the interpretation of x and digits.

Usage

## S3 method for class 'mpl'
print(x, digits, ...)

Details

The function defaults to:

print.mpl(x, digits = max(3, getOption("digits")-3), \dots)

See Also

mpl.object, print, print.default

Use print() on a ‘nlreg’ object

Description

This is a method for the function print() for objects inheriting from class nlreg. See print and print.default for the general behaviour of this function and for the interpretation of digits.

Usage

## S3 method for class 'nlreg'
print(x, digits = max(3, getOption("digits")-3), ...)

Details

The method defaults to:

plot.nlreg(x, digits = max(3, getOption("digits")-3), \dots)

See Also

nlreg.object, print, print.default

Use print() on a ‘nlreg.contours’ object

Description

Objects of class nlreg.contours have no proper print method. They are plotted instead. See plot.nlreg.contours for details.

Usage

## S3 method for class 'nlreg.contours'
print(x, alpha = c(0.1, 0.05), drawlabels = FALSE, lwd1 = 1, lwd2 = 1, 
     lty1 = "solid", lty2 = "solid", cl1 = "blue", cl2 = "red", col = "black", 
     pch1 = 1, pch2 = 16, cex = 0.5, ...)

Details

The function defaults to:

plot.nlreg.contours(x, alpha = c(0.1, 0.05), drawlabels = FALSE, lwd1 = 1, lwd2 = 1, 
                    lty1 = "solid", lty2 = "solid", cl1 = "blue", cl2 = "red", 
                    col = "black", pch1 = 1, pch2 = 16, cex = 0.5, \dots)

See Also

nlreg.contours.object, plot.nlreg.contours

Use print() on a ‘nlreg.profile’ and ‘all.nlreg.profiles’ object

Description

Objects of class nlreg.profile and all.nlreg.profiles have no proper print method. They are plotted instead. See plot.nlreg.profile and plot.all.nlreg.profiles for details.

Usage

## S3 method for class 'nlreg.profile'
print(x = stop("nothing to plot"), alpha = 0.05, add.leg = FALSE, stats = c("sk", "fr"),
     cex = 0.7, cex.lab = 1, cex.axis = 1, cex.main = 1, lwd1 = 1, lwd2 = 2, 
     lty1 = "solid", lty2 = "solid", cl1 = "blue", cl2 = "red", col = "black", 
     ylim = c(-3,3), ...)
## S3 method for class 'all.nlreg.profiles'
print(x = stop("nothing to plot"), nframe, alpha = 0.05, stats = c("sk", "fr"),
     cex = 0.7, cex.lab = 1, cex.axis = 1, cex.main = 1, lwd1 = 1, lwd2 = 2, 
     lty1 = "solid", lty2 = "solid", cl1 = "blue", cl2 = "red", col = "black", 
     ylim = c(-3,3), ...)

Details

The function defaults to:

plot.nlreg.profile(x = stop("nothing to plot"), alpha = 0.05, add.leg = FALSE, 
                   stats = c("sk", "fr"), cex = 0.7, cex.lab = 1, cex.axis = 1, 
                   cex.main = 1, lwd1 = 1, lwd2 = 2, lty1 = "solid", lty2 = "solid", 
                   cl1 = "blue", cl2 = "red", col = "black", ylim = c(-3,3), \dots)

plot.all.nlreg.profiles(x = stop("nothing to plot"), nframe, alpha = 0.05, 
                        stats = c("sk", "fr"), cex = 0.7, cex.lab = 1, cex.axis = 1, 
                        cex.main = 1, lwd1 = 1, lwd2 = 2, lty1 = "solid", lty2 = "solid", 
                        cl1 = "blue", cl2 = "red", col = "black", ylim = c(-3,3), \dots)

See Also

nlreg.profile.objects, plot.nlreg.profile, plot.all.nlreg.profiles

Use print() on a ‘summary.mpl’ object

Description

This is a method for the function print() for objects inheriting from class summary.mpl. See print and print.default for the general behaviour of this function and for the interpretation of x and digits.

Usage

## S3 method for class 'summary.mpl'
print(x, corr = FALSE, digits, ...)

Details

The function defaults to:

print.summary.mpl(x, corr = FALSE, 
                  digits = if(!is.null(x$digits)) x$digits 
                           else max(3, getOption("digits")-3), \dots)

The corr argument is a logical value indicating whether the (asymptotic) correlation matrix of the MLEs of the regression coefficients should be printed.

See Also

summary.mpl, print, print.default

Use print() on a ‘summary.nlreg’ object

Description

This is a method for the function print() for objects inheriting from class summary.nlreg. See print and print.default for the general behaviour of this function and for the interpretation of x, digits, signif.stars and quote.

Usage

## S3 method for class 'summary.nlreg'
print(x, digits, signif.stars, quote = TRUE, ...)

Details

The function defaults to:

print.summary.nlreg(x, digits = max(3, getOption("digits")-3), 
                     signif.stars = getOption("show.signif.stars"), 
                     quote = TRUE, \dots)

See Also

summary.nlreg, print, print.default

Use print() on a ‘summary.nlreg.profile’ and ‘summary.all.nlreg.profiles’ object

Description

This is a method for the function print() for objects inheriting from class summary.nlreg.profile and summary.all.nlreg.profiles. See print and print.default for the general behaviour of this function and for the interpretation of x and digits.

Usage

## S3 method for class 'summary.nlreg.profile'
print(x, digits, ...)
## S3 method for class 'summary.all.nlreg.profiles'
print(x, digits, ...)

Details

The function defaults to:

print.summary.nlreg.profile(x, 
                            digits = if(!is.null(x$digits)) x$digits
                                     else max(3, getOption("digits")-3), \dots)

print.summary.all.nlreg.profiles(x, 
                                 digits = if(!is.null(x$digits)) x$digits
                                          else max(3, getOption("digits")-3), \dots)

See Also

summary.nlreg.profile, summary.all.nlreg.profiles, print, print.default,

Profile Method for ‘nlreg’ Objects

Description

Returns a list of elements for profiling a nonlinear heteroscedastic model.

Usage

## S3 method for class 'nlreg'
profile(fitted, offset = "all", hoa = TRUE, precision = 6, 
        signif = 30, n = 50, omit = 0.5, trace = FALSE, md, vd, 
        all = FALSE, ...)

Arguments

Details

The function profile.nlreg calculates all elements necessary for profiling a scalar parameter of interest or all model parameters. The model formula must contain more than one regression coefficient.

A classical profile plot (Bates and Watts, 1988, Section 6.1.2) is a plot of the likelihood root statistic and of the Wald statistic against a range of values for the interest parameter. It provides a means to assess the accuracy of the normal approximation to the distribution of both statistics: the closer the corresponding curves are, the better the approximations. Confidence intervals can easily be read off for any desired level: the confidence bounds identify with the values on the x-axis at which the curves intersect the horizontal lines representing the standard normal quantiles of the desired level.

Profiling is performed by updating a fitted nonlinear heteroscedastic model. All statistics are calculated exactly for at maximum signif equally spaced points distributed around the MLE. To save execution time, the iterations start with a value close to the MLE and proceed in the two directions MLE \pm \delta, until the absolute value of all statistics exceeds the threshold 2.4. The step size \delta is defined by the signif argument. A spline interpolation is used to extend them over the whole interval of interest. A range of values, defined by the omit argument is omitted to avoid numerical instabilities around the MLE. All results are stored in an object of class nlreg.profile or all.nlreg.profiles depending on the value assumed by the offset argument. The summary and plot method functions must be used to examine the output or represent it graphically. No print method is available.

If hoa = TRUE, profile.nlreg produces an enhanced version of the classical profile plots by including the third order modified likelihood root statistic r^*. More precisely, it implements two approximations to Barndorff-Nielsens's (1991) original formulation where the sample space derivatives are replaced by respectively the approximations proposed in Skovgaard (1996) and Fraser, Reid and Wu (1999). The idea is to provide insight into the behaviour of first order methods, such as detecting possible bias of the estimates or the influence of the model curvature.

The theory and statistics used are summarized in Brazzale (2000, Chapters 2 and 3). More details of the implementation are given in Brazzale (1999; 2000, Section 6.3.2).

Value

a list of elements of class nlreg.profile or, if offset = "all", of class all.nlreg.profiles for profiling a nonlinear heteroscedastic model. The nlreg.profile class considers a scalar parameter of interest, while the all.nlreg.profiles class ontains the profiles of all parameters – regression coefficients and variance parameters.

Side Effects

If trace = TRUE, the parameter which is currently profiled and the corresponding value are printed.

Note

profile.nlreg is a method for the generic function profile for class nlreg. It can be invoked by calling profile for an object of the appropriate class, or directly by calling profile.nlreg.

To obtain the profiles of the different statistics considered, the model is refitted several times while keeping the value of the parameter of interest fixed. Although rarely, convergence problems may occur as the starting values are chosen in an automatic way. A try construct is used to prevent the profile.nlreg method from breaking down. Hence, the values of the statistics are not available where a convergence problem was encountered. A warning is issued whenever this occurs.

References

Barndorff-Nielsen, O. E. (1991) Modified signed log likelihood ratio. Biometrika, 78, 557–564.

Bates, D. M. and Watts, D. G. (1988) Nonlinear Regression Analysis and Its Applications. New York: Wiley.

Brazzale, A. R. (2000) Practical Small-Sample Parametric Inference. Ph.D. Thesis N. 2230, Department of Mathematics, Swiss Federal Institute of Technology Lausanne.

Fraser, D.A.S., Reid, N. and Wu, J. (1999). A simple general formula for tail probabilities for frequentist and Bayesian inference. Biometrika, 86, 249–264.

Skovgaard, I. (1996) An explicit large-deviation approximation to one-parameter tests. Bernoulli, 2, 145–165.

See Also

nlreg.profile.object, all.nlreg.profiles.object, profile

Examples

## Not run: 
data(metsulfuron)
metsulfuron.nl <- 
    nlreg( formula = log(area) ~ log( b1+(b2-b1) / (1+(dose/b4)^b3) ), 
           weights = ~ ( 1+dose^exp(g) )^2, data = metsulfuron, 
           start = c(b1 = 138, b2 = 2470, b3 = 2, b4 = 0.07, g = log(0.3)),
           hoa = TRUE )
##
metsulfuron.prof <- profile( metsulfuron.nl, offset = g, trace = TRUE )
plot( metsulfuron.prof, lwd2 = 2 )
#
metsulfuron.prof <- profile( metsulfuron.nl, trace = TRUE )
plot( metsulfuron.prof, nframe = c(2,3), lwd2 = 2 )
## End(Not run)

Support for ‘nlreg’ package of ‘hoa’ bundle

Description

Support routines for various functions in nlreg package

Usage

qhat.nlreg(nlregObj1, nlregObj0, par.1, par.0, mu.1, mu.0, v.1, v.0, 
           m1.1 = NULL, v1.1 = NULL)
Shat.nlreg(nlregObj1, nlregObj0, par.1, par.0, mu.1, mu.0, v.1, v.0, 
           m1.1 = NULL, m1.0 = NULL, v1.1 = NULL, v1.0 = NULL)
theta.deriv(nlregObj, par, mu, v, m1 = NULL, v1 = NULL)

Details

qhat.nlreg and Shat.nlreg are support routines for nlreg.diag, profile.nlreg and, the latter, mpl.nlreg, and should not be used on their own.

theta.deriv is a support routine for expInfo.nlreg, qhat.nlreg and Shat.nlreg, and should not be used on its own.

Use residuals() on a ‘nlreg’ object

Description

This is a method for the function residuals() for objects inheriting from class nlreg. It extracts the standardized residuals from a nonlinear model fit. See residuals for the general behavior of this function and for the interpretation of object. A wider choice of residuals is available through the nlreg.diag function.

Usage

## S3 method for class 'nlreg'
residuals(object, ...)
## S3 method for class 'nlreg'
resid(object, ...)

See Also

nlreg.object, nlreg.diag, residuals

Radioimmunoassay Data

Description

The ria data frame has 16 rows and 2 columns.

Run of a radioimmunoassay (RIA) to estimate the concentrations of a drug in samples of porcine serum. The experiment consists of 16 observations made at 8 different drug levels with two replications at each level.

Usage

data(ria)

Format

This data frame contains the following columns:

conc

the drug concentration (ng/ml);

count

the observed percentage of radioactive gamma counts.

Source

The data were obtained from

Belanger, B. A., Davidian, M. and Giltinan, D. M. (1996) The effect of variance function estimation on nonlinear calibration inference in immunoassay data. Biometrics, 52, 158–175. Table 1, first two columns.

References

Brazzale, A. R. (2000) Practical Small-Sample Parametric Inference. Ph.D. Thesis N. 2230, Department of Mathematics, Swiss Federal Institute of Technology Lausanne. Section 5.3, Example 6.

Examples

data(ria)
attach(ria)
plot(conc, count, xlab="drug concentration (ng/ml)", ylab="gamma counts (%)")
detach()

Summary Method for Objects of Class ‘all.nlreg.profiles’

Description

Returns a summary list for objects of class all.nlreg.profiles.

Usage

## S3 method for class 'all.nlreg.profiles'
summary(object, alpha = 0.05, twoside = TRUE, digits = NULL, ...)

Arguments

Details

This function is a method for the generic function summary for objects of class all.nlreg.profiles. It can be invoked by calling summary or directly summary.all.nlreg.profiles for an object of the appropriate class.

Value

A list is returned where the first components are named after the parameters of the nonlinear model that was profiled. Each component represents a matrix with k\dim(\code{alpha}) rows and 2 columns, where k equals 2 or 4 depending on whether hoa = TRUE in the call that generated the nlreg.profile object. This matrix contains the upper and lower confidence bounds for the test statistics considered and for the confidence levels defined through alpha. The remaining components are the following:

See Also

nlreg.profile.objects, profile.nlreg, summary

Examples

## Not run: 
data(metsulfuron)
metsulfuron.nl <- 
    nlreg( formula = log(area) ~ log( b1+(b2-b1) / (1+(dose/b4)^b3) ), 
           weights = ~ ( 1+dose^exp(g) )^2, data = metsulfuron, 
           start = c(b1 = 138, b2 = 2470, b3 = 2, b4 = 0.07, g = log(0.3)),
           hoa = TRUE )
##
metsulfuron.prof <- profile( metsulfuron.nl, trace = TRUE )
summary( metsulfuron.prof, alpha = c(0.1, 0.05) )
## End(Not run)

Summary Method for ‘mpl’ Objects

Description

Returns a summary list for objects of class mpl.

Usage

## S3 method for class 'mpl'
summary(object, correlation = FALSE, digits = NULL, ...)

Arguments

Details

This function is a method for the generic function summary for class mpl. It can be invoked by calling summary for an object of the appropriate class, or directly by calling summary.mpl regardless of the class of the object.

Value

A list is returned with the following components:

See Also

mpl.object, nlreg.object, summary

Examples

data(metsulfuron)
metsulfuron.nl <- 
    nlreg( formula = log(area) ~ log( b1+(b2-b1) / (1+(dose/b4)^b3) ), 
           weights = ~ ( 1+dose^exp(g) )^2, data = metsulfuron, hoa = TRUE, 
           start = c(b1 = 138, b2 = 2470, b3 = 2, b4 = 0.07, g = log(0.3)) )
##
metsulfuron.mpl <- mpl( metsulfuron.nl, trace = TRUE )
summary( metsulfuron.mpl, corr = FALSE )

Summary Method for Nonlinear Heteroscedastic Models

Description

Returns a summary list for a fitted nonlinear heteroscedastic model.

Usage

## S3 method for class 'nlreg'
summary(object, observed = TRUE, correlation = FALSE, 
        digits = NULL, ...)

Arguments

Details

This function is a method for the generic function summary for class nlreg. It can be invoked by calling summary for an object of the appropriate class, or directly by calling summary.nlreg regardless of the class of the object.

Value

A list is returned with the following components:

See Also

nlreg.object, summary

Examples

data(metsulfuron)
metsulfuron.nl <- 
    nlreg( formula = log(area) ~ log( b1+(b2-b1) / (1+(dose/b4)^b3) ), 
           weights = ~ ( 1+dose^exp(g) )^2, data = metsulfuron, 
           start = c(b1 = 138, b2 = 2470, b3 = 2, b4 = 0.07, g = log(0.3)),
           hoa = TRUE )
##
summary( metsulfuron.nl, digits = 3 )
##
print( summary( metsulfuron.nl )$cov, digits = 3 )
print( summary( metsulfuron.nl, observed = FALSE )$cov, digits = 3 )

Summary Method for Objects of Class ‘nlreg.profile’

Description

Returns a summary list for objects of class nlreg.profile.

Usage

## S3 method for class 'nlreg.profile'
summary(object, alpha = 0.05, twoside = TRUE, digits = NULL, ...)

Arguments

Details

This function is a method for the generic function summary for objects of class nlreg.profile. It can be invoked by calling summary or directly summary.nlreg.profile for an object of the appropriate class.

Value

A list is returned with the following components:

References

Fraser, D.A.S., Reid, N. and Wu, J. (1999). A simple general formula for tail probabilities for frequentist and Bayesian inference. Biometrika, 86, 249–264.

Skovgaard, I. (1996) An explicit large-deviation approximation to one-parameter tests. Bernoulli, 2, 145–165.

See Also

nlreg.profile.object, profile.nlreg, summary

Examples

data(metsulfuron)
metsulfuron.nl <- 
    nlreg( formula = log(area) ~ log( b1+(b2-b1) / (1+(dose/b4)^b3) ), 
           weights = ~ ( 1+dose^exp(g) )^2, data = metsulfuron, 
           start = c(b1 = 138, b2 = 2470, b3 = 2, b4 = 0.07, g = log(0.3)),
           hoa = TRUE )
##
metsulfuron.prof <- profile( metsulfuron.nl, offset = g, trace = TRUE )
summary( metsulfuron.prof, alpha = c(0.9, 0.95) )

Convert Covariance Matrix to Correlation Matrix — Generic Function

Description

This function converts the covariance matrix from a fitted model into the correlation matrix.

Usage

var2cor(object, ...)

Arguments

Details

This function is generic (see methods); method functions can be written to handle specific classes of data. Classes which already have methods for this function include: nlreg, summary.nlreg, mpl and summary.mpl.

Value

the correlation matrix of the estimates from a fitted model.

See Also

var2cor.nlreg, var2cor.mpl, methods

Examples

data(metsulfuron)
metsulfuron.nl <- 
    nlreg( log(area) ~ log( b1+(b2-b1) / (1+(dose/b4)^b3) ), 
           weights = ~( 1+dose^exp(g) )^2, data = metsulfuron, 
           start = c(b1 = 138, b2 = 2470, b3 = 2, b4 = 0.07, g = log(0.3)),
           hoa = TRUE )
var2cor( metsulfuron.nl )
##
metsulfuron.sum <- summary( metsulfuron.nl, corr = FALSE )
var2cor( metsulfuron.sum )

Use var2cor() on a ‘nlreg’ and ‘mpl’ object

Description

This is a method for the function var2cor() for objects inheriting from class nlreg or mpl. See var2cor for the general behavior of this function and for the interpretation of object.

Usage

## S3 method for class 'mpl'
var2cor(object, ...)
## S3 method for class 'nlreg'
var2cor(object, ...)
## S3 method for class 'summary.mpl'
var2cor(object, ...)
## S3 method for class 'summary.nlreg'
var2cor(object, ...)

See Also

mpl.object, nlreg.object, var2cor