Generate Constraint Matrices

Description

Generate a constraint matrix with a predefined structure

Usage

constrMat(n, type = c("monotone", "control", "average", "laverage",
  "uaverage", "caverage"), base = 1)

Arguments

Value

a constraint matrix

See Also

orlm, constrSet

Examples

n <- c(10,20,30,40)
constrMat(n, type="monotone")
constrMat(n, type="control", base=2)
constrMat(n, type="average")
constrMat(n, type="laverage")
constrMat(n, type="uaverage")
constrMat(n, type="caverage", base=2)

Generate Constraint Sets

Description

Generate sets of constraint matrices (constr), right hand side elements, and numbers of equality constraints (nec) with a predefined structure

Usage

constrSet(n, set = c("sequence", "seqcontrol", "lplateau", "uplateau",
  "downturn", "williams"), direction = c("increase", "decrease"), base = 1)

Arguments

Value

a list with slots constr, rhs, and nec for each constraint definition

See Also

orlm, constrMat

Examples

n <- c(10,20,30,40)
constrSet(n, set="sequence")
constrSet(n, set="seqcontrol")
constrSet(n, set="lplateau")
constrSet(n, set="uplateau")
constrSet(n, set="downturn")
constrSet(n, set="williams")

Calculate GORIC

Description

The goric function calculates the order-restricted log likelihood, the penalty of the generalised order restricted information criterion (GORIC), the GORIC values, differences to the minimum GORIC value, and the GORIC weights for a set of hypotheses, where the penalty is based on iter iterations. The hypothesis with the lowest GORIC value is the preferred one. The GORIC weights reflect the support of each hypothesis in the set. To compare two hypotheses (and not one to the whole set), one should examine the ratio of the two corresponding GORIC weights. To safequard for weak hypotheses (i.e., hypotheses not supported by the data), one should include a model with no constraints (the so-called unconstrained model).

Usage

goric(object, ..., iter = 1e+05, type = "GORIC", dispersion = 1,
  mc.cores = 1)

## S3 method for class 'orlm'
goric(object, ..., iter = 1e+05, type = "GORIC",
  mc.cores = 1)

## S3 method for class 'orgls'
goric(object, ..., iter = 1e+05, type = "GORIC",
  mc.cores = 1)

## S3 method for class 'list'
goric(object, ..., iter = 1e+05, type = "GORIC",
  dispersion = 1, mc.cores = 1)

## S3 method for class 'orglm'
goric(object, ..., iter = 1e+05, type = "GORIC",
  dispersion = 1, mc.cores = 1)

Arguments

Value

a data.frame with the information criteria or a single penalty term

References

• Kuiper R.M., Hoijtink H., Silvapulle M.J. (2011). An Akaike-type Information Criterion for Model Selection Under Inequality Constraints. Biometrika, 98, 495–501.

• Kuiper R.M., Hoijtink H., Silvapulle M.J. (2012). Generalization of the Order-Restricted Information Criterion for Multivariate Normal Linear Models. Journal of Statistical Planning and Inference, 142, 2454-2463. doi:10.1016/j.jspi.2012.03.007.

• Kuiper R.M. and Hoijtink H. (submitted). A Fortran 90 Program for the Generalization of the Order-Restricted Information Criterion. Journal of Statictical Software.

See Also

orlm, orgls

Examples

## Example from Kuiper, R.M. and Hoijtink, H. (Unpublished).
# A Fortran 90 program for the generalization of the 
# order restricted information criterion.
# constraint definition
cmat <- cbind(diag(3), 0) + cbind(0, -diag(3))
constr <- kronecker(diag(3), cmat)
constr

# no effect model
(fm0 <- orlm(cbind(SDH, SGOT, SGPT) ~ dose-1, data=vinylidene,
            constr=constr, rhs=rep(0, nrow(constr)), nec=nrow(constr)))

# order constrained model (increasing serum levels with increasing doses)
fm1 <- orlm(cbind(SDH, SGOT, SGPT) ~ dose-1, data=vinylidene,
            constr=constr, rhs=rep(0, nrow(constr)), nec=0)
summary(fm1)

# unconstrained model
(fmunc <- orlm(cbind(SDH, SGOT, SGPT) ~ dose-1, data=vinylidene,
              constr=matrix(0, nrow=1, ncol=12), rhs=0, nec=0))

# calculate GORIC
# (only small number of iterations to decrease computation time, default: iter=100000)
goric(fm0, fm1, fmunc, iter=1000)

GORIC penalty term

Description

Calculates the GORIC penalty term (level probability) by Monte-Carlo simulation.

Usage

goric_penalty(object, iter = 1e+05, type = "GORIC", mc.cores = 1)

orglm_penalty(object, iter = 1e+05, type = "GORIC", mc.cores = 1)

Arguments

See Also

orlm, orgls, orglm

Fitting Order-Restricted Generalised Linear Models

Description

orglm is used to fit generalised linear models with restrictions on the parameters, specified by giving a description of the linear predictor, a description of the error distribution, and a description of a matrix with linear constraints. The quadprog package is used to apply linear constraints on the parameter vector.

Usage

orglm(formula, family = gaussian, data, weights, subset, na.action,
  start = NULL, etastart, mustart, offset, control = list(...),
  model = TRUE, method = "orglm.fit", x = FALSE, y = TRUE,
  contrasts = NULL, constr, rhs, nec, ...)

orglm.fit(x, y, weights = rep(1, nobs), start = NULL, etastart = NULL,
  mustart = NULL, offset = rep(0, nobs), family = gaussian(),
  control = list(), intercept = TRUE, constr, rhs, nec)

Arguments

Details

Non-NULL weights can be used to indicate that different observations have different dispersions (with the values in weights being inversely proportional to the dispersions); or equivalently, when the elements of weights are positive integers w_i, that each response y_i is the mean of w_i unit-weight observations. For a binomial GLM prior weights are used to give the number of trials when the response is the proportion of successes: they would rarely be used for a Poisson GLM. If more than one of etastart, start and mustart is specified, the first in the list will be used. It is often advisable to supply starting values for a quasi family, and also for families with unusual links such as gaussian("log"). For the background to warning messages about ‘fitted probabilities numerically 0 or 1 occurred’ for binomial GLMs, see Venables & Ripley (2002, pp. 197–8).

Value

An object of class "orglm" is a list containing at least the following components:

coefficients

a named vector of coefficients

residuals

the working residuals, that is the residuals in the final iteration of the IWLS fit. Since cases with zero weights are omitted, their working residuals are NA.

fitted.values

the fitted mean values, obtained by transforming the linear predictors by the inverse of the link function.

rank

the numeric rank of the fitted linear model.

family

the family object used.

linear.predictors

the linear fit on link scale.

deviance

up to a constant, minus twice the maximized log-likelihood. Where sensible, the constant is chosen so that a saturated model has deviance zero.

null.deviance

The deviance for the null model, comparable with deviance. The null model will include the offset, and an intercept if there is one in the model. Note that this will be incorrect if the link function depends on the data other than through the fitted mean: specify a zero offset to force a correct calculation.

iter

the number of iterations of IWLS used.

weights

the working weights, that is the weights in the final iteration of the IWLS fit.

prior.weights

the weights initially supplied, a vector of 1s if none were.

df.residual

the residual degrees of freedom of the unconstrained model.

df.null

the residual degrees of freedom for the null model.

y

if requested (the default) the y vector used. (It is a vector even for a binomial model.)

converged

logical. Was the IWLS algorithm judged to have converged?

boundary

logical. Is the fitted value on the boundary of the attainable values?

Author(s)

Modification of the original glm.fit by Daniel Gerhard. The original R implementation of glm was written by Simon Davies working for Ross Ihaka at the University of Auckland, but has since been extensively re-written by members of the R Core team. The design was inspired by the S function of the same name described in Hastie & Pregibon (1992).

References

• Dobson, A. J. (1990) An Introduction to Generalized Linear Models. London: Chapman and Hall.

• Hastie, T. J. and Pregibon, D. (1992) Generalized linear models. Chapter 6 of Statistical Models in S eds J. M. Chambers and T. J. Hastie, Wadsworth & Brooks//Cole.

• McCullagh P. and Nelder, J. A. (1989) Generalized Linear Models. London: Chapman and Hall.

• Venables, W. N. and Ripley, B. D. (2002) Modern Applied Statistics with S. New York: Springer.

See Also

glm, solve.QP

Fitting generalized least squares regression models with order restrictions

Description

orgls is used to fit generalised least square models analogously to the function gls in package nlme but with order restrictions on the parameters.

Usage

orgls(formula, data, constr, rhs, nec, weights = NULL, correlation = NULL,
  control = orlmcontrol())

## S3 method for class 'formula'
orgls(formula, data, constr, rhs, nec, weights = NULL,
  correlation = NULL, control = orlmcontrol())

Arguments

Details

The contraints in the hypothesis of interest are defined by constr, rhs, and nec. The first nec constraints are the equality contraints: Constr[1:nec, 1:tk] \theta = rhs[1:nec]; and the remaing ones are the inequality contraints: Constr[nec+1:c_m, 1:tk] \theta \geq rhs[nec+1:c_m]. Two requirements should be met:

1. The first nec constraints must be the equality contraints (i.e., Constr[1:nec, 1:tk] \theta = rhs[1:nec]) and the remaining ones the inequality contraints (i.e., Constr[nec+1:c_m, 1:tk] \theta \geq rhs[nec+1:c_m]).

2. When rhs is not zero, Constr should be of full rank (after discarding redundant restrictions).

Value

an object of class orgls

References

• Kuiper R.M., Hoijtink H., Silvapulle M.J. (2011). An Akaike-type Information Criterion for Model Selection Under Inequality Constraints. Biometrika, 98, 495–501.

• Kuiper R.M., Hoijtink H., Silvapulle M.J. (2012). Generalization of the Order-Restricted Information Criterion for Multivariate Normal Linear Models. Journal of Statistical Planning and Inference, 142, 2454-2463. doi:10.1016//j.jspi.2012.03.007.

• Kuiper R.M. and Hoijtink H. (submitted). A Fortran 90 Program for the Generalization of the Order-Restricted Information Criterion. Journal of Statictical Software.

See Also

solve.QP, goric

Examples

# generating example data
library(mvtnorm)
# group means
m <- c(0,5,5,7)
# compound symmetry structure of residuals
# (10 individuals per group, rho=0.7) 
cormat <- kronecker(diag(length(m)*10), matrix(0.7, nrow=length(m), ncol=length(m)))
diag(cormat) <- 1
# different variances per group
sds <- rep(c(1,2,0.5,1), times=10*length(m))
sigma <- crossprod(diag(sds), crossprod(cormat, diag(sds)))
response <- as.vector(rmvnorm(1, rep(m, times=10*length(m)), sigma=sigma))
dat <- data.frame(response,
                  grp=rep(LETTERS[1:length(m)], times=10*length(m)), 
                  ID=as.factor(rep(1:(10*length(m)), each=length(m))))
                  
## set of gls models:
# unconstrained model
m1 <- orgls(response ~ grp-1, data = dat,
            constr=rbind(c(0,0,0,0)), rhs=0, nec=0,
            weights=varIdent(form=~1|grp),
            correlation=corCompSymm(form=~1|ID))

# simple order
m2 <- orgls(response ~ grp-1, data = dat,
            constr=rbind(c(-1,1,0,0),c(0,-1,1,0),c(0,0,-1,1)), rhs=c(0,0,0), nec=0,
            weights=varIdent(form=~1|grp),
            correlation=corCompSymm(form=~1|ID))

# equality constraints
m3 <- orgls(response ~ grp-1, data = dat,
            constr=rbind(c(-1,1,0,0),c(0,-1,1,0),c(0,0,-1,1)), rhs=c(0,0,0), nec=3,
            weights=varIdent(form=~1|grp),
            correlation=corCompSymm(form=~1|ID))

Set of generalised least-squares models

Description

Fitting a specific set of generalisd least-squares models with order restrictions.

Usage

orglsSet(formula, data, weights = NULL, correlation = NULL, set,
  direction = "increase", n = NULL, base = 1, control = orlmcontrol())

Arguments

Details

This function is just a wrapper for repeated calls of orgls with different constraint definitions. Predefined lists with constraint-sets can be constructed with function constrSet.

Value

a list with orgls objects

See Also

orgls, constrSet, goric

Fitting multivariate regression models with order restrictions

Description

This is a modification of the lm function, fitting (multivariate) linear models with order constraints on the model coefficients.

Usage

orlm(formula, data, constr, rhs, nec, control = orlmcontrol())

## S3 method for class 'formula'
orlm(formula, data, constr, rhs, nec,
  control = orlmcontrol())

Arguments

Details

The contraints in the hypothesis of interest are defined by Constr, rhs, and nec. The first nec constraints are the equality contraints: Constr[1:nec, 1:tk] \theta = rhs[1:nec]; and the remaing ones are the inequality contraints: Constr[nec+1:c_m, 1:tk] \theta \geq rhs[nec+1:c_m]. Two requirements should be met:

1. The first nec constraints must be the equality contraints (i.e., Constr[1:nec, 1:tk] \theta = rhs[1:nec]) and the remaining ones the inequality contraints (i.e., Constr[nec+1:c_m, 1:tk] \theta \geq rhs[nec+1:c_m]).

2. When rhs is not zero, Constr should be of full rank (after discarding redundant restrictions).

Value

an object of class orlm

References

• Kuiper R.M., Hoijtink H., Silvapulle M.J. (2011). An Akaike-type Information Criterion for Model Selection Under Inequality Constraints. Biometrika, 98, 495–501.

• Kuiper R.M., Hoijtink H., Silvapulle M.J. (2012). Generalization of the Order-Restricted Information Criterion for Multivariate Normal Linear Models. Journal of Statistical Planning and Inference, 142, 2454-2463. doi:10.1016//j.jspi.2012.03.007.

• Kuiper R.M. and Hoijtink H. (submitted). A Fortran 90 Program for the Generalization of the Order-Restricted Information Criterion. Journal of Statictical Software.

See Also

solve.QP, goric

Examples

########################
## Artificial example ##
########################
n <- 10
m <- c(1,2,1,5)
nm <- length(m)
dat <- data.frame(grp=as.factor(rep(1:nm, each=n)),
                  y=rnorm(n*nm, rep(m, each=n), 1))

# unrestricted linear model
cm1 <- matrix(0, nrow=1, ncol=4)
fm1 <- orlm(y ~ grp-1, data=dat, constr=cm1, rhs=0, nec=0)

# order restriction (increasing means)
cm2 <- rbind(c(-1,1,0,0),
             c(0,-1,1,0),
             c(0,0,-1,1))
fm2 <- orlm(y ~ grp-1, data=dat, constr=cm2,
            rhs=rep(0,nrow(cm2)), nec=0)

# order restriction (increasing at least by delta=1)
fm3 <- orlm(y ~ grp-1, data=dat, constr=cm2,
            rhs=rep(1,nrow(cm2)), nec=0)

# larger than average of the neighboring first 2 parameters
cm4 <- rbind(c(-0.5,-0.5,1,0),
             c(0,-0.5,-0.5,1))
fm4 <- orlm(y ~ grp-1, data=dat, constr=cm4,
            rhs=rep(0,nrow(cm4)), nec=0)

# equality constraints (all parameters equal)
fm5 <- orlm(y ~ grp-1, data=dat, constr=cm2,
            rhs=rep(0,nrow(cm2)), nec=nrow(cm2))
            
# alternatively
fm5 <- orlm(y ~ grp-1, data=dat, constr=cm2,
            rhs=rep(0,nrow(cm2)), nec=c(TRUE,TRUE,TRUE))
            
# constraining the 1st and the 4th parameter
# to their true values, and the 2nd and 3rd between them
cm6 <- rbind(c( 1,0,0,0),
             c(-1,1,0,0),
             c(0,-1,0,1),
             c(-1,0,1,0),
             c(0,0,-1,1),
             c(0,0, 0,1))
fm6 <- orlm(y ~ grp-1, data=dat, constr=cm6,
            rhs=c(1,rep(0,4),5), nec=c(TRUE,rep(FALSE,4),TRUE))
            
            
###############################################################
## Example from Kuiper, R.M. and Hoijtink, H. (Unpublished). ##
## A Fortran 90 program for the generalization of the        ##
## order restricted information criterion.                   ##
###############################################################

# constraint definition
cmat <- cbind(diag(3), 0) + cbind(0, -diag(3))
constr <- kronecker(diag(3), cmat)

# no effect model
(fm0 <- orlm(cbind(SDH, SGOT, SGPT) ~ dose-1, data=vinylidene,
             constr=constr, rhs=rep(0, nrow(constr)), nec=nrow(constr)))

# order constrained model (increasing serum levels with increasing doses)
fm1 <- orlm(cbind(SDH, SGOT, SGPT) ~ dose-1, data=vinylidene,
            constr=constr, rhs=rep(0, nrow(constr)), nec=0)
summary(fm1)

# unconstrained model
(fmunc <- orlm(cbind(SDH, SGOT, SGPT) ~ dose-1, data=vinylidene,
               constr=matrix(0, nrow=1, ncol=12), rhs=0, nec=0))

Set of multivariate regression models

Description

Fitting a specific set of multivariate regression models with order restrictions.

Usage

orlmSet(formula, data, set, direction = "increase", n = NULL, base = 1,
  control = orlmcontrol())

Arguments

Details

This function is just a wrapper for repeated calls of orlm with different constraint definitions. Predefined lists with constraint-sets can be constructed with function constrSet.

Value

a list with orlm objects

See Also

orlm, constrSet, goric

Examples

########################
## Artificial example ##
########################

n <- 10
m <- c(1,2,4,5,2,1)
nm <- length(m)
dat <- data.frame(grp=as.factor(rep(1:nm, each=n)),
                  y=rnorm(n*nm, rep(m, each=n), 1))

(cs <- constrSet(table(dat$grp), set="sequence"))
(oss <- orlmSet(y ~ grp-1, data=dat, set=cs))

# the same as:
oss <- orlmSet(y ~ grp-1, data=dat, set="sequence")

Control arguments for the orlm function.

Description

A list with control arguments controlling the orlm function

Usage

orlmcontrol(maxiter = 10000, absval = 1e-04)

Arguments

Value

a list with control arguments

See Also

orlm

Simulation from order restricted linear models

Description

Simulation function for orlm and orgls objects

Usage

sim(object, n.sims)

## S3 method for class 'orlm'
sim(object, n.sims)

## S3 method for class 'orgls'
sim(object, n.sims)

Arguments

Details

Given the estimated coefficients of a orlm or orgls model, a set new parameters are generated. n.sims new sets of observations are generated based on the unrestricted model; these new datasets are used to estimate a new set of model coefficients incorporating the given order restrictions.

Value

a list with sets of simulated parameters.

See Also

orlm, orgls

Examples

########################
## Artificial example ##
########################
n <- 10
m <- c(1,1,2)
dat <- data.frame(grp=as.factor(rep(1:length(m), each=n)),
                  y=rnorm(n*length(m), rep(m, each=n), 1))
cm <- rbind(c(-1,1,0),
            c(0,-1,1))
fm <- orlm(y ~ grp-1, data=dat, constr=cm, rhs=rep(0,nrow(cm)), nec=0)
b <- sim(fm, n.sims=1000)$coef
pairs(t(b), cex=0.3)

Effect of vinylidene fluoride on liver cancer

Description

Real data which are available on page 10 of Silvapulle and Sen (2005) and in a report prepared by Litton Bionetics Inc in 1984. These data were used in an experiment to find out whether vinylidene fluoride gives rise to liver damage. Since increased levels of serum enzyme are inherent in liver damage, the focus is on whether enzyme levels are affected by vinylidene fluoride. The variable of interest is the serum enzyme level. Three types of enzymes are inspected, namely SDH, SGOT, and SGPT. To study whether vinylidene fluoride has an influence on the three serum enzymes, four dosages of this substance are examined. In each of these four treatment groups, ten male Fischer-344 rats received the substance.

Usage

vinylidene

Format

A data frame with 40 observations on the following 4 variables.

SDH

serum enzyme level of enzyme type SDH.

SGOT

serum enzyme level of enzyme type SGOT.

SGPT

serum enzyme level of enzyme type SGPT.

dose

factor with 4 levels (d1-d4) representing the 4 vinylidene fluoride concentrations.

References

Silvapulle MJ, Sen PK (2005). Constrained Statistical Inference. New Jersey: Wiley.