Description Usage Arguments Details Value References See Also Examples
This is a modification of the lm
function, fitting (multivariate) linear models with order constraints on the model coefficients.
1 2 3 4 5 | orlm(formula, data, constr, rhs, nec, control = orlmcontrol())
## S3 method for class 'formula'
orlm(formula, data, constr, rhs, nec,
control = orlmcontrol())
|
formula |
an object of class "formula" (or one that can be coerced to that class): a symbolic description of the model to be fitted. |
data |
an optional data frame, list or environment (or object coercible by as.data.frame to a data frame) containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment from which lm is called. |
constr |
matrix with constraints; with rows as constraint definition, columns should be in line with the parameters of the model |
rhs |
vector of right hand side elements; Constr \; θ ≥q rhs; number should equal the number of rows of the constr matrix |
nec |
number of equality constraints; a numeric value treating the first nec constr rows as equality constraints, or a logical vector with |
control |
a list of control arguments; see |
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] θ = rhs[1:nec]; and the remaing ones are the inequality contraints: Constr[nec+1:c_m, 1:tk] θ ≥q rhs[nec+1:c_m]. Two requirements should be met:
The first nec constraints must be the equality contraints (i.e., Constr[1:nec, 1:tk] θ = rhs[1:nec]) and the remaining ones the inequality contraints (i.e., Constr[nec+1:c_m, 1:tk] θ ≥q rhs[nec+1:c_m]).
When rhs is not zero, Constr should be of full rank (after discarding redundant restrictions).
an object of class orlm
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.
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## 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))
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