| gsci.ratio {mratios} | R Documentation |
Simultaneous confidence intervals for ratios of linear combinations of parameters.
Description
This function calculates simultaneous confidence intervals for ratios of user-defined linear combinations, given a vector parameter estiamtes and a corresponding variance-covariance matrix. Beside unadjusted intervals, multiplicity adjustments are available using quantiles of a multivariate Normal- or t-distribution. The function provides a more general, but less user-friendly function to calculate ratios of mean parameters from linear (mixed models).
Usage
gsci.ratio(est, vcmat, Num.Contrast, Den.Contrast,
degfree = NULL, conf.level = 0.95, alternative = "two.sided",
adjusted = TRUE)
Arguments
est |
A numeric vector of parameter estimates, for example coefficients of a linear model |
vcmat |
The corresponding variance-covariance matrix (Number of rows and columns should be the same as the length of the parameter vector) |
Num.Contrast |
Numerator contrast matrix, where the number of columns must be the same as the length of the parameter vector, and each row represents one contrast |
Den.Contrast |
Denominator contrast matrix, where the number of columns must be the same as the length of the parameter vector, and each row represents one contrast |
degfree |
Degrees of freedom used for calculating quantiles of a (multivariate) t-distribution. If NULL, Normal approximations are used |
conf.level |
Simultaneous confidence level in case of adjusted == TRUE, and comparisonwise confidence level in case of adjusted == FALSE |
alternative |
a character string: "two.sided" for two-sided intervals, "less" for upper confidence limits, "greater" for lower confidence limits |
adjusted |
If TRUE, the simultaneous confidence level is controlled, otherwise the comparisonwise confidence level is used |
Details
Given a parameter vector and its corresponding covariance matrix from a linera model fit, approximate simultaneous confidence intervals for several ratios of linear combinations of these parameters are calculated. For simultaneous confidence intervals (adjusted=TRUE) the plug-in method is used (plugging the maximum likelihood estimates of the ratios to obtain the correlation matrix for calculating quantiles of a multivariate t or normal distribution).
Linear combinations can be defined by providing matrices for the nominator and the denominator; some pre-defined contrasts can be constructed by the function contrMatRatio. (These may be weighted for different sample sizes.)
Value
An object of class "sci.ratio" and "gsci.ratio", containing a list with elements:
estimate |
point estimates of the ratios |
CorrMat.est |
estimate of the correlation matrix |
Num.Contrast |
matrix of contrasts used for the numerator of ratios |
Den.Contrast |
matrix of contrasts used for the denominator of ratios |
conf.int |
confidence interval estimates of the ratios |
And some further elements to be passed to print and summary functions.
Author(s)
Daniel Gerhard & Frank Schaarschmidt adapting code of Gemechis Dilba Djira
References
The general methodology of constructing inference for ratios of linear model parameters can be found in:
Zerbe G.O., (1978): On Fieller's Theorem and the General Linear Model. The American Statistician 32(3), 103-105.
Young D.A., Zerbe G.O., Hay W.W. (1997): Fieller's Theorem, Scheffe's simultaneous confidence intervals, and ratios of parameters of linear and nonlinear mixed-effect models. Biometrics 53(3), 835-847.
Djira G.D.(2010): Relative Potency Estimation in Parallel-Line Assays - Method Comparison and Some Extensions. Communications in Statistics - Theory and Methods 39(7), 1180-1189.
However, when adjusted=TRUE, the quantiles are not obtained as described in Zerbe(1978) or Young et al. (1997), but by adapting the 'plug-in' method described for the completely randomized one-way layout in
Dilba, G., Bretz, F., and Guiard, V. (2006): Simultaneous confidence sets and confidence intervals for multiple ratios. Journal of Statistical Planning and Inference 136, 2640-2658.
A simulation study of the performance of these methods in linear mixed models:
Schaarschmidt and Djira(2016): Simultaneous Confidence Intervals for Ratios of Fixed Effect Parameters in Linear Mixed Models. Communications in Statistics - Simulation and Computation 45:5, 1704-1717. DOI: 10.1080/03610918.2013.849741
See Also
glht(multcomp) for simultaneous CI of differences of means, and function sci.ratio.gen(mratios)
Examples
library(mratios)
##############################################################
# A 90-days chronic toxicity assay:
# Which of the doses (groups 2,3,4) do not show a decrease in
# bodyweight more pronounced than 90 percent of the bodyweight
# in the control group?
#############################################################
data(BW)
boxplot(Weight~Dose,data=BW)
lmfit <- lm(Weight~Dose-1, data=BW)
est <- coefficients(lmfit)
vc <- vcov(lmfit)
CMAT <- contrMatRatio(table(BW$Dose), type="Dunnett")
BWnoninf <- gsci.ratio(est, vc, CMAT$numC, CMAT$denC,
alternative="greater", degfree=lmfit$df.residual)
BWnoninf
# Plot
plot(BWnoninf, rho0=0.9)
##############################################################
#### Mixed Model Example
##############################################################
library("nlme")
data(Milk)
# Fit a linear mixed model (maybe there are nicer models available!)
lmefit <- lme(protein ~ Diet-1, data=Milk,
random=~Time|Cow, correlation=corAR1(form=~Time|Cow))
# Extract the parameter estimates and the corresponding
# variance-covariance matrix
estm <- fixef(lmefit)
vcm <- vcov(lmefit)
# Define the matrices defining the ratios of interest for
# all-pair comparisons: CM is the numerator matrix and
# DM is the denominator matrix.
CM <- rbind(c(1,0,0),
c(1,0,0),
c(0,1,0))
DM <- rbind(c(0,1,0),
c(0,0,1),
c(0,0,1))
# Add some row names (This is optional!)
rownames(CM) <- c("b/b+l", "b/l", "b+l/l")
# Calculate and plot simultaneous confidence intervals:
gscimix <- gsci.ratio(estm, vcm, CM, DM, degfree=anova(lmefit)[,2])
plot(gscimix)