mratios (version 1.4.2)

gsci.ratio: 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

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.

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.)

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

# NOT RUN {
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)

# }