gsci.ratio: Simultaneous confidence intervals for ratios of linear...
In mratios: Ratios of Coefficients in the General Linear Model

Description Usage Arguments Details Value Author(s) References See Also Examples

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).

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gsci.ratio(est, vcmat, Num.Contrast, Den.Contrast,
 degfree = NULL, conf.level = 0.95, alternative = "two.sided",
 adjusted = TRUE)

`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

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.)

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.

Daniel Gerhard & Frank Schaarschmidt adapting code of Gemechis Dilba Djira

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

glht(multcomp) for simultaneous CI of differences of means, and function sci.ratio.gen(mratios)

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)