R: Simultaneous confidence intervals for ratios of linear...

sci.ratio {mratios} | R Documentation

## Simultaneous confidence intervals for ratios of linear combinations of means

### Description

This function constructs simultaneous confidence intervals for ratios of linear combinations of normal means in a one-way ANOVA model. Different methods are available for multiplicity adjustment.

### Usage

    sci.ratio(formula, data, type = "Dunnett", base = 1,
     method = "Plug", Num.Contrast = NULL, Den.Contrast = NULL,
     alternative = "two.sided", conf.level = 0.95, names=TRUE)

### Arguments

`formula` | A formula specifying a numerical response and a grouping factor as e.g. `response ~ treatment`
`data` | A dataframe containing the response and group variable
`type` | type of contrast, with the following options: "Dunnett": many-to-one comparisons, with the control group in the denominator "Tukey": all-pair comparisons "Sequen": comparison of consecutive groups, where the group with lower order is the denominator "AVE": comparison of each group with average of all others, where the average is taken as denominator "GrandMean": comparison of each group with grand mean of all groups, where the grand mean is taken as denominator "Changepoint": ratio of averages of groups of higher order divided by averages of groups of lower order "Marcus": Marcus contrasts as ratios "McDermott": McDermott contrasts as ratios "Williams": Williams contrasts as ratios "UmbrellaWilliams": Umbrella-protected Williams contrasts as ratios Note: `type` is ignored, if `Num.Contrast` and `Den.Contrast` are specified by the user (See below).
`base` | a single integer specifying the control (i.e. denominator) group for the Dunnett contrasts, ignored otherwise
`method` | character string specifying the method to be used for confidence interval construction: "Plug": Plug-in of ratio estimates in the correlation matrix of the multivariate t distribution. This method is the default. "Bonf": Simple Bonferroni-adjustment of Fieller confidence intervals for the ratios "MtI": Sidak or Slepian- adjustment for two-sided and one-sided confidence intervals, respectively "Unadj": Unadjusted Fieller confidence intervals for the ratios (i.e. with comparisonwise confidence level = conf.level)
`Num.Contrast` | Numerator contrast matrix, where columns correspond to groups and rows correspond to contrasts
`Den.Contrast` | Denominator contrast matrix, where columns correspond to groups and rows correspond to contrasts
`alternative` | a character string: "two.sided" for two-sided intervals, "less" for upper confidence limits, "greater" for lower confidence limits
`conf.level` | simultaneous confidence level in case of method="Plug","Bonf", or "MtI", and comparisonwise confidence level in case of method="Unadj"
`names` | logical, indicating whether rownames of the contrast matrices shall be retained in the output

### Details

Given a one-way ANOVA model, the interest is in simultaneous confidence intervals for several ratios of linear combinations of the treatment means. It is assumed that the responses are normally distributed with homogeneous variances. Unlike in multiple testing for ratios, the joint distribution of the likelihood ratio statistics has a multivariate t-distribution the correlation matrix of which depends on the unknown ratios. This means that the critical point needed for CI calculations also depends on the ratios. There are various methods of dealing with this problem (for example, see Dilba et al., 2006). The methods include (i) the unadjusted intervals (Fieller confidence intervals without multiplicity adjustments), (ii) Bonferroni (Fieller intervals with simple Bonferroni adjustments), (iii) MtI (a method based on Sidak and Slepian inequalities for two- and one-sided confidence intervals, respectively), and (iv) plug-in (plugging the maximum likelihood estimates of the ratios in the unknown correlation matrix). The latter method is known to have good simultaneous coverage probabilities. The MtI method consists of replacing the unknown correlation matrix of the multivariate t by an identity matrix of the same dimension.

See the examples for the usage of Numerator and Denominator contrasts. Note that the argument names Num.Contrast and Den.Contrast need to be specified. If numerator and denominator contrasts are plugged in without their argument names, they will not be recognized.

### Value

An object of class "sci.ratio", containing a list with elements:

`estimate` | point estimates of the ratios
`CorrMat.est` | estimate of the correlation matrix (for the plug-in approach)
`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)

Gemechis Dilba Djira

### References

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.

### See Also

`glht(multcomp)` for simultaneous CI of differences of means, plot.sci.ratio for a plotting function of the intervals

### Examples

    # # #

    # Antibiotic activity of 8 different strains of a micro organisms.
    # (Horn and Vollandt, 1995):

    data(Penicillin)

    boxplot(diameter~strain, data=Penicillin)

    allpairs<-sci.ratio(diameter~strain, data=Penicillin, type="Tukey")
    plot(allpairs)
    summary(allpairs)

    # Comparison to the grand mean of all strains:

    CGM<-sci.ratio(diameter~strain, data=Penicillin, type="GrandMean")
    plot(CGM)
    summary(CGM)

    # # #

    # 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)

    BWnoninf <- sci.ratio(Weight~Dose, data=BW, type="Dunnett",
     alternative="greater")

    plot(BWnoninf, rho0=0.9)

[Package mratios version 1.3.17 Index]