R: Simultaneous tests for ratios of normal means

simtest.ratio {mratios} | R Documentation

## Simultaneous tests for ratios of normal means

### Description

Performs simultaneous tests for several ratios of linear combinations of treatment means in the normal one-way ANOVA model with homogeneous variances.

### Usage

    simtest.ratio(formula, data, type = "Dunnett", base = 1,
     alternative = "two.sided", Margin.vec = NULL, FWER = 0.05,
     Num.Contrast = NULL, Den.Contrast = NULL, names = TRUE)

### Arguments

`formula` | A formula specifying a numerical response and a grouping factor (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 control in the denominator "Tukey": all-pair comparisons "Sequen": comparison of consecutive groups, where the group with lower order is the denomniator "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
`alternative` | a character string: "two.sided": for two-sided tests "less": for lower tail tests "greater": for upper tail tests
`Margin.vec` | a single numerical value or vector of Margins under the null hypotheses, default is 1
`FWER` | a single numeric value specifying the family-wise error rate to be controlled
`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
`names` | a logical value: if TRUE, the output will be named according to names of user defined contrast or factor levels

### Details

Given a one-way ANOVA model, the interest is in simultaneous tests for several ratios of linear combinations of the treatment means. Let us denote the ratios by γ_i, i=1,...,r, and let ψ_i, i=1,...,r, denote the relative margins against which we compare the ratios. For example, upper-tail simultaneous tests for the ratios are stated as

H_0i: γ_i <= ψ_i

versus

H_1i: γ_i > ψ_i, i=1,...,r

.

The associated likelihood ratio test statistic T_i has a t-distribution. For multiplicity adjustments, we use the joint distribution of the T_i , i=1,...,r, which under the null hypotheses follows a central r-variate t-distribution. Adjusted p-values can be calculated by adapting the results of Westfall et al. (1999) for ratio formatted hypotheses.

### Value

An object of class simtest.ratio containing:

`estimate` | a (named) vector of estimated ratios
`teststat` | a (named) vector of the calculated test statistics
`Num.Contrast` | the numerator contrast matrix
`Den.Contrast` | the denominator contrast matrix
`CorrMat` | the correlation matrix of the multivariate t-distribution calculated under the null hypotheses
`critical.pt` | the equicoordinate critical value of the multi-variate t-distribution for a specified FWER
`p.value.raw` | a (named) vector of unadjusted p-values
`p.value.adj` | a (named) vector of p-values adjusted for multiplicity
`Margin.vec` | the vector of margins under the null hypotheses

and some other input arguments.

### 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.

Westfall, P.H., Tobias, R.D., Rom, D., Wolfinger, R.D., and Hochberg, Y. (1999). Multiple comparisons and multiple tests using the SAS system. SAS Institute Inc. Cary, NC, 65-81.

### See Also

While print.simtest.ratio produces a small default print-out of the results,

summary.simtest.ratio can be used to produce a more detailed print-out, which is recommended if user-defined contrasts are used,

sci.ratio for constructing simultaneous confidence intervals for ratios in oneway layout

See `summary.glht(multcomp)` for multiple tests for parameters of lm, glm.

### Examples

    library(mratios)

    ########################################################

    # User-defined contrasts for comparisons
    # between Active control, Placebo and three dosage groups:

    data(AP)
    AP
    boxplot(prepost~treatment, data=AP)

    # Test whether the differences of doses 50, 100, 150 vs. Placebo
    # are non-inferior to the difference of Active control vs. Placebo

    # User-defined contrasts:

    # Numerator Contrasts:

    NC <- rbind(
    "(D100-D0)" = c(0,-1,1,0,0),
    "(D150-D0)" = c(0,-1,0,1,0),
     "(D50-D0)" = c(0,-1,0,0,1))

    # Denominator Contrasts:

    DC <- rbind(
    "(AC-D0)" = c(1,-1,0,0,0),
    "(AC-D0)" = c(1,-1,0,0,0),
    "(AC-D0)" = c(1,-1,0,0,0))

    NC
    DC

    noninf <- simtest.ratio(prepost ~ treatment, data=AP,
     Num.Contrast=NC, Den.Contrast=DC, Margin.vec=c(0.9,0.9,0.9),
     alternative="greater")

    summary( noninf )

    #########################################################

    # Some more examples on standard multiple comparison procedures
    # stated in terms of ratio hypotheses:

    # Comparisons vs. Control:

    many21 <- simtest.ratio(prepost ~ treatment, data=AP,
     type="Dunnett")

    summary(many21)

    # Let the Placebo be the control group, which is the second level
    # in alpha-numeric order. A simultaneous test for superiority of
    # the three doses and the Active control vs. Placebo could be
    # done as:

    many21P <- simtest.ratio(prepost ~ treatment, data=AP,
     type="Dunnett", base=2, alternative="greater", Margin.vec=1.1)
    summary(many21P)

    # All pairwise comparisons:

    allpairs <- simtest.ratio(prepost ~ treatment, data=AP,
     type="Tukey")

    summary(allpairs)

    #######################################################

    # Comparison to grand mean of all strains
    # in the Penicillin example:

    data(Penicillin)

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

[Package mratios version 1.3.17 Index]