# Simultaneous Confidence Intervals for General Contrasts (Ratios) of Means of Multiple Endpoints Simultaneous confidence intervals for general contrasts (linear functions) of normal means (e.g., "Dunnett", "Tukey", "Williams" ect.), and for single or multiple endpoints (primary response variables) simultaneously. The procedure of Hasler and Hothorn (2012) is applied for ratios of means of normally distributed data. The variances/ covariance matrices of the treatment groups (containing the covariances between the endpoints) may be assumed to be equal or possibly unequal for the different groups (Hasler, 2014 ). For the case of only a single endpoint and unequal covariance matrices (variances), the procedure coincides with the PI procedure of Hasler and Hothorn (2008) . ```r ## Default S3 method: SimCiRat(data, grp, resp = NULL, na.action = "na.error", type = "Dunnett", base = 1, Num.Contrast = NULL, Den.Contrast = NULL, alternative = "two.sided", covar.equal = FALSE, conf.level = 0.95, CorrMatDat = NULL, ...) ## S3 method for class 'formula' SimCiRat(formula, ...) ``` ## Arguments - `data`: a data frame containing a grouping variable and the endpoints as columns - `grp`: a character string with the name of the grouping variable - `resp`: a vector of character strings with the names of the endpoints; if `resp=NULL` (default), all column names of the data frame without the grouping variable are chosen automatically - `formula`: a formula specifying a numerical response and a grouping factor (e.g. response ~ treatment ) - `na.action`: a character string indicating what should happen when the data contain `NAs`; if `na.action="na.error"` (default) the procedure stops with an error message; if `na.action="multi.df"` a new experimental version is used (details will follow soon) - `type`: a character string, defining the type of contrast, with the following options: * "Dunnett": many-to-one comparisons * "Tukey": all-pair comparisons * "Sequen": comparisons of consecutive groups * "AVE": comparison of each group with average of all others * "GrandMean": comparison of each group with grand mean of all groups * "Changepoint": differences of averages of groups of higher order to averages of groups of lower order * "Marcus": Marcus contrasts * "McDermott": McDermott contrasts * "Williams": Williams trend tests * "UmbrellaWilliams": Umbrella-protected Williams trend tests note that `type` is ignored if `Num.Contrast` or `Den.Contrast` is specified by the user (see below) - `base`: a single integer specifying the control group for Dunnett contrasts, ignored otherwise - `Num.Contrast`: a numerator contrast matrix, where columns correspond to groups and rows correspond to contrasts - `Den.Contrast`: a denominator contrast matrix, where columns correspond to groups and rows correspond to contrasts - `alternative`: a character string specifying the alternative hypothesis, must be one of `"two.sided"` (default), `"greater"` or `"less"` - `covar.equal`: a logical variable indicating whether to treat the variances/ covariance matrices of the treatment groups (containing the covariances between the endpoints) as being equal; if `TRUE` then the pooled variance/ covariance matrix is used, otherwise the Satterthwaite approximation to the degrees of freedom is used - `conf.level`: a numeric value defining the simultaneous confidence level - `CorrMatDat`: a correlation matrix of the endpoints, if `NULL` (default) it is estimated from the data - ``...``: arguments to be passed to SimCiRat.default ## Details The interest is in simultaneous confidence intervals for several linear combinations (contrasts) of treatment means in a one-way ANOVA model, and for single or multiple endpoints simultaneously. For example, corresponding intervals for the all- pair comparison of Tukey (1953) and the many-to-one comparison of Dunnett (1955) are implemented, but allowing for heteroscedasticity and multiple endpoints, and in terms of ratios of means. The user is also free to create other interesting problem-specific contrasts. Approximate multivariate **t**-distributions are used to calculate lower and upper limits (Hasler and Hothorn, 2012 ). Simultaneous tests based on these intervals control the familywise error rate in admissible ranges and in the strong sense. The variances/ covariance matrices of the treatment groups (containing the covariances between the endpoints) can be assumed to be equal (`covar.equal=TRUE`) or unequal (`covar.equal=FALSE`). If being equal, the pooled variance/ covariance matrix is used, otherwise approximations to the degrees of freedom (Satterthwaite, 1946) are used (Hasler, 2014 ; Hasler and Hothorn, 2008 ). Unequal covariance matrices occure if variances or correlations of some endpoints differ depending on the treatment groups. ## Returns An object of class SimCi containing: - **estimate**: a matrix of estimated ratios - **lower.raw**: a matrix of raw (unadjusted) lower limits - **upper.raw**: a matrix of raw (unadjusted) upper limits - **lower**: a matrix of lower limits adjusted for multiplicity - **upper**: a matrix of upper limits adjusted for multiplicity - **CorrMatDat**: if not prespecified by `CorrMatDat`, either the estimated common correlation matrix of the endpoints (`covar.equal=TRUE`) or a list of different (one for each treatment) estimated correlation matrices of the endpoints (`covar.equal=FALSE`) - **CorrMatComp**: the estimated correlation matrix of the comparisons - **degr.fr**: a matrix of degrees of freedom ## Note By default (`na.action="na.error"`), the procedure stops if there are missing values. A new experimental version for missing values is used if `na.action="multi.df"`. If `covar.equal=TRUE`, the number of endpoints must not be greater than the total sample size minus the number of treatment groups. If `covar.equal=FALSE`, the number of endpoints must not be greater than the minimal sample size minus 1. Otherwise the procedure stops. All intervals have the same direction for all comparisons and endpoints (`alternative="..."`). In case of doubt, use `"two.sided"`. The correlation matrix for the multivariate **t**-distribution also depends on the unknown ratios. The same problem also arises for the degrees of freedom if the covariance matrices for the different groups are assumed to be unequal (`covar.equal=FALSE`). Both problems are handled by a plug-in approach, see the references therefore. ## References Hasler, M. (2014): Multiple contrast tests for multiple endpoints in the presence of heteroscedasticity. **The International Journal of Biostatistics** 10, 17--28, . Hasler, M. and Hothorn, L.A. (2012): A multivariate Williams-type trend procedure. **Statistics in Biopharmaceutical Research** 4, 57--65, . Hasler, M. and Hothorn, L.A. (2008): Multiple contrast tests in the presence of heteroscedasticity. **Biometrical Journal** 50, 793--800, . Dilba, G. et al. (2006): Simultaneous confidence sets and confidence intervals for multiple ratios. **Journal of Statistical Planning and Inference** 136, 2640--2658, . ## Author(s) Mario Hasler ## See Also `SimTestRat`, `SimTestDiff`, `SimCiDiff` ## Examples ```r # Example 1: # Simultaneous confidence intervals related to a comparison of the groups # B and H against the standard S, for endpoint Thromb.count, assuming unequal # variances for the groups. This is an extension of the well-known Dunnett- # intervals to the case of heteroscedasticity and in terms of ratios of means # instead of differences. data(coagulation) interv1 <- SimCiRat(data=coagulation, grp="Group", resp="Thromb.count", type="Dunnett", base=3, alternative="greater", covar.equal=FALSE) interv1 plot(interv1) # Example 2: # Simultaneous confidence intervals related to a comparisons of the groups # B and H against the standard S, simultaneously for all endpoints, assuming # unequal covariance matrices for the groups. This is an extension of the well- # known Dunnett-intervals to the case of heteroscedasticity and multiple # endpoints and in terms of ratios of means instead of differences. data(coagulation) interv2 <- SimCiRat(data=coagulation, grp="Group", resp=c("Thromb.count","ADP","TRAP"), type="Dunnett", base=3, alternative="greater", covar.equal=FALSE) summary(interv2) plot(interv2) ```

SimCiRat function