Simultaneous Confidence Intervals for Ratios of Means of Multiple Endpoints

Description

Simultaneous confidence intervals for ratios of contrasts (linear functions) of normal means (e.g., "Dunnett", "Tukey", "Williams" ect.) when there is more than one primary response variable (endpoint). The procedure of Hasler and Hothorn (2012) is applied for ratios of means of normally distributed data. The covariance matrices (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).

Usage

SimCiRat(data, grp, resp = NULL, type = "Dunnett", base = 1, Num.Contrast = NULL,
         Den.Contrast = NULL, alternative = "two.sided", covar.equal = FALSE,
         conf.level = 0.95)

Arguments

Details

The interest is in simultaneous confidence intervals for several ratios of linear combinations (contrasts) of treatment means in a one-way ANOVA model, and simultaneously for multiple endpoints. For example, corresponding intervals for the all-pair comparison of Tukey (1953) and the many-to-one comparison of Dunnett (1955) for ratios of means are implemented, but allowing for multiple endpoints. Also, the user is free to create other interesting problem-specific contrasts. An approximate multivariate t-distribution is used to calculate lower and upper limits (see Hasler and Hothorn, 2012). Simultaneous tests based on these intervals control the familywise error rate in an admissible range and in the strong sense. The 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 covariance matrix is used, otherwise approximations to the degrees of freedom (Satterthwaite, 1946) are used (see Hasler, 2014). Unequal covariance matrices occure if variances or correlations of some endpoints differ depending on the treatment groups.

Value

An object of class SimCi containing:

Note

All measurement objects of each treatment group must have values for each endpoint. If there are missing values then the procedure stops. If covar.equal=TRUE, then 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".

In contrast to simultaneous confidence intervals for differences, the correlation matrix for the multivariate t-distribution 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.

Author(s)

Mario Hasler

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.

Satterthwaite, F.E. (1946): An approximate distribution of estimates of variance components. Biometrics 2, 110–114.

See Also

SimCiDiff, SimTestRat, SimTestDiff

Examples

# Example 1:
# Simultaneous confidence intervals for ratios of means, related to a
# comparison of the groups B and H against the standard S, on 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

# Example 2:
# Simultaneous confidence intervals for ratios of means, related to a
# comparison of the groups B and H against the standard S, simultaneously on
# 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)

Results

R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
Copyright (C) 2016 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)

R is free software and comes with ABSOLUTELY NO WARRANTY.
You are welcome to redistribute it under certain conditions.
Type 'license()' or 'licence()' for distribution details.

R is a collaborative project with many contributors.
Type 'contributors()' for more information and
'citation()' on how to cite R or R packages in publications.

Type 'demo()' for some demos, 'help()' for on-line help, or
'help.start()' for an HTML browser interface to help.
Type 'q()' to quit R.

> library(SimComp)
> png(filename="/home/ddbj/snapshot/RGM3/R_CC/result/SimComp/SimCiRat.Rd_%03d_medium.png", width=480, height=480)
> ### Name: SimCiRat
> ### Title: Simultaneous Confidence Intervals for Ratios of Means of
> ###   Multiple Endpoints
> ### Aliases: SimCiRat
> ### Keywords: htest
> 
> ### ** Examples
> 
> # Example 1:
> # Simultaneous confidence intervals for ratios of means, related to a
> # comparison of the groups B and H against the standard S, on 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
 
Simultaneous 95% confidence intervals for ratios of means of multiple endpoints 
Assumption: Heterogeneous covariance matrices for the groups 
 
  comparison     endpoint estimate lower.raw upper.raw  lower upper
1        B/S Thromb.count    1.139    0.9609       Inf 0.9287   Inf
2        H/S Thromb.count    1.050    0.8533       Inf 0.8175   Inf
 
> 
> # Example 2:
> # Simultaneous confidence intervals for ratios of means, related to a
> # comparison of the groups B and H against the standard S, simultaneously on
> # 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)
 
Numerator contrast matrix: 
    B H S
B/S 1 0 0
H/S 0 1 0
 
Denominator contrast matrix: 
    B H S
B/S 0 0 1
H/S 0 0 1
 
Estimated covariance matrices of the data: 
$B
             Thromb.count    ADP    TRAP
Thromb.count       0.0626 0.0565 -0.0102
ADP                0.0565 0.0638  0.0054
TRAP              -0.0102 0.0054  0.0963

$H
             Thromb.count    ADP   TRAP
Thromb.count       0.0943 0.0637 0.0663
ADP                0.0637 0.0518 0.0446
TRAP               0.0663 0.0446 0.1157

$S
             Thromb.count    ADP   TRAP
Thromb.count       0.0318 0.0132 0.0598
ADP                0.0132 0.0079 0.0269
TRAP               0.0598 0.0269 0.1376

 
Estimated correlation matrices of the data: 
$B
             Thromb.count    ADP    TRAP
Thromb.count       1.0000 0.8937 -0.1314
ADP                0.8937 1.0000  0.0687
TRAP              -0.1314 0.0687  1.0000

$H
             Thromb.count    ADP   TRAP
Thromb.count       1.0000 0.9121 0.6348
ADP                0.9121 1.0000 0.5770
TRAP               0.6348 0.5770 1.0000

$S
             Thromb.count    ADP   TRAP
Thromb.count       1.0000 0.8338 0.9033
ADP                0.8338 1.0000 0.8161
TRAP               0.9033 0.8161 1.0000

 
Estimated correlation matrix of the comparisons: 
       [,1]   [,2]   [,3]   [,4]   [,5]   [,6]
[1,] 1.0000 0.8494 0.3781 0.3198 0.2025 0.4258
[2,] 0.8494 1.0000 0.2918 0.1697 0.1545 0.2448
[3,] 0.3781 0.2918 1.0000 0.3740 0.2566 0.6102
[4,] 0.3198 0.1697 0.3740 1.0000 0.8869 0.7084
[5,] 0.2025 0.1545 0.2566 0.8869 1.0000 0.5874
[6,] 0.4258 0.2448 0.6102 0.7084 0.5874 1.0000
 
  comparison     endpoint estimate lower.raw upper.raw  lower upper
1        B/S Thromb.count    1.139    0.9609       Inf 0.8848   Inf
2        B/S          ADP    1.262    1.0853       Inf 1.0116   Inf
3        B/S         TRAP    1.145    0.8370       Inf 0.7231   Inf
4        H/S Thromb.count    1.050    0.8533       Inf 0.7712   Inf
5        H/S          ADP    1.104    0.9522       Inf 0.8893   Inf
6        H/S         TRAP    1.098    0.7907       Inf 0.6771   Inf
 
> 
> 
> 
> 
> 
> dev.off()
null device 
          1 
>