DfSattRat {SimComp}

Degrees of Freedom Accoding to Satterthwaite (1946) for Ratios of Means

Description

Degrees of freedom accoding to Satterthwaite (1946) for (multivariate) t-distributions related to multiple contrast tests or corresponding simultaneous confidence intervals for ratios of means. For contrasts representing a two-sample t-test, the degree of freedom coincides with the one of Welch (1938).

Usage

DfSattRat(n, sd, type = "Dunnett", base = 1, Num.Contrast = NULL, Den.Contrast = NULL,
          Margin = 1)

Arguments

n

a vector of numbers of observations

sd

a vector of standard deviations

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

Margin

a single numeric value, or a numeric vector with length equal to the number of contrasts, default is 1

Details

The calculation of critical values or (adjusted) p-values related to multiple contrast tests or corresponding simultaneous confidence intervals is based on a multivariate t-distribution. For homoscedastic data, the respective degree of freedom only depends on the total sample size and the number of groups. A simple and well-known special case is the usual t-test. If the data are heteroscedastic, however, the degree of freedom of a t-test must be decreased according to Welch (1938) to come to an approximate solution. Degrees of freedom according to Satterthwaite (1946) refer to any linear combinations (contrasts) of normal means. They are applied, for example, when doing multiple contrast tests for heteroscedastic data according to Hasler and Hothorn (2008) <doi:10.1002/bimj.200710466> or Hasler (2014) <doi:10.1515/ijb-2012-0015>. Like Welch (1938), Satterthwaite (1946) approximated the degree of freedom by matching first and second moments. The resulting degree of freedom then depends on the numerator contrast, the denominator contrast, the (relative) margin to test against, and on the sample sizes and sample variances per group. If Margin=1 or Margin=NULL (default), the result coincides with the result of DfSattDiff().

Value

A vector of degrees of freedom.

Note

The commands SimTestRat() and SimCiRat() use these degrees of freedom automatically if covar.equal=FALSE (default). You don't need to apply DfSattRat() additionally.

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, <doi:10.1515/ijb-2012-0015>.

Hasler, M. and Hothorn, L.A. (2008): Multiple contrast tests in the presence of heteroscedasticity. Biometrical Journal 50, 793–800, <doi:10.1002/bimj.200710466>.

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

Welch, B.L. (1938): The significance of the difference between two means when the population variances are unequal. Biometrika 29, 350–362.

See Also

DfSattDiff

Examples

# Example 1:
# Degrees of freedom for a non-inferiority test of group two and three against group one,
# assuming unequal standard deviations for the groups. This is an extension for the well-
# known Dunnett-test to the case of heteroscedasticity and in terms of ratios of means
# instead of differences.

# Either by specifying the type of contrast:
DfSattRat(n=c(10,6,6), sd=c(1,3,6), type="Dunnett", base=1, Margin=0.8)

# Or by specifying the contrast matrices:
DfSattRat(n=c(10,6,6), sd=c(1,3,6), Num.Contrast=rbind(c(0,1,0),c(0,0,1)),
  Den.Contrast=rbind(c(1,0,0),c(1,0,0)), Margin=0.8)

# Example 2:
# Degrees of freedom for an all-pair comparison of the groups B, H and S on endpoint ADP,
# assuming unequal standard deviations for the groups. This is an extension for the well-
# known Tukey-test to the case of heteroscedasticity and in terms of ratios of means
# instead of differences. The same degrees of freedom are used automatically by command 
# \code{SimTestRat()}.

data(coagulation)

DfSattRat(n=tapply(X=coagulation$ADP, INDEX=coagulation$Group, FUN=length),
  sd=tapply(X=coagulation$ADP, INDEX=coagulation$Group, FUN=sd),
  type="Tukey")