DfSattDiff {SimComp}

Degrees of Freedom Accoding to Satterthwaite (1946) for Differences 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 differences of means. For contrasts representing a two-sample t-test, the degree of freedom coincides with the one of Welch (1938).

Usage

DfSattDiff(n, sd, type = "Dunnett", base = 1, ContrastMat = NULL)

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 ContrastMat is specified by the user (see below)

base

a single integer specifying the control group for Dunnett contrasts, ignored otherwise

ContrastMat

a contrast matrix, where columns correspond to groups and rows correspond to contrasts

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 contrast and on the sample sizes and sample variances per group.

Value

A vector of degrees of freedom.

Note

The commands SimTestDiff() and SimCiDiff() use these degrees of freedom automatically if covar.equal=FALSE (default). You don't need to apply DfSattDiff() 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

DfSattRat

Examples

# Example 1:
# Degrees of freedom for a comparison 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.

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

# Or by specifying the contrast matrix:
DfSattDiff(n=c(10,6,6), sd=c(1,3,6), ContrastMat=rbind(c(-1,1,0),c(-1,0,1)))

# 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. The same degrees of freedom are 
# used automatically by command \code{SimTestDiff()}.

data(coagulation)

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