etc.diff {ETC}

Simultaneous equivalence to control for differences of means

Description

Performs simultaneous equivalence tests and related confidence intervals for differences to control in a one-way layout.

Usage

etc.diff(formula, data, base = 1, margin.up = NULL, margin.lo = -margin.up,
         method = "var.unequal", FWER = 0.05)

Arguments

formula

a formula specifying a numerical response and a grouping factor (e.g., response ~ treatment)

data

a data frame containing the response and group variable as columns

base

a single integer specifying the control group

margin.up

a single numerical value or a numeric vector (of lenght equal to the number of comparisons) for absolute upper margins under the null hypotheses

margin.lo

a single numerical value or a numeric vector (of lenght equal to the number of comparisons) for absolute lower margins under the null hypotheses, set to -margin.up by default if not given

method

a character string:

• "var.unequal": possibly unequal group variances,

• "var.equal": equal group variances,

• "Bofinger": equal group variances and equal sample sizes for the non-control groups

• "non.par": non-normally distributed data

FWER

a single numeric value specifying the familywise error rate to be controlled by the simultaneous confidence intervals

Details

Having several treatment groups and a control in a one-way layout, the object is to simultaneously select those treatments being equivalent to the control. Bonferroni adjusted "two one-sided t-tests" (TOST) and related simultaneous confidence intervals are used for differences of means of normally distributed data with equal group variances (method="var.equal"). A pooled sample variance over all treatments is taken in this case. Welch-t-Tests are applied for unequal variances (method="var.unequal"). If the sample sizes of the treatment groups are balanced in the case of equal variances, the single-step procedure of Bofinger and Bofinger (1995) can be chosen with p-values and quantiles coming from a multivariate t-distribution (method="Bofinger"). A warning is given in the output if the Bofinger method is applied for unbalanced treatment groups. For non-normal data, tests based on wilcox.test(..., exact=FALSE, correct=TRUE, ...) are used (method="non.par").

Value

An object of class etc.diff containing:

estimate

a (named) vector of estimated differences

test.stat

a (named) vector of the calculated test statistics

degr.fr

either a single degree of freedom (method="var.equal" and method="Bofinger") or a (named) vector of degrees of freedom (method="var.unequal")

corr.mat

if method="Bofinger", the correlation matrix of the multivariate t-distribution

crit.value

either a single critical value (method="var.equal" and method="Bofinger") or a (named) vector of critical values (method="var.unequal")

p.value

a (named) vector of p-values adjusted for multiplicity

conf.int

a (named) matrix of simultaneous confidence intervals

Note

The confidence intervals related to the TOST method (method="var.equal", method="var.unequal", and method="non.par") have simultaneous coverage probability (1-2alpha), while the intervals according to Bofinger and Bofinger (1995) (method="Bofinger") have (1-alpha). All intervals are "expanded", see Bofinger (1985).

Author(s)

Mario Hasler

References

Hothorn, L.A. and Hasler, M. (2008): Proof of hazard and proof of safety in toxicological studies using simultaneous confidence intervals for differences and ratios to control, Journal of Biopharmaceutical Statistics 18, 915-933;

Bofinger, E. and Bofinger M. (1995): Equivalence with Respect to a Control: Stepwise Tests, Journal of the Royal Statistical Society B 57 (4), 721-733;

Bofinger, E. (1985): Expanded confidence intervals, Communications in Statistics - Theory and Methods 14 (8), 1849-1864

See Also

etc.rat

Examples

data(BW)

comp <- etc.diff(formula=Weight~Dose, data=BW, margin.up=30, method="Bofinger")
summary(comp)