farrington.manning {DescrTab2}

Farrington-Manning test for rate difference

Description

The Farrington-Manning test for rate differences can be used to compare the rate difference of successes between two groups to a preset value. It uses an explicit formula for the standard deviation of the test statistic under the null hypothesis [1].

Usage

farrington.manning(
  group1,
  group2,
  delta = 0,
  alternative = "greater",
  alpha = 0.025
)

Arguments

group1

a logical vector of data from group 1, where TRUE indicates a success

group2

a logical vector of data from group 2, where TRUE indicates a success

delta

the rate difference under the null hypothesis

alternative

character string indicating the alternative to use, either of "two.sided", "less", "greater"

alpha

the significance level (acceptable error of the first kind), a two-sided confidence intnerval is returned with confidence level 1 - 2*alpha, such that the lower bound is a valid one sided confidence interval at the confidence level 1 - alpha.

Details

The Farrington-Maning test for rate differences test the null hypothesis of

H_{0}: p_{1} - p_{2} = \delta

for the "two.sided" alternative (or \geq for the "greater" respectively \leq for the "less" alternative). This formulation allows to specify non-inferiority and superiority test in a consistent manner:

non-inferiority

for delta < 0 and alternative == "greater" the null hypothesis reads H_{0}: p_{1} - p_{2} \geq \delta and consequently rejection allows concluding that p_1 \geq p_2 + \delta i.e. that the rate of success in group one is at least the success rate in group two plus delta - as delta is negagtive this is equivalent to the success rate of group 1 being at worst |delta| smaller than that of group 2.

superiority

for delta >= 0 and alternative == "greater" the null hypothesis reads H_{0}: p_{1} - p_{2} \geq \delta and consequently rejection allows concluding that p_1 \geq p_2 + \delta i.e. that the rate of success in group one is at least delta greater than the success rate in group two.

The confidence interval is always computed as two-sided, but with 1-2\alpha confidence level in case of a one-sided hypthesis. This means that the lower or upper vound are valid one-sided confidence bounds at level \alpha in this case. The confidence interval is constructed by inverting the two-sided test directly.

Value

A list of class "htest" containing the following components:

statistic:	the value of the Z-statistic
parameter:	delta, rate difference (group 1 - group 2) under the null hypothesis
p.value:	the p-value for the Farrington-Manning test
null.value:	rate difference (group 1 - group 2) under the null
alternative:	a character string indicating the alternative hypothesis
method:	a character string indicating the exact method employed
data.name:	a character string giving the names of the data used
estimate:	the estimated rate difference (maximum likelihood)
conf.int:	a confidence interval for the rate difference
sample.size:	the total sample size used for the test

Author(s)

Kevin Kunzmann

References

[1] Farrington, Conor P., and Godfrey Manning. "Test statistics and sample size formulae for comparative binomial trials with null hypothesis of non-zero risk difference or non-unity relative risk." Statistics in medicine 9.12 (1990): 1447-1454.

Examples

x <- c(rep(TRUE, 20), rep(FALSE, 15))
y <- c(rep(TRUE, 30), rep(FALSE, 25))

farrington.manning(x, y, -.3)