binomMeld.test {exact2x2}

Melded Binomial Confidence Intervals and Tests

Description

Creates tests to compare two binomials, giving confidence intervals for either the difference in proportions, the rate ratio, or the odds ratio. The 95 percent confidence intervals have been shown to guarantee nominal coverage by extensive numerical calculations. It has been theoretically proven that the p-values from the one-sided tests on the null hypothesis of equality match Fisher's exact p-values.

Usage

binomMeld.test(x1, n1, x2, n2, nullparm = NULL, 
    parmtype = c("difference", "oddsratio", "ratio"), 
    conf.level = 0.95, conf.int=TRUE, 
    alternative = c("two.sided", "less", "greater"), 
    midp=FALSE, nmc=0, eps=10^-8)

Arguments

x1

number of events in group 1

n1

sample size in group 1

x2

number of events in group 2

n2

sample size in group 2

nullparm

value of the parameter of interest at null, default of NULL gives 0 for parmtype='difference' and 1 for parmtype='ratio' or 'oddsratio'

parmtype

type of parameter of interest, one of "difference", "ratio" or "oddsratio" (see details)

conf.level

confidence level

conf.int

logical, calculate confidence intervals?

alternative

alternative hypothesis, one of "two.sided", "less", or "greater" (see details)

midp

logical, do mid-p version of p-value and confidence intervals?

nmc

integer, number of Monte Carlo replications for p-value and CI calculations, 0 (default) means calculate by numeric integration instead

eps

small number used to adjust numeric integration (see note)

Details

Assume X1~ Binomial(n1,p1) and X2~Binomial(n2,p2). We want to test hypotheses on a function of p1 and p2. The functions are given by parmtype: difference tests p2-p1, ratio tests p2/p1, and odds ratio tests p2(1-p1)/(p1(1-p2)). Let g(p1,p2) be one of the three functions. So when alternative is "less" we test H0: g(p1,p2) >= nullparm vs. H1: g(p1,p2)<nullparm.

For details when midp=FALSE see Fay, Proschan, and Brittain (2015).

When midp=TRUE, the method performs the mid-p version on the p-value and the associated confidence intervals. This means that we replace the confidence distribution random variables in the p-value and CI calculations with a random variable that is a mixture of the lower and upper CD random variables. For example, if W1L and W1U are the lower and upper confidence distribution random variables for group 1, then we replace those values in all calculations with W1midp = U1*W1L + (1-U1)*W1U, where U1 is a Bernoulli with parameter 0.5. For a discussion of mid-p p-values and the associated confidence intervals in a closely related context, see the vignette on mid p-values or Fay and Brittain (2016, especially the Appendix).

Value

An object of class 'htest'. A list with elements

statistic

proportion of events in group 1

parameter

proportion of events in group 2

p.value

p-value

conf.int

confidence interval

estimate

estimate of g(p1,p2) by plugging in sample proportions, i.e., unconditional MLE

null.value

value of g(p1,p2) under null

alternative

type of alternative hypothesis

method

description of test

data.name

character explicit description of data

Note

For numeric integration, the integrate function may have problems if nearly all of the integrand values are about 0 within the range of integration. Because of this, we use the eps value to make sure we integrate over ranges in which the integrand is nontrivially greater than 0. We restrict the range then add eps back to the p-value so that if the integrate function works perfectly, then the p-values would be very slightly conservative (for very small eps). There is no need to adjust the eps value. See code for detailed description of how eps is used in the calculation before changing it from the default.

An alternative method of calculation is to use Monte Carlo simulation (option with nmc>0). This provides a check of the numeric integration. There is no need to do Monte Carlo simulations for routine use. Please inform the package maintainer if the p-values or confidence intervals are substantially different when nmc=0 and nmc=10^7.

Author(s)

Michael P. Fay

References

Fay, MP, Proschan, MA, and Brittain, E (2015) Combining One Sample Confidence Procedures for Inferences in the Two Sample Case. Biometrics 71: 146-156.

Fay, Michael P., and Erica H. Brittain. (2016). Finite sample pointwise confidence intervals for a survival distribution with right-censored data. Statistics in medicine. 35: 2726-2740.

Examples

# Note the p-value for all tests of equality 
# (Null Hypthesis: true prop 1=true prop 2)
# are the same, and equal to the 
# Fisher's exact (central) p-value
binomMeld.test(3,5,1,8,parmtype="difference")
binomMeld.test(3,5,1,8,parmtype="ratio")
# note that binomMeld.test gives the unconditional MLE 
# for the odds ratio, while fisher.test and exact2x2 
# gives the conditional MLE for the odds ratio
# (also fisher.test gives the odds ratio defined as 
#  the inverse of how it is defined in binomMeld.test)
binomMeld.test(3,5,1,8,parmtype="oddsratio")
exact2x2(matrix(c(1,8-1,3,5-3),2,2),tsmethod="central")