abcnonHtest function

Nonparametric ABC (Approximate Bootstrap Confidence) intervals.

A hypothesis testing function using the nonparametric ABC intervals.

abcnonHtest(x, tt, nullValue = NULL, conf.level = 0.95, 
   alternative = c("two.sided", "less", "greater"), epsilon = 0.001, minp = 0.001)

Arguments

• x: the data. Must be either a vector, or a matrix whose rows are the observations

• tt: function defining the parameter in the resampling form tt(p,x), where p is the vector of proportions and x

  is the data

• nullValue: null value of the parameter for the two-sided hypothesis test, or boundary of null parameter space for one-sided ones

• conf.level: confidence level for interval

• alternative: a character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater" or "less". You can specify just the initial letter.

• epsilon: optional argument specifying step size for finite difference calculations

• minp: minimum p-value (used in uniroot search to give a bound, toe two.sided alternatives actual minimum is 2*minp)

Details

Calculates the nonparametric ABC confidence interval of DiCiccio and Efron (1992). See also Efron and Tibshirani (1993).

The p-values are calculated by solving for confidence limit that just touches the nullValue. If it is outside of the range (minp, 1-minp) for one-sided p-values, then it is set to minp. If it is outside the range (2minp, 1- 2minp) for two-sided p-values, then it is set to 2*minp.

Returns

A value of class "htest" containing the following components: - p.value: p-value for test defined by alternative and nullValue

• estimate: estimate of the parameter, calculated using x and the tt function

• conf.int: confidence interval for the parameter associated with tt

• null.value: the nullValue (or null boundary) for the hypothesis test

• alternative: a character string describing the alternative hypothesis

• method: a character string describing the kind of test

• data.name: a character string giving the name of the data and the function

References

DiCiccio, T and Efron, B (1992). More accurate confidence intervals in exponential families. Biometrika 79: 231-245.

Efron, B and Tibshirani, RJ (1993). An introduction to the bootstrap. Chapman and Hall: New York.

Author(s)

the function is modification of abcnon in the bootstrap R package, originally written by Rob Tibshirani, modifications by M.P. Fay

See Also

See also abcnon.

Examples

# compute abc intervals for the mean
x <- c(2,4,12,4,6,3,5,7,6)
theta <- function(p,x) {sum(p*x)/sum(p)}
## smallest p-value is 2*minp for two-sided alternatives
abcnonHtest(x, theta, nullValue=0)  
## test null at 95% confidence limit is like just barely
## rejecting at the two-sided 5% level, so p-value is 0.05
abcnonHtest(x, theta, nullValue=4.072772)  
# compute abc intervals for the correlation
set.seed(1)
x <- matrix(rnorm(20),ncol=2)
theta <- function(p, x)
{
    x1m <- sum(p * x[, 1])/sum(p)
    x2m <- sum(p * x[, 2])/sum(p)
    num <- sum(p * (x[, 1] - x1m) * (x[, 2] - x2m))
    den <- sqrt(sum(p * (x[, 1] - x1m)^2) *
              sum(p * (x[, 2] - x2m)^2))
    return(num/den)
}
abcnonHtest(x, theta) 
## compare with 
## Not run:

library(bootstrap)
abcnon(x, theta, alpha=c(.025,.975))$limits[,"abc"]
## End(Not run)