normDiffCI function

Confidence Intervals for Difference of Means

This function can be used to compute confidence intervals for difference of means assuming normal distributions.

normDiffCI(x, y, conf.level = 0.95, paired = FALSE, method = "welch", 
           boot = FALSE, R = 9999, bootci.type = "all", na.rm = TRUE, 
           alternative = c("two.sided", "less", "greater"), ...)
meanDiffCI(x, y, conf.level = 0.95, paired = FALSE, method = "welch", 
           boot = FALSE, R = 9999, bootci.type = "all", na.rm = TRUE, 
           alternative = c("two.sided", "less", "greater"), ...)

Arguments

• x: numeric vector of data values of group 1.
• y: numeric vector of data values of group 2.
• conf.level: confidence level.
• paired: a logical value indicating whether the two groups are paired.
• method: a character string specifing which method to use in the unpaired case; see details.
• boot: a logical value indicating whether bootstrapped confidence intervals shall be computed.
• R: number of bootstrap replicates.
• bootci.type: type of bootstrap interval; see boot.ci.
• na.rm: a logical value indicating whether NA values should be stripped before the computation proceeds.
• alternative: a character string specifying one- or two-sided confidence intervals. Must be one of "two.sided" (default), "greater" or "less" (one-sided intervals). You can specify just the initial letter.
• ...: further arguments passed to function boot, e.g. for parallel computing.

Details

The standard confidence intervals for the difference of means are computed that can be found in many textbooks, e.g. Chapter 4 in Altman et al. (2000).

The method "classical" assumes equal variances whereas methods "welch" and "hsu" allow for unequal variances. The latter two methods use different formulas for computing the degrees of freedom of the respective t-distribution providing the quantiles in the confidence interval. Instead of the Welch-Satterhwaite equation the method of Hsu uses the minimum of the group sample sizes minus 1; see Section 6.8.3 of Hedderich and Sachs (2018).

Returns

A list with class "confint" containing the following components: - estimate: point estimate (mean of differences or difference in means).

• conf.int: confidence interval.

• Infos: additional information.

References

D. Altman, D. Machin, T. Bryant, M. Gardner (eds). Statistics with Confidence: Confidence Intervals and Statistical Guidelines, 2nd edition. John Wiley and Sons 2000.

J. Hedderich, L. Sachs. Angewandte Statistik: Methodensammlung mit R. Springer 2018.

Author(s)

Matthias Kohl Matthias.Kohl@stamats.de

Examples

x <- rnorm(100)
y <- rnorm(100, sd = 2)
## paired
normDiffCI(x, y, paired = TRUE)
## (R = 999 to reduce computation time for R checks)
normDiffCI(x, y, paired = TRUE, boot = TRUE, R = 999)
## compare
normCI(x-y)
## (R = 999 to reduce computation time for R checks)
normCI(x-y, boot = TRUE, R = 999)

## unpaired
y <- rnorm(90, mean = 1, sd = 2)
## classical
normDiffCI(x, y, method = "classical")
## (R = 999 to reduce computation time for R checks)
normDiffCI(x, y, method = "classical", boot = TRUE, R = 999)
## Welch (default as in case of function t.test)
normDiffCI(x, y, method = "welch")
## (R = 999 to reduce computation time for R checks)
normDiffCI(x, y, method = "welch", boot = TRUE, R = 999)
## Hsu
normDiffCI(x, y, method = "hsu")
## In case of bootstrap there is no difference between welch and hsu
## (R = 999 to reduce computation time for R checks)
normDiffCI(x, y, method = "hsu", boot = TRUE, R = 999)

## one-sided
normDiffCI(x, y, alternative = "less")
normDiffCI(x, y, boot = TRUE, R = 999, alternative = "greater")

## parallel computing for bootstrap
normDiffCI(x, y, method = "welch", boot = TRUE, R = 9999, 
           parallel = "multicore", ncpus = 2)


## Monte-Carlo simulation: coverage probability
M <- 100 # increase for more stable/realistic results!
CIhsu <- CIwelch <- CIclass <- matrix(NA, nrow = M, ncol = 2)
for(i in 1:M){
  x <- rnorm(10)
  y <- rnorm(30, sd = 0.1)
  CIclass[i,] <- normDiffCI(x, y, method = "classical")$conf.int
  CIwelch[i,] <- normDiffCI(x, y, method = "welch")$conf.int
  CIhsu[i,] <- normDiffCI(x, y, method = "hsu")$conf.int
}
## coverage probabilies
## classical
sum(CIclass[,1] < 0 & 0 < CIclass[,2])/M
## Welch
sum(CIwelch[,1] < 0 & 0 < CIwelch[,2])/M
## Hsu
sum(CIhsu[,1] < 0 & 0 < CIhsu[,2])/M