# Confidence Intervals for Difference of Means This function can be used to compute confidence intervals for difference of means assuming normal distributions. ```r 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 ```r 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 ```

normDiffCI function