Dear R Team!
There was an ongoing debate on Research Gate about the “Welch” option in your base R t.test command. A user noticed that the correction of the degrees of freedom labeled as “Welch Two Sample t-test”, if you choose var.equal = TRUE in the R t.test command, differs from the output of the Stata analysis, which is also labeled as “Welch's degrees of freedom”. Confusingly enough, the R output coincided with the Stata result labeled as “Satterthwaite's degrees of freedom”. Unfortunately, the R documentation wasn’t clear either, since it lacks any specific reference and the formulation is ambiguous: “If TRUE then the pooled variance is used to estimate the variance otherwise the Welch (or Satterthwaite) approximation to the degrees of freedom is used." It rather sounds as if both options are available and not that both authors proposed the same correction separately. After doing some research and looking into the R code, we found a solution and would like to suggest an update to the R documentation, to make it more clear (you can find the similar proposal to the Stata list here: https://www.statalist.org/forums/forum/general-stata-discussion/general/1734987-unequal-vs-welch-options-for-ttest-why-no-mention-of-welch-1938-in-the-documentation ) What is called “Welch Two Sample t-test” in the t.test command refers to two publications (see links below) with the same correction, namely Welch (1938) and Satterthwaite (1946). Hence, you also find "Welch–Satterthwaite" correction as a description in the literature for this (which is the aforementioned “Satterthwaite's degrees of freedom” correction in Stata). But there is also another correction proposed by Welch (1947), which has slightly different denominators (see code below), which is called “Welch's degrees of freedom” in Stata. This option is not available in R so far. Therefore, we suggest a) to cite the appropriate references in the documentation (at least Welch (1938) and Satterthwaite (1946)), b) adapt the output to something like “Welch-Satterthwaite adjusted Two Sample t-test” and maybe c) to incorporate the third option for the Welch (1947) adjustment, where the Welch-Satterthwaite correction should be the default option (Aspin & Welch, 1949). Code proposal below for the df correction. Best wishes, Rainer Düsing 1. · https://www.jstor.org/stable/2332010 <https://www.researchgate.net/deref/https%3A%2F%2Fwww.jstor.org%2Fstable%2F2332010?_tp=eyJjb250ZXh0Ijp7ImZpcnN0UGFnZSI6InF1ZXN0aW9uIiwicGFnZSI6InF1ZXN0aW9uIiwicG9zaXRpb24iOiJwYWdlQ29udGVudCJ9fQ> (Welch, 1938) 2. · https://www.jstor.org/stable/3002019 <https://www.researchgate.net/deref/https%3A%2F%2Fwww.jstor.org%2Fstable%2F3002019?_tp=eyJjb250ZXh0Ijp7ImZpcnN0UGFnZSI6InF1ZXN0aW9uIiwicGFnZSI6InF1ZXN0aW9uIiwicG9zaXRpb24iOiJwYWdlQ29udGVudCJ9fQ> (Satterthwaite, 1946) 3. · https://www.jstor.org/stable/2332510 <https://www.researchgate.net/deref/https%3A%2F%2Fwww.jstor.org%2Fstable%2F2332510?_tp=eyJjb250ZXh0Ijp7ImZpcnN0UGFnZSI6InF1ZXN0aW9uIiwicGFnZSI6InF1ZXN0aW9uIiwicG9zaXRpb24iOiJwYWdlQ29udGVudCJ9fQ> (Welch, 1947) 4. · Aspin, Alice A., and B. L. Welch. “Tables for Use in Comparisons Whose Accuracy Involves Two Variances, Separately Estimated.” *Biometrika* 36, no. 3/4 (1949): 290–96. https://doi.org/10.2307/2332668 <https://www.researchgate.net/deref/https%3A%2F%2Fdoi.org%2F10.2307%2F2332668?_tp=eyJjb250ZXh0Ijp7ImZpcnN0UGFnZSI6InF1ZXN0aW9uIiwicGFnZSI6InF1ZXN0aW9uIiwicG9zaXRpb24iOiJwYWdlQ29udGVudCJ9fQ>. [see point 4 in the Appendix by Welch] var.equal = "yes" var.equal = "Welch" var.equal = "W-S" vx <- var(x) nx <- length(x) vy <- var(y) ny <- length(y) if (var.equal == "yes") { df <- nx + ny - 2 v <- 0 if (nx > 1) v <- v + (nx - 1) * vx if (ny > 1) v <- v + (ny - 1) * vy v <- v/df stderr <- sqrt(v * (1/nx + 1/ny)) } else if (var.equal == "Welch") { stderrx <- sqrt(vx/nx) stderry <- sqrt(vy/ny) stderr <- sqrt(stderrx^2 + stderry^2) df <- -2+(stderr^4/(stderrx^4/(nx + 1) + stderry^4/(ny +1))) } else { stderrx <- sqrt(vx/nx) stderry <- sqrt(vy/ny) stderr <- sqrt(stderrx^2 + stderry^2) df <- stderr^4/(stderrx^4/(nx - 1) + stderry^4/(ny -1)) } -- *Dr. rer. nat. Rainer Düsing, Dipl.-Psych. * Universität Osnabrück Institut für Psychologie Fachgebiet Forschungsmethodik, Diagnostik und Evaluation Lise-Meitner-Str. 3 49076 Osnabrück Raum 75/222 Tel: +49-541 969 7734 Email: radues...@uos.de <rdues...@uos.de> [[alternative HTML version deleted]] ______________________________________________ R-help@r-project.org mailing list -- To UNSUBSCRIBE and more, see https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.
