Under what circumstances should the degrees of freedom for a Welch's t-test be N-1? I recently tried to replicate the analyses of a colleague, and want to confirm that I have found an error. They used Welch's 2-sample t-test, but their results show a t-statistic with n-1 degrees of freedom (in this case df = 153).
My own test found a df that aligns much more with what I'd expect from Welch's test -- I got df = 245.91.
I know they performed the test in STATA, and I'm thinking this disparity in the degrees of freedom alone suggests they hit the wrong button and performed a single-sample t-test. Is this assumption fair? Are there any circumstances in which a two-sample Welch's t-test would have n-1 degrees of freedom?
 A: It is possible, but unlikely.  If both sample sizes are $n$ then the effective number of degrees of freedom is bounded between between $n-1$ and $2n-2$, and near to the top end either if the variances are close to each other or if the variances are assumed to be equal.  I might suspect your friend may have done a paired $t$-test (i.e. a one-sample $t$-test of the differences).
But you could make $n-1$ appear (up to rounding) if one variance was very much larger than the other. For example, using R rather than Stata, here is an extreme case with $N=6$:
> x <- c(3,3,3,3,3,1000)
> y <- c(1,1,1,2,2,2)
> t.test(x, y,  paired = FALSE, var.equal = FALSE)

        Welch Two Sample t-test

data:  x and y
t = 1.009, df = 5, p-value = 0.3593
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -259.4783  594.8116
sample estimates:
mean of x mean of y 
 169.1667    1.5000 

Change the 1000 to 1.658and you would get a rounded df = 10 (and a very much smaller $p$-value despite the means being much closer and the two samples overlapping)
