# Generalization of one-sample t-test for multiple groups

If I have some data*, and want to test the hypothesis that it has a given mean $$\mu_0$$, I know of the possibility of using a one-sample t-test. Here, $$\mu_0=0$$ could be the null-hypothesis of a treatment having no effect, for example. (*let’s assume the data is normally distributed around its mean)

Now, suppose I now have several such datasets, and am interested in the question “Which of these groups have a mean different to $$\mu_0$$”. I could run several of these t-tests against $$\mu_0$$, but then I increase Type-I errors due to the multiple comparisons. A Bonferroni correction seems like a valid but very inelegant and too conservative solution.

I learned of the ANOVA as a generalization of the two-sample t-tests to compare their means, but here the question is different: “Do any two of these groups have different means?”

My question is: are there any generalizations of one-sample t-tests that control for multiple comparisons, other than a Bonferroni correction?

The Hotelling's T squared is a multivariate version of the T test. If you assume your data to be jointly normal (note: even if each marginal distribution is normal, that does not guarantee joint normality) then the Hotelling's T can let you test an entire vector of means against some claimed value in one go. There's one sample and two sample versions (to compare two vectors of means against each other) out there, and an R package was recently developed. It's just a scaled F test with different degrees of freedom.

• Thanks! I had not considered the possibility of multivariate testing. This now begs the next question: if there are differences, how do I find out which groups have different means from $\mu_0$ without increasing my error rate? Dec 10 '18 at 15:45
• @PurpleRover , use simulatenous confidence intervals and see if any of them include the null value. This Penn State course basically takes you through the entire thing. But I think it's over complicated--if post hoc tests are important to you, i think you should read more onTukey's HSD, and ask around more about how to get it to test against a claimed value. That's much easier than a Hotelling's T Squared. (i was merely answering your specific question about a one sample t test generalization). Dec 10 '18 at 16:33