Two sample t-test with mean centralized data

Question

I have two sets of equal sized and normally distributed data, which are output from two different experiments. I wish to do a two-sample t-test with these two data sets to see if the results from the two experiments agree.

Is it appropriate to mean (or median) centralize the data sets before doing the t-test?

The variance of the data are equal and passes the var.test. My understanding is that if I centralize the data and the mean will be the same for the two data sets and the result of a t-test will always favor the null hypothesis of equal means.

Can the same thing be said about centralizing data or not for a one-way ANOVA if I have more than two datasets?

Answer 1

Imagine you have two samples, one is distributed as $N(0,1)$, mean 0 and variance 1, and the second is distributed as $N(1,1)$, mean 1 and variance 1. The null hypothesis is obviously false here.

Now if we do operate a mean-centering, we will transform the second sample into something that has a distribution very close to $N(0,1)$. Intuitively, we wipe out the differences between the samples by mean- or median-centering, so this is not appropriate for testing.

Formally, you do not even need to test for equality of the means by a t-test if samples are mean-centered, because you know that they have the same mean (namely 0). Any rejection of the null hypothesis in this case is a false positive.