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I am interested whether or not there is a difference in a (presumably) normally distributed response variable $x$ in two conditions. There are at least two ways to tackle this: Given a set of measurements for the two conditions, I get some number of measurements $x1$ and $x2$ and then...

  1. ...one computes a one-sample t-test for the ratio $r=x1 / x2$ (edit: usually, log transformed) assuming $H_0:\mu_r=0$

  2. ...one computes a 2-sample t-test assuming $H_0:\mu_{x1} = \mu_{x2}$

Intuitively, I would assume that 2. gives in general better results since the mean and variance are not "pooled" as in the first case. However, I couldn't really find deeper resources (with possibly missing keywords to search for).

My questions are:

  1. Should I prefer one method over the other?
  2. Does the answer depend on the sample size?

Edit: After posting the question I actually found the following resources (sorry!):

Kam Sen
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    What do you mean by conditions? Are x1 and x2 normal random variables? If so x1/x2 does not have a t distribution. So the one-sample t test would not apply. – Michael R. Chernick Apr 06 '17 at 18:15
  • 'Conditions' refers here to a biological set-up but I was trying to think more generally. In the particular case that lead to the question, I have a number of measurements for one cell line and a number of measurements for another cell line and simply want to know if there differences in the response. – Kam Sen Apr 06 '17 at 18:33

1 Answers1

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The answer is neither. You want a paired t-test.

  1. is simply wrong. First off, I assume you mean $H_0: \mu_r = 1$; if the two samples have equal means, their ratios will cluster around 1, not around 0. But that isn't enough to fix this approach. The ratio of two normally distributed variates is not normally distributed, so the fundamental assumptions of the one-sample t-test are likely to be grossly violated.

  2. is better. If the two samples were independent, it would be the right approach. But, at least as I read your question, your two sample are actually measuring responses from the same individuals, just under different conditions, so the sample are not independent.

  3. A paired t-test is designed for precisely this circumstance. As a bonus, if any shift induced by the treatment is small compared to the spread among individuals, a paired t-test will have a better chance of finding it than a two-sample t-test that ignores pairing.

David Wright
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  • @ 1: A little detail slipped my description, normally logs are used so they would cluster aroud 0...@2: no that is not necessarily the case. – Kam Sen Apr 07 '17 at 14:32