1
$\begingroup$

I am measuring two unpaired variables $x$ and $y$ in two different conditions ($x$ and $y$ are magnitudes of some special magnetic signals). In the first condition, my hypothesis is that $\bar{x} > \bar{y}$ and in the second condition that $\bar{x} < \bar{y}$. Now that I have $N$ samples from both variables, how can I test whether my hypotheses are true? I am not sure if I can safely assume that $x$ and $y$ are independent from each other. Neither do I know from what kind of distributions they are sampled from. The sample size I have is small. I have read several introductions to statistics for the past few days, but never saw a worked out example for this kind of situations. All help appreciated.

Edit: Like Michael Mayer wrote, there is a binary grouping variable "condition". Sorry for a bit unclear question.

Improve this question
$\endgroup$
9
  • 1
    $\begingroup$ I have a high degree of belief that exactly one of your hypotheses is true :-). Your data might be able to cast doubt on one of them. But what is "small"? To some people that means $N\le 2$; to others, it might mean $N\le 10^6$. $\endgroup$
    – whuber ♦
    Commented Oct 7, 2013 at 21:34
  • $\begingroup$ My sample size is $N = 10$. $\endgroup$
    – user31197
    Commented Oct 7, 2013 at 21:39
  • 1
    $\begingroup$ Hypotheses are about populations, not samples. You can check whether $\bar x < \bar y$ at a glance - no need for p-values or anything. $\endgroup$
    – Glen_b
    Commented Oct 7, 2013 at 23:20
  • $\begingroup$ Given your small sample and lack of knowledge about the distributions, I'd suggest a permutation test. $\endgroup$
    – Peter Flom
    Commented Oct 7, 2013 at 23:53
  • 1
    $\begingroup$ @Peter See stats.stackexchange.com/a/1836/919 for why $N=1$ can work. $N=2$ is usually needed in order to estimate the variability. In applications where observations are sufficiently expensive, $N \gt 2$ may be considered extremely large. (I work in a field where (a) private parties pay for observations which are (b) required by government regulations that (c) are viewed as a burden and, in the worst situations, (d) an observation (actually a monitoring station) can cost \$100K or more. If you want to tell my clients they need a larger $N$, you had better have a great reason!) $\endgroup$
    – whuber ♦
    Commented Oct 8, 2013 at 3:10

1 Answer 1

Reset to default
1
$\begingroup$

A simple approach would be the following:

1) Take all observations sampled at random under condition A and obtain the relevant one sided p-value from Wilcoxon's rank sum test.

2) Do the same for the observations sampled under condition B.

3) If the smaller of the two p-values is below the level $\alpha/2$ and the other p-value is below $\alpha$, then your claim holds at the $\alpha$ level. (This would be the Bonferroni-Holm correction for multiple testing.)

Since the sample sizes are extremely low, you will get a "significant" result only if the signal is very strong.

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.