# appropriate hypothesis test for non-negative quantity

This question is intentionally written in a very abstract way, because I am interested in a generic solution, or for pointers to different possible concrete cases in different fields:

I am measuring a particular quantity in a sample of subjects taken from a given population, and I want to apply a null hypothesis test to check whether there is evidence for "an effect" in that population. Under most circumstances, I would apply a one-sample t-test to test whether the population mean is different from zero.

In this case, however, I know that the underlying quantity can only be zero or positive, but never negative. In my view this invalidates the null hypothesis of a population mean of zero, because as soon as there is any variation across subjects in the population that mean has to be non-zero anyway. Or to put it differently: A t-test applied in such a situation effectively tests whether the variance in the population is non-zero, which however we usually assume to be the case anyway, because no person is exactly like the other.

The situation is complicated by the fact that I can only obtain estimates of that underlying non-negative quantity, where the sampling distribution can be safely assumed to be normal, so that in contrast to the true value these estimates can and will often be below zero.

Two questions:

• Is this a known situation with a generic approach? I couldn't find anything in standard textbooks on hypothesis testing.

• Are you aware of instances of this situation? It might help me get to the core of the problem if I can compare my concrete case to other concrete cases in different disciplines.

• I like the question: can you say more about the kind of data other than that they are non-negative? For example, are you explicitly referring to count data? Not that you are, but if you were, you could perform hypothesis tests based on the Poisson distribution (or negative binomial distribution), rather than the t distribution. Also: I took the liberty of edit the last sentence in your first paragraph to avoid a misinterpretation of non-parametric tests. – Alexis Jul 9 '14 at 16:39
• @Alexis, if there's a misunderstanding on my part, please point it out to me, but: The nonparametric analogue I have in mind, the sign-permutation test, actually does test whether the mean is different from 0. Since this part is not essential, and might be misleading, I just deleted the parentheses. Ok? – A. Donda Jul 9 '14 at 16:42
• You are mistaken. The sign test is a test for stochastic dominance, not a test for mean difference, its H$_0\text{: }\text{P}\left(X_{A} > X_{B}\right) = 0.5$ is explicitly a hypothesis about stochastic dominance, not about the mean. – Alexis Jul 9 '14 at 16:44
• @Alexis, I fear that if I give more details I'd receive answers that focus too much on technicalities, so I'd rather keep it like it is for the time being. But thank you for pointing out counts as a possible example. I'll read up on that case, but if you have the time it would be interesting to have a full answer from you on that case. – A. Donda Jul 9 '14 at 16:44
• Well, if you are being vague about the nature of your data, I would not expect much beyond vague answers: (1) Yes, it is known. The "standard approach" is to apply a test based on an appropriate distribution. (2) Yes, I am aware of instances. I just gave you one. :) – Alexis Jul 9 '14 at 16:48