# How do I estimate sample size for a single-sample equivalence test?

• From literature I have a theoretically proposed value for human performance in a particular task.
• I want to experimentally measure performance in this task across a sample of humans and test whether the result I get is equivalent to the theoretical value.
• It seems that a Single Sample Equivalence Test can do this.

Given that I do not know population variance, or mean, only sample variance and mean, how can I go about a power analysis to calculate the sample size I need to use?

I have so far looked into the R packages PowerTOST and Equivalence, but have been confused somewhat by the fact that "single-sample" TOST is quite rare, I don't understand what the "theta0" value is and the fact co-efficient of variance seems to be calculated from population rather than the sample. If possible I would like to use R to do this, though I am open to other free solutions.

• Why do you want a single sample equivalence test instead of a single sample t-test? In any case, in any power analysis, the answer to your question is "you make an intelligent guess". – Peter Flom - Reinstate Monica Nov 6 '12 at 17:42
• A t-test allows us to either reject or not reject the possibility that the mean equals a given value. This is different to accepting that a mean is equal to a given value; to "not reject" is different to "accept". Using a statistical power of 0.8 and an alpha significance of 0.05, and a "range of equivalence" used to calculate the estimated effect size, I understand it should be possible to estimate the number of participants needed. I'm just not sure how this is actually done in practice. – user16548 Nov 9 '12 at 15:46
• The single sample TOST is quite common if one reflects that the paired t test is a single sample test: there is a single sample of $d$, where $d = x_{\text{A}}-x_{\text{B}}$, and the test for difference has H$_{0}\text{: }\delta = 0$. Paired t tests for equivalence are single sample tests. – Alexis May 4 '14 at 14:56