# Correct variance for minimum detectable difference

I have a question regarding variance, paired testing and minimum detectable difference (MDD).

Paired samples: $$MDD (δ) = \sqrt{ \frac{σ^2}{n} (t_{(α/2,n-1)} + t_{(1-β, n-1)})}$$ I have a set of baseline samples ($A$) sampled at time 0. Because acquiring those samples is destructive, I’m unable to re-sample the same sample twice (similar to for example a soil sample). Therefore I cannot assume that a future re-sampling will have the same variance as the baseline sampling.

To measure how big this error is I have taken a second set of samples ($B$) in close proximity to the first samples ($A$), which gives me a number ($n = 15$) of $A-B$ pairs. Sample $B$ is also taken at time 0. If I run a paired t-test on these two sample populations I fail to reject the null hypothesis ($H_0: A\neq B$).

I should probably mention that both sample population $A$, $B$ and the difference ($A-B$) are normally distributed.

Q1: If I want to estimate the MDD for the baseline mean, how do I integrate this sample error into the equation?

Q2: Can I simply take the variance from the baseline sample and add the variance of the differences ($A-B$)?

Q3: What if the samples don't reject the null hypothesis? Can I then assume that the samples are independent and use a pooled variance?

• Thanks, the formula is correct. I have re-labeled my question. Hope that helps. – Nordenskiold Aug 7 '12 at 19:40
• What is this MDD? Are you talking about the smallest difference you have a high probability of detecting (i.e. rejecting the null hypothesis) or the smallest measurement that you can take that is >0? I doubt it is the latter because that would not relate to t distribution percentiles and sample standard errors. – Michael R. Chernick Aug 7 '12 at 20:21
• @MichaelChernick I am talking about the smallest detectable difference in the mean of the base samples(A). Since i am unable to re-take this samples, I have to consider that the new samples i take will have a slightly different variance than the base samples. This leads me to think that I would need more samples to cover for this location error than if i could re-take the baseline samples. To investigate how big this location error is i have the B samples. – Nordenskiold Aug 7 '12 at 20:41