Calculating variance between paired samples, where one sample is constrained to always be the lower bound?

Question

I'm sure this is a solved question, but I haven't been able to hit on the right search terms.

Suppose I have paired samples A and B. A represents a derived variable (say distance "as-the-crow-flies" between two geolocations), and B represents a calculated variable (say distance calculated on the basis of two or more geolocations). A will always be less than or equal to B.

Taking what I assume is the simplest case, let's say I have three paired values for A and B:

set	A	B
1	200	400
2	200	500
3	200	200

If I didn't know that A would always be lower, I could calculate the standard deviation on the differences, but that seems wrong somehow.

My goal is to provide a very simple example of calibrating datasets with known measurement differences, but I think I've outfoxed myself all the same. Can someone help set me on the right path with some terminology?

Edit: To give an example on what I mean regarding calibrating data sets, in case this provides some helpful information

Assume I'm measuring someone's travel behavior over three days using two different instruments: a self-report survey, and an app that records their location. On each day, the person travels between the same locations and . The derived distance would be the same (e.g., 200 meters), whereas the calculated distance may differ if they take a slightly different route (e.g., 200, 400, or 500 meters).

Suppose I wanted to use this relationship to say something about a secondary data set where there was no calculated distance, only the derived distance. One sensible way of doing this might be to regress B on A, then use this model to predict B^* from the known value of A^* in the new dataset. Probably here it would be wise if not strictly speaking necessary to consider the distribution of my response, since it will always be strictly positive.

Anyway, while this is possible, my instinct is that there exists a simple way to calculate the upper boundary given the constraint that $A \le B$. If it helps to vary the values of A, that's fine as well.

set	A	B
1	200	400
2	300	500
3	200	200

Answer 1

The fact that A is always lower than B does not mean you cannot calculate the standard deviation of the difference (or the variance of the difference). But the fact that A is always 200 means that it adds nothing to the analysis. The SD of the difference will be the same as the SD of B by itself.

I'm not sure exactly what you mean by "calibrating datasets". I Googled it, and it appears to mean several different things. However, whatever it means, there's no information in A so, there's nothing to calibrate.

You need new data.

Answer 2

Assuming that $A$ is not a constant value (as in your 2nd example, not as in the 1st), i.e. $A$ carries some information, and also assuming that we have $A<=B$ for all pairs, simply compute the paired differences $B-A$, then compute a 90% CI (I assume you are using the "traditonal" $\alpha=.05$) for the mean (the paired differences do not need to be normal; it is the sampling distribution of the mean which needs to be). Replace the low bound with 0, and you now have a non-symmetrical 95% CI of mean differences. Alternatively, you can try different values of $\alpha$, until you get the low bound equal to 0; say this happens e.g. for $\alpha=.08$. You now have a symmetrical 96% CI for the mean.