Testing for difference in means between two groups for which each measurement has a unique associated uncertainty

Question

I am wondering what type of statistical test would be appropriate to test for a difference in means between two groups for which each measurement has a different uncertainty. For example, I want to see if there is a significant difference in height when comparing men and women, but each person was measured using a different type of ruler (let's say one ruler only measures up to a precision of 1 cm, another to 1 mm, etc). So, I would have a set of measurements like:

m = [170 cm, 150 cm, 200 cm]

w = [150 cm, 120 cm, 170 cm]

u_m = [1 cm, 0.5 cm, 0.1 cm]

u_w = [0.1 cm, 0.25 cm, 1 cm]

where order is preserved (i.e., u_m[1] is the uncertainty of m[1]).

What type of test could tell me if there is a significant difference in means (outputting a significance value) with the associated uncertainties taken into account?

Answer 1

If it's precision in the sense of "could be any value between a and b" (and the ends of the intervals aren't necessarily less likely than the middle) with a and b different for each observation, then over could treat this as interval censored data. If you'd like to treat this as measurements e.g. with a normally distributed error that's sometimes wider or narrower, then a normal likelihood with known for SD works. There's specialized software for all of these (e.g. survival analysis tools for the first, meta regression for the second). Alternatively, the brms R package covers both. Within these you uses a regression with suitable terms such as group (plus the assignment mechanism to group, e.g. randomisation strata or propensity scores).

Thereafter, you just use a likelihood ratio test or the like based on the likelihood, or in a Bayesian setting look at the posterior for the group coefficient.