Comparing two groups of measurements with missing values

There are two groups of normally distributed measurements, carried out under the same conditions: $$x_i = \theta_{x,i} + \epsilon_x$$, $$y_i = \theta_{y,i} + \epsilon_y$$. I would like to test the hypothesis that $$\theta_{x,i} = \theta_{y,i}$$. I cannot use the two-sided t-test, because the means are not constant (they depend on $$i$$). However I could use a one-sided t-test for $$x_i-y_i$$ with the same effect (testing that the mean of $$x_i - y_i$$ is zero).

The problem: There are missing data in these two sets of measurements. If the means were constant, I could have used t-test for two samples with unequal sizes (Welsh's test). Here I however end up with a one-sided test with the size corresponding to that of the smaller sample. It seems like I am losing lots of useful information about the variance. Any suggestions?

What you have is called "partially paired data" and with that as your search term in scholar.google (or similar) you will find a number of publications on different ways of performing such t-tests. This may be a point of entry:

However I am not aware of implementations in standard statistics software so you may have to do some programming.

• Thanks, the right keyword will do it! Commented Nov 21, 2019 at 12:45
• You're welcome. Please come back and post if you find an implementation in whatever statistics software you're using. Commented Nov 21, 2019 at 12:47