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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?

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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:

https://journals.sagepub.com/doi/pdf/10.1177/0962280215577111 or http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.865.734&rep=rep1&type=pdf or this: https://www.tandfonline.com/doi/abs/10.1080/02664763.2013.830285

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

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    $\begingroup$ Thanks, the right keyword will do it! $\endgroup$ – Vadim Nov 21 '19 at 12:45
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    $\begingroup$ You're welcome. Please come back and post if you find an implementation in whatever statistics software you're using. $\endgroup$ – Bernhard Nov 21 '19 at 12:47

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