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I am running a behavioral experiment where a number of subjects (=20 in my case) perform a simple cognitive task. The experiment consists of a fixed number of trials (say, 40 in my case). During each trial, the participant executes a single key-press and the response time (RT) is recorded.

So my recorded data looks like this:

subject-1   -> [Trial-1 RT, Trial-2 RT ... Trial-40 RT]              # (40 trials)
subject-2   -> [Trial-1 RT, Trial-2 RT ... Trial-40 RT]              # (40 trials)
...        ...       ...
subject-20  -> [Trial-1 RT, Trial-2 RT ... Trial-40 RT]              # (40 trials)

Now based on some RT criterion, a few trials are removed for each subject. This resulting data looks like this:

subject-1   -> [Trial-1 RT, Trial-3 RT ... Trial-40 RT]              # (32 trials)
subject-2   -> [Trial-1 RT, Trial-2 RT ... Trial-38 RT]              # (36 trials)
...        ...       ...
subject-20  -> [Trial-3 RT, Trial-8 RT ... Trial-40 RT]              # (28 trials)

The removal of a few trials result in a non-uniform number of data-points for each subject. Eg. subject-1, subject-2 and subject-20 have 32, 36 and 28 trials respectively.

Now I want to test the effect of trials on response times, what statistical methods should I use?

I know that when I am not removing the data, I have a nice 20*40 (subjects * trials) data matrix on which I can perform repeated measures ANOVA (within-subject) to see the effect of trials on response times. But how should I go about it if I am removing a couple of trials?

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Mixed effects regression is somewhat related to repeated measures ANOVA and can handle missing data. However, they assume the missing data are missing at random.

Depending on what type of rejection criteria you are using, that assumption may be broken, in which case I think you're pretty much out of luck without a method to account for that. It may be that it is a sufficiently reasonable assumption, however, even if not strictly warranted.

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