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Suppose I have $N=10$ participants.

For each participant, I have $M=100$ consecutive answers, which are either correct or incorrect. My hypothesis is that the probability of giving a correct answers decreases with time. Can I analyze the data with logistic regression? I would like to do the analysis separately for each participant.

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Sure you can. The model then looks like

$$logit(P(trueAnswer)) = \beta_0 + \beta_1 * time$$

with $logit(x)=log(\frac{x}{1-x})$

I would still do a two-sided test for $\beta_1$(since you never know performance may improve over time). If it is negative and significant you have proven your point. If you have not recorded the exact times I think it is still ok to use time = 1,..., M in this case.

It is fine to do the analysis separately per participant since M is large enough, though you may gain power by using a glmm with a random effect for participant.

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