One-way ANOVA controlling for session effects I have a dataset collected from an experiment which has a simple structure that lends itself to a one-way ANOVA analysis (categorical IV, continuous DV).  The data are collected over 20 experiment sessions so I think it will be good to conduct the one-way ANOVA while controlling for session effects.  Could anyone please tell me if there is a way to do so?
Thank you very much!
 A: Response to comment:
Let's make sure we're on the same page: Factor 1 has levels A and B, Factor 2 (sessions) has levels 1 through 20. Ten sessions consider only level A and the other ten sessions consider only level B. Say Session 1 does A and 2 does B. And so on.
If that's true you can't know whether the difference between results for sessions 1 and 2 is due to how the session was conducted or due to choice of A vs B. This is not necessarily a catastrophe.

Example: In an (unusually small) clinical trial with 20 subjects, 10 might be assigned at random to a new Drug under study and the other 10 to placebo (or to the drug in current use). 'Control' over inter-subject differences is done by randomization. Doctors or drug company representatives aren't allowed to 'hand pick' which subjects get the new drug. Assignment is done according to a random mechanism. Also, according to a protocol, subjects may be similar to one another in ways that might matter (age, early or late stage of disease, motivation to take the drug exactly as prescribed, etc.)

Somewhat similarly, for your study, I suggest two precautions:
(1) Try to make conditions for sessions as much alike as possible (type of room, time of day, testing routine, expertise of technician, whatever might matter for your experiment). This might help to cut down on any 'session effect'.
(2) Then for randomization, maybe have 20 cards in a hat, 10 marked A, 10 marked B. Before each session draw a card at random to determine whether it will be an A-session or a B-session. This helps to avoid subconsciously or accidentally assigning 'more favorable' sessions to A and 'less favorable' sessions to B.
Because I have no idea what kind of data collection goes on within each session, I can't give details of the ANOVA model. But maybe this much is a start.
