What methods of statistical analysis can be used for time series data? I have done many 1-sample T-tests before, but I can't figure out if I am able to use one in this situation. In our experiment, we took 12 individual insects and placed them in a chamber where they could choose to be on whatever side they pleased (One side had sugar in it and the other did not). We recorded the number of insects on each side at 30 second intervals for 20 minutes.
Is it possible to use a 1-sample T test here? It seems the data wouldn't be random, violating one of the fundamental assumptions for the test. The number of insects on one side at any time strongly influences the number that will be there at the time of the next reading.
What exactly is data like this called? How can I analyze it and potentially reject the null hypothesis that the insects have no preference to the side with sugar vs. the side without?
 A: My thought is that you could define a new variable 
diff(t) = num_on_sugar_side(t) - num_on_sugar_side(t-30s)

If the insects do not care about the sugar, then these counts should be randomly distributed, with the average change relating to the probability that a bug chooses to move over a 30s period. The counts should go up or down with equal probability (everything else being equal). 
If the insects do care about the sugar, then these counts should be lopsided: i.e. they should be more likely to increase if bugs like sugar, and more likely to decrease otherwise.
If our null hypothesis is that the bugs don't care about sugar, then we aught to be able to test it here.
I might opt to use a non-parametric test because I'm not too sure about how this would be distributed. Maybe Wilcoxon Rank Sum? Perhaps someone who knows more about these things can jump in...
A: One method you might consider for time-course data is Growth curve analysis, a technique that was developed to track agricultural interventions from year to year. More recently it has been applied to the finer time-scale of eye-movements on a display (i.e. milliseconds). 
This site provides a thorough treatment of the technique in the latter context - http://www.danmirman.org/gca 
Also available there is R code to implement the technique with your data.
I have a basic understanding of R and got this code to work within a couple hours of work.
From your description, it sounds like your data would fit well with this method and would give you a much richer understanding of the effects occuring throughout the time-course, versus using time-bins and repeated t-tests, which have the obvious drawback of imposing arbitirary distinctions in your data (i.e. where to draw bin boundaries) just to get a significant effect.
