Statistical testing on discrete time series data? I am trying to analyse some data which has 40 different subjects, 1 categorical factor with two levels (20 male and 20 female). The rate of tree cutting was recorded at 4 different time points for each individual (one value per individual per time point). So 4 cutting rate values per person.
Upon visualizing the data in plots it seems that there is a negative relationship for one sex and a positive for the other (as in felling rate lets larger gradually between time periods for one sex and gets gradually lower for one). I want to know if there a difference between the sexes with respect to total cutting rates and b) is there a difference in tree cutting rates depending on time of day, and if yes, is the relationship the same for both genders? 
What statistical tests would be appropriate to test this sort of data and questions?
 A: This seems to be a design with repeated measures within subjects.  This means that measurements within a particular subject are likely to be more similar to each other than to other subjects. That is, there will be correlation within subjects. One way to proceed with this is to use a mixed effects model and fit random intercepts for subject.
In the common notation used in the lme package in R and also by other packages that fit mixed models, the model formula could be:
CuttingRate ~ Sex * Time + (1 | Subject)

The software will estimate a variance for the random intercepts for SubjectID. This is usually assumed to be normally distributed with a mean of zero, so together with the fixed intercept each subject can be considered to have it's own intercept. 
The fixed effect for Sex will answer the question: is the sex of the subject associated with different cutting rates?
The fixed effect for Time will answer the question: does the cutting rate vary at different times of measurement? Since you have only 4 measurements per subject you could encode time categorically, rather than continuously, although you might want to try both. Just bear in mind that if you treat it as continuous you are only estimating a single (linear) association over time, whereas if it is categorical you will allow for a non linear association since each time point will have it's own estimate. If you do code it as continuous then to estimate a nonlinear association for time, you would need to add higher order (eg quadratic) terms or use splines, but this will perhaps hamper the interpretation of the interaction, since you will have to decide whether to interact the higher order terms (or splines) with sex. So in the first instance I would suggest coding time as categorical. 
The fixed effect for the interaction Sex:Time will answer the question: does the variation in the cutting rate over at different time points differ between sexes? 
