How to choose df for comparisons between summary statistics (e.g. slope values)? In order to correlate or compare means of two dependent variables. 
In my case, I need to correlate individual (e.g. subjects=30) slope values from different conditions (e.g. conditions=4), and each slope value summarizes the relation between the dependent variable (e.g. measured 4 times in each level of the independent variable) and the independent variable (e.g. 5 levels).
How to correct the df of the comparison to reflect the fact that each data point (slope value) summarizes many measurements?
Note: I am not asking how to do a regression between slope values. I already did regression in a within subject design, minimized euclidean distance regression etc.
 A: Here's how I have understood your question:


*

*you have two groups of participants

*Five observations per participant

*Based on the five observations, you can extract a single summary statistic (e.g., if the five observations were performance over five time points, the summary statistic might be the slope of the regression line predicting performance from time)


General points:


*

*If you want to test whether there are differences between groups on the summary statistic, you can do a standard t-test with standard degrees of freedom.

*Having ore observations per individual will increase the reliability with which you measure the summary statistic.

*Greater reliability of measurement means larger expected group differences and thus greater statistical power (see reliability attenuation). 


Very similar points could be made if instead of having two groups you had a numeric variable measured once on each participant, such as age, and you wanted to correlate this with your summary statistic.
There are many ways to measure something on a set of participants. You just happened to have applied an algorithm (e.g., a linear regression leading to a slope) to a set of observations to derive your measure.
A: This a very simple multi-level (a.k.a. hierarchical) model. Douglas Bate his currently working on a book on the subject (draft avalaible here: http://lme4.r-forge.r-project.org/book/). While there are many books on this subject, Doug's has the added benefit of being designed arround the 'lme' R package, a very handy pakage designed to fit such model. I think it is best for you to go and read the first chapter of that book as well as practice the exemples provided there inside R. You can always come back with more specific questions.
A: Concerning your more specific question (i.e. how many degrees of freedom): the question is how many replicates do you have. Look at the early pages of chapter 19 of the R book for examples and guidelines for such accounting.
We could do the accounting here but i don't understand the design of your experiment (probably due to difference in vocabulary, it could be easier if you explained it in formal (i.e. math) script with care to define the indices).
You might also want to Check the following paper 
Hurlbert, S.H. (1984) Pseudoreplication and the design of ecological field experiments. Ecological Monographs, 54, 187–211.
