How to test for relationship between cumulative intake and outcome over time in single arm study? I have a one group trial with n = 100. I want to analyze the relationship between the accumulated amount of drug intake (continuous) and the effect (measured by symptom score).
For example, for different time points for patient 1:
t1: d_amount 10, S_score 50
t2: d_amount 15, S_score 45
t3: d_amount 20, S_score 40

For example, for different time points for patient 2:
t1: d_amount  5, S_score 50
t2: d_amount 10, S_score 45
t3: d_amount 30, S_score 40

How would I construct a (formally correct) hypothesis proving the statement: "There is a significant negative relation between the amount of drug taken (over time) and the symptom score"?
 A: If we neglect the time variable, the problem can be addressed in a more straightforward way. Your question, both on the title and at the end of your post, relegate the time variable to a secondary position, so I do the same.
If you want to test the correlation between two variables, you can use Pearson's correlation coefficient, or Spearman's rank. You can calculate for each individual a value; then you can test how significant is this value by either


*

*taking a look at the distribution the indicators follow, and calculate the probability of having such a value for the number of points obtained.

*doing a Monte Carlo in which you shuffle the values in the data pairs a large number of times, calculate the indicator value for each of these datasets and with this get the distribution of indicator values for datasets with the same properties as yours, but that are by construction not correlated. You can then calculate what is the probability of getting you indicator value as the result of a chance alignment of the points in your shuffled dataset.


Note that these tests can be done individual by individual. You can after evaluate how the correlation in different individuals compares. However, if for each individual you only have 3 datapoints (i.e. pairs drug intake - response) then it will be hard to make any assessment on the correlation significance, because fortuitous alignments for such low number of points are still very likely. If that is the case, you should use the same kind of principle to take a look at how different is the population of correlation coefficients in your sample with those in an uncorrelated population.  
