First I show an example of some hypothetical data and the type of question/analysis I am interested in. Below it I try to explain my question in words (for those who want additional info)
Suppose we give experiment participants a shot that is supposed to have some numerically measurable effect (for this example, lets assume the shot is supposed to changes how many times a heart beats per minute (BPM))
Suppose further, that different dosages of the shot can be applied.
In other words, suppose that we have some data that looks like this (this is made-up data)
Subject_id dosage BPM 1 50cc 120 1 60cc 125 1 70cc 130 1 80cc 135 1 90 cc 140 ---------------------------- 2 50cc 130 2 60cc 125 2 70cc 120 2 80cc 115 2 90cc 110 ----------------------------- 3 50cc 120 3 60cc 125 3 70cc 130 3 80cc 130 3 90cc 130
Suppose I think that for some subjects the drug will have a positive effect (more dosage causes more BPM) but for other subjects it will have a negative effect (more dosage causes less BPM).
- What are some approaches to testing this? (perhaps Anova or using random slopes in a linear model?)
I was thinking maybe I could categorize each individual (i.e. create a variable for increasing/decreasing) and then use this as a dummy variable in a regression but
- I don't know how to do this in a significant way (by significant I mean I don't know how I could be sure an increase is significant different than zero)
- This approach wouldn't work with subjects where there is no clear increasing/decreasing response?
Suppose that we have data that gives an outcome for every treatment that a subject faces, and that there are a large amount of subjects.
- (for example,) suppose that we have 1000 subjects, and each subject faces 10 treatments
Also suppose that we are interested in whether an increase in the level of treatment leads to an increase in the level of outcome.
- (this seems like a problem that can easily be analyzed using panel data techniques.)
Instead suppose that we think an increase in the level of treatment leads to an increase in outcome for individuals of "TYPE A", but a decrease for individuals of "TYPE B". However we do not know what these types are.
Is there a way to test this?
- My naive intuition say we could look at the data on an individual level, and try to analyze this. However, we only have 10 observations per subject. I don't know what statistical tests would work with such a small sample size (10 observations)
A concern might be "Well, the outcome has to either decrease or increase in response to the treatment... duh". But here the goal is to answer: does an increased treatment level lead to an increased or decreased outcome level (depending on subject type), as opposed to an increased treatment level sometimes causing an increased outcome level, and sometimes a decreased outcome level, for the same subject
- Basically, the goal is to test for monotonicity at the subject level
If we knew the types we could include a dummy variable (if we were doing a regression), or simply partition the sample (i.e. if type A was "Male" and type b was "female" we could include a "Sex" dummy.). But here we are uncertain of an indviduals type.
Note: I have asked a similar question earlier (a few weeks back). This question differs though because here the goal is analysis with a small sample (alternatively, analysis with unknown types)