Testing for monotonicity at the subject level (i.e. testing if increased treatment levels have increased/decreased outcome levels at subject level)

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)

Example:

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?

Words:

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)

• Please link to your earlier question! Please give context: what is your "treatments"? what means "10 treatments"---treatments at 10 different times, or ten different kinds of treatments? This isn't really answerable without some such information! – kjetil b halvorsen Mar 9 at 13:57
• @kjetilbhalvorsen I added example data and tried to clarify. I hope it helps. Basically, treatments are dosages (of a shot). By 10 treatments I mean 10 different dosages (in example there are only 5 treatments, for readability). I believe that is considered ten different kinds of treatments. The main feature I am trying to illustrate is that there are 2 or more groups in the sample, and the treatment (dosage of shot) is hypothesized to have different effect for the different groups, but it is not known what subject belong to which group – user106860 Apr 12 at 18:18
• My concern with standard estimation in this case -- such as some sort of regression -- is the following: regressions estimate a conditional mean, but if there are two groups for which the effect is significant but opposite, the conditional mean might be indistinguishable from 0 if we don't condition on which group an observation comes from. (which we can't do since we don't know which subjects belong to which group. Unless perhaps we somehow infer the group from the data) – user106860 Apr 12 at 18:23

You can model this as a hierarchical/mixed model, where the regression slopes are themselves distributed as a latent mixture of Gaussians. This is a complicated, tricky analysis, and will require quite a bit of research, so I'll just give an overview here, and point to to some resources below.

For each subject, you have a linear model:

$$y = \alpha_s + \beta_s x + \epsilon$$

where $$x$$ is dose, $$y$$ is BPM at that dose, $$\alpha$$ and $$\beta$$ are the intercept and slope (and $$s$$ indicates the subject).

In a standard mixed model, individual subjects' slopes are assumed to be normally distributed around the mean slope $$\mu_{\beta}$$.

$$\alpha_s \sim \text{Normal}(\mu_{\alpha}, \sigma_{\alpha})\\ \beta_s \sim \text{Normal}(\mu_{\beta}, \sigma_{\beta})\\$$

Here, instead, you assume that there are two distributions: a positive distribution with mean $$\mu_{\beta(+)}$$, and a negative distribution with mean $$\mu_{\beta(-)}$$. Each subject is either a $$(+)$$ or $$(-)$$ responder.

$$\beta_s \sim \text{Normal}(\mu_{\beta(s)}, \sigma_{\beta})\\ \mu_{\beta(s)} = \begin{cases} \mu_{\beta(+)} \text{ if s is (+)}\\ \mu_{\beta(-)} \text{ if s is (-)}\\ \end{cases}$$

Thus, in the full model, you'll be estimating:

• $$\mu_{\beta(+)}$$, the average slope for positive responders
• $$\mu_{\beta(-)}$$, the average slope for negative responders
• The probability that each subject is a positive responder
• The proportion of positive and negative responders in the population.
• If you have any participant-level predictors, you can investigate how these affect the likelihood of someone being a positive or negative responder.