Medical experimental design with non-comparable control My population consists of people at high risk of cardiovascular disease who are identified through standard blood tests. These people, when identified, are prescribed some standard medication.
I will randomly select some of these high-risk people and perform an ultrasound scan to determine if they have plaque or not, the control will just get the standard medication with no ultrasound. The people who received the ultrasound will get to know the results, that is whether they have plaque or not and I cannot randomize who gets to know the results and who doesn't (ethics and so on). Knowing whether they have plaque or not is the treatment, that is I am testing whether people will behave differently knowing this additional information, whether they will take their medication more seriously.
I would like to test the effectiveness of the medication on the people who have plaque (and know it) versus the control, since these people are really at risk and are of interest. My issue is that I am comparing a subgroup within the people who got the ultrasound (those who have plaque) versus the control which (presumably) contains both groups (plaque and no plaque), but since no ultrasound was done on the control I do not know the people who have it (if they do). Is there a way to account for this in the analysis so that I am comparing apples to apples?

Let's call those with no ultrasound the Group $C$ (as in Control), the group with Ultrasound and plaques the $A_p$ ($p$ as in positive) and those with ultrasound and no plaque the $A_n$ ($n$ as in negative).
We can compare the drug adherence in $A_p$ and $A_n$ to investigate, whether having plaques has an influence on drug adherence. Maybe those with plaques feel some extra symptoms and thus adhere more or less to their prescription.
If having plaques does not influence drug adherence, then you can go and compare the $A_p$ to the $C$ group without problems.
If having plaques does influence drug adherence, then you will have to change your design to a waiting group design. Those in $A = A_p \cup A_n$ will have an ultrasound and then be observed for drug adherence whilst the $C$ will be observed for drug adherence then then be ultrasounded later. And adherence can probably measured within weeks whilst plaques probably develop over years, so a later ultrasound is still valid to retrospectively divide $C$ into $C_p$ and $C_n$ so that a comparison of $A_p$ to $C_p$ becomes feasible.
Alternatively: You might assume that the effect of having an ultrasound at all and the effect of having and ultrasound and knowing there are plaques are somewhat additive effects so that drug adherence in $C$ is drug adherence in $A$ plus some extra and that some extra is the difference in drug adherence in $A_p$ and $A_n$ but you want to accept such assumptions if a simple waiting group design can give a far more reliable experimental answer?
