In the context of clinical trials, the monotonicity means there are no subjects who do not take the experimental treatment when in the experimental group but do take the experimental treatment when in the control group.  Similarly, there are no subjects who take the experimental treatment when in the control group but do not take the experimental treatment when in the experimental group.  

University of Michigan - Ann Arbor

The intervention causes the variable to become unrelated to factors that usually affect it so the arrows between these factors and the variables will not exist in the causal diagram with the intervention.

Intervention in Causal Diagrams

Pearl, J., & Mackenzie, D. (2018). $\mathit{The \ Book \ of \ Why: \ The \ New \ Science \ of \ Cause \ and \ Effect.}$ Basic Books.

The Book of Why

In conditional probability, when we want to find the likelihood of an occurence given we **know** an *intervention* occured and the value of that intervention, we use the do() operator. 

`do()` operator

In a causal diagram, an instrumental variable is a variable that satisfies the following properties:

- The instrumental variable has a causal effect on the treatment variable.  
- The instrumental variable is not associated with unmeasured confounder(s).
- The instrumental variable affects the outcome only via the treatment variable; i.e., there is no direct effect of the instrumental variable on the outcome variable.
- Assumptions: linearity or monotonicity

Instrumental Variable

Monotonicity Assumption in Clinical Trials

If sprinklers are seasonally on (that is they are only turned on during spring time to water plants), then the sprinklers being on would give you the abililty to make a backward inference about what season it is (seen in the causal model on the left by the arrow connecting season with sprinkler). Contrastly, if a person were to turn on the sprinkler on random days throughout the year the same backward inference can not be made as to what season it is (seen in the causal model on the left by the absence of an arrow between season and sprinkler). This is the model representation of do(sprinkler = on) as opposed to sprinkler = on.

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