Causal inference - knowledge related to the definition of causal effects

Treatment variable (Treatment variable) A : is a dichotomous variable (dichotomous variable), where 1 means treated and 0 means untreated.
Outcome variable Y : is a dichotomous variable, where 1 means death (the outcome occurred), and 0 means survival (the outcome did not occur).
According to the above two definitions, we can express that in processing $a$下YYThe value of$Y$$a=$In the case of$1$$Y^{a=1}$ . Sometimes we also want to express in more detail, for example, to indicate that an individual has an outcome Y under treatment a, you can use$Y_i^a=1$ , for each object$Y_i^{a=1}\ne Y_i^{a=0}$. variable $Y^{a=1}$和$Y^{a=0}$ is called a potential or counterfactual outcome.

average causal effect

Individual causal effects include:

• a result of interest;
• Handle the comparison of a=1 and a=0;
• Individual outcome $Y^{a=1}$和$Y^{a=0}$ comparison.

The average causal effect contains:

• a result of interest;
• Handle the comparison of a=1 and a=0;
• The result of crowd $Y^{a=1}$和$Y^{a=0}$ comparison.

Result $E[{AND}^{a=1}]\ne E\left[{AND}^{a=0}\right]$, it means that there is an average causal effect in the population. The average causal effect can be calculated using risk difference, risk ratio and odds ratio.

• $Pr[Y^{a=1}=1]-Pr[Y^{a=0}=1]=0$
• $\frac{Pr[Y^{a=1}=1]}{Pr[Y^{a=0}=1]}=1$
• $\frac{Pr[Y^{a=1}=1]/Pr[Y^{a=1}=0]}{Pr[Y^{a=0}=1]/Pr[Y^{a=0}=0]}$

We generally cannot know the results of personal causal effects at the same time, such as the change in blood pressure after a person took medicine in a trial, and the change in blood pressure if the person did not take medicine in this trial. We need to use the average causal effect to measure causality (if the test group and control group are selected completely at random, the causal effect can be accounted for by calculating the difference in RD, RR, OR values ​​between the two groups).