## Marginal structural model (MSM) :

A marginal structural model (MSM) is a statistical method used in causal inference that allows researchers to estimate the effects of an intervention on a population. This is often done by using data from randomized controlled trials, but can also be applied to observational data.

MSMs are used to estimate the average causal effect (ACE) of an intervention on a population by taking into account the time-varying effects of confounders on the outcome of interest. This is important because confounders can bias the estimate of the intervention’s effect if they are not properly controlled for.

To illustrate this, consider the following example:

Suppose that a researcher is interested in studying the effects of a new medication on blood pressure. In this study, the researcher randomly assigns participants to either receive the medication or a placebo. After several weeks of treatment, the researcher measures the participants’ blood pressure and compares the results between the two groups.

However, this simple study design may not accurately reflect the true effect of the medication on blood pressure because there may be other factors that influence blood pressure, such as diet, exercise, and stress levels. These factors are known as confounders, and if they are not controlled for, they can bias the estimate of the medication’s effect on blood pressure.

To control for these confounders, the researcher can use an MSM. In this case, the researcher would first specify the variables that are likely to be confounders, such as diet, exercise, and stress levels. Then, the researcher would use statistical techniques to estimate the average causal effect of the medication on blood pressure, taking into account the time-varying effects of these confounders on blood pressure.

Here is another example:

Suppose that a researcher is interested in studying the effects of a new educational program on student achievement. In this study, the researcher randomly assigns schools to either implement the program or continue with their current curriculum. After one year, the researcher measures the students’ achievement scores and compares the results between the two groups.

Again, this simple study design may not accurately reflect the true effect of the program on student achievement because there may be other factors that influence achievement, such as parental education level, socioeconomic status, and prior achievement. To control for these confounders, the researcher can use an MSM. In this case, the researcher would first specify the variables that are likely to be confounders, such as parental education level, socioeconomic status, and prior achievement. Then, the researcher would use statistical techniques to estimate the average causal effect of the program on student achievement, taking into account the time-varying effects of these confounders on achievement.

In summary, marginal structural models are statistical methods used in causal inference to estimate the average causal effect of an intervention on a population. They do this by controlling for the time-varying effects of confounders on the outcome of interest, which can help to reduce bias in the estimate of the intervention’s effect.