When studying the effect of a treatment (or an intervention) on an outcome, we should keep in mind that it will probably not be the same for everyone. In other words, each person will likely experience a different effect of the same treatment — we say that the treatment has a *heterogeneous effect*.

We can describe this heterogeneous treatment effect in 8 different ways. None of these is necessarily better than the others, but different treatment effects are good for different things. So, here’s a table to help you understand and compare them:

Type of treatment effect | Definition | Example of a question that it answers |
---|---|---|

Average treatment effect (ATE) | Effect of the treatment on the average person in the population | How protective is the COVID-19 vaccine on average? |

Conditional average treatment effect (CATE) | Effect of the treatment in a given subgroup | Is the COVID-19 vaccine effective for children? |

Average treatment effect on the treated (ATT) | Effect of the treatment among those who received it | How protective was the COVID-19 vaccine for those who took it? |

Average treatment effect on the untreated (ATUT) | Effect of the treatment on the untreated group | How protective could the COVID-19 vaccine be for those who chose not to take it? |

Marginal treatment effect | Effect of the treatment if we treat one additional person | Should we vaccinate more people? |

Weighted average treatment effect | Effect of the treatment on a group where some individuals are more important than others | What is the age-adjusted effect of the COVID-19 vaccine on mortality? * |

Intention-to-treat effect (ITT) | Effect of assigning a treatment to a group where not everyone complies with the treatment | What was the effect of recommending a third booster dose of the vaccine? |

Local average treatment effect (LATE) | Effect of the treatment among compliers (those who took the treatment when assigned to them) | How protective was the COVID-19 vaccine among those who complied with the vaccination guidelines? |

*** Why adjusting for age gave us the weighted average treatment effect?**

Let’s start with: Why would we need to adjust for age in the first place?

Age is a confounding variable since it is a common cause of the treatment and the effect (i.e. older age causes people to get the vaccine, and it also affects mortality). So, including age in the linear regression model will help us control for its confounding effect:

`Mortality = β`_{0} + β_{1} Vaccine + β_{2} Age

In this model, β_{1} will be the age-adjusted effect of the vaccine on mortality. Technically, the interpretation of the coefficient β_{1} is:

*“For a given age, receiving the vaccine (changing the variable Vaccine from 0 to 1) changes the mortality by β _{1} units.”*

Now consider the extreme case where all of the older population got vaccinated, but only half of the younger generation got vaccinated:

- For the older adults, the effect of changing the Vaccine variable from 0 to 1 cannot be calculated since everyone has Vaccine = 1.
- For the younger generation, this effect can be calculated since half of them have Vaccine = 1 and the other half has Vaccine = 0.

So, in this case, the effect β_{1} reflects the effect of the vaccine in young adults only. In other words, only young adults contributed to the calculation of the effect of the vaccine on mortality.

More generally, when we adjust for a variable, the subgroup that has more variation in the treatment will be weighted more than the subgroup with less variation — getting the *weighted average treatment effect*.

## References

- Huntington-Klein N.
*The Effect*. 1st edition. Routledge; 2022. - Wikipedia contributors. Local average treatment effect. Wikipedia, The Free Encyclopedia. October 11, 2022, 12:38 UTC. Available at: https://en.wikipedia.org/w/index.php?title=Local_average_treatment_effect&oldid=1115431310.
- Wikipedia contributors. Average treatment effect. Wikipedia, The Free Encyclopedia. September 28, 2022, 05:22 UTC. Available at: https://en.wikipedia.org/w/index.php?title=Average_treatment_effect&oldid=1112798948.