# What does "randomly assigned conditional on some observable" mean intuitively?

From my textbook it say that "If the treatment in a quasi-experiment is "as if" randomly assigned, conditional on some observed variables w, then the treatment effect can be estimated using differences regression."

I am interpreting "the treatment is "as if" randomly assigned, conditional on some observed variables w" as the following.

An individual receives the treatment or they do not receive the treatment. The probability of receiving the treatment is not impacted by their observable characteristics w which could be gender, income etc. So a man with high wages is no more or less likely to have received the treatment than a women with low wages etc.

Is this a correct interpretation?

In this example, gender is our $$W$$. Note that, contrary to what you say, $$W$$ does affect the probability of receiving the treatment --- women are more likely to take the drug than men. By itself, this would not be a problem. But $$W$$ also affects our outcome. This means that $$W$$ is a confounder, it induces a non-causal association between treatment and outcome, so if we try to measure the treatment effect from the aggregate data it will be biased.
However, in our simplified example, we are assuming the only factor that affects both treatment selection and the outcome is $$W$$. So let us split men and women, and analyze each group separately. Now, given our assumptions, within each group, there are no other factors that affect both the likelihood of getting treatment and the outcome. That is, although the probability of receiving treatment does depend on covariates $$W$$, if you look at individuals with the same $$W =w$$, the probability of receiving treatment does not depend on how they potentially respond to treatment (their potential outcomes).
In a real research context, you need to defend that $$W$$ actually makes your treatment "as-if random". To do that, you need to formally articulate the relationships you believe to hold between the variables you measure, and causal diagrams are a formal and intuitive way to help you decide whether your set $$W$$ will in fact satisfy the "as-if random" conditions, as explained in this other question. Intuitively, $$W$$ will render your treatment "as-if" random if it blocks the spurious associations created by common causes of treatment and outcome and if does not create any other spurious associations. This means you have no unobserved confounders besides $$W$$ and that the set $$W$$ satisfies what we call the backdoor criterion.