# Difference between marginal and conditional treatment effect? Relating to regression vs. propensity score methods

Peter Austin has a nice introduction to propensity score methods (citation below), and he states that one of the differences between PS methods and plain regression is that PS methods give you a marginal treatment effect, while regression gives you a conditional treatment effect. He defines conditional and marginal treatment effects as thus:

"A conditional treatment effect is the average effect of treatment on the individual. A marginal treatment effect is the average effect of treatment on the population."

OK, I understand his definition, but why does regression give you the treatment effect on the individual, and what are the practical implications of that when a clinician is interpreting one study that estimated treatment effect with regression vs. another study that estimated treatment effect with PS methods?

Austin, Peter C. "An introduction to propensity score methods for reducing the effects of confounding in observational studies." Multivariate Behavioral Research 46.3 (2011): 399-424.

• He does not seem to use the words "marginal" and "conditional" in the statistical sense? Also, what is the set-up? If the data does not come from a randomized experiment linear regression is likely to not give you any sort of treatment effect at all.
– Andy
Commented Dec 10, 2014 at 22:13
• Well the paper is specifically talking about estimating treatment effect from observational data. He compares and contrasts multivariable regression vs. propensity score methods as ways to estimate treatment effect (but obviously more biased than an RCT).
– JJM
Commented Dec 10, 2014 at 22:22
• Can I get a pdf of both of these articles? stats.stackexchange.com/questions/127570/… stats.stackexchange.com/questions/182761/… Commented Jan 19, 2017 at 16:02

What do marginal and conditional relate to?

Assuming the treatment effects are accurately estimated, the conditional treatment effect relates to the estimated effect on an individual whereas the marginal treatment effect relates to the effect on the entire population.

When do the estimates differ?

It sounds odd that the two estimates can differ, but they can in certain situations. The most commonly encountered situations are when the treatment effect is an odds ratio or hazard ratio (HR). Note that the marginal and conditional estimates are equal with risk ratios or with linear regressions. The scenarios where marginal and conditional (odds ratios or HRs) estimates differ most tend to coincide with scenarios when the difference between HRs and risk ratios are greatest. This is when the outcome is "common" and the covariates included in the multivariate regression model are highly predictive of the outcome.

How does this affect a clinician's interpretation?

If the conditional HR is 0.7 then you can say that giving drug A rather than drug B will lower the hazard in the patient sitting in front of you by 30%. Whereas for a marginal HR of 0.7 you could say that if you gave the entire population drug A rather than drug B you will lower the entire population's hazard by 30% (this can be useful for healthcare planners or decision makers). Note that the conditional HR is usually less precise and tends to give larger treatment effects (further from the null).

Why the difference between PS and regression?

When you use multivariate regression the interpretation of the coefficients is "the estimated change in outcome whilst holding all other variables constant". This may help you understand why this treatment is conditional-it is whilst conditioning on the other covariates in the model (ie comparing two patients with the same set of characteristics). In contrast, say you use inverse probability weighting with the propensity score. You just compare outcome rates in two (weighted) population, without reference to each individual's characteristics. This gives you a marginal HR. Coincidentally, this is the type of HR you get from a typical randomized clinical trial (with an primary analysis which does not adjust for covariates).